nonlinear evolution equations and wave phenomena ...cobweb.cs.uga.edu/~thiab/book_waves2019.pdf ·...

The Eleventh IMACS International Conference on

NONLINEAR EVOLUTION EQUATIONS AND WAVE

PHENOMENA: COMPUTATION AND THEORY

April 17–19, 2019

Georgia Center for Continuing Education

University of Georgia, Athens, GA, USA

http://waves2019.uga.edu

Edited by Gino Biondini and Thiab Taha

Book of Abstracts

The Eleventh IMACS International Conference On

Nonlinear Evolution Equations and Wave Phenomena:

Computation and Theory

Athens, Georgia

April 17—19, 2019

Sponsored by

The International Association for Mathematics and Computers in Simulation (IMACS)

The Computer Science Department, University of Georgia

Edited by Gino Biondini and Thiab Taha

Sponsors

International Association for Mathematics and

Computers in Simulation (IMACS)

Computer Science Department at UGA

Organization

T. Taha (USA), General Chair & Conference Coordinator

G. Biondini (USA), Co-chair

J. Bona (USA), Co-chair

R. Vichnevetsky (USA),

Honorary President of IMACS, Honorary Chair

Scientific program committee

Bedros Afeyan (USA)

David Amrbrose (USA)

Stephen Anco (Canada)

Andrea Barreiro (USA)

Gino Biondini (USA)

Lorena Bociu (USA)

Jerry Bona (USA)

Jared Bronski(USA)

Robert Buckingham (USA)

Annalisa Calini (USA)

Ricardo Carretero (USA)

John Carter (USA)

Efstathios G. Charalampidis (USA)

Min Chen (USA)

Guangye Chen (USA)

Wooyoung Choi (USA)

Antoine Cerfon (USA)

Anton Dzhamay (USA)

Anna Ghazaryan (USA)

Alex Himonas (USA)

Curtis Holliman (USA)

Pedro Jordan (USA)

Nalini Joshi (Australia)

Kenji Kajiwara (USA)

Henrik Kalisch (Norway)

David Kaup (USA)

Panayotis Kevrekidis (USA)

Alexander Korotkevic (USA)

Gregor Kovacic (USA)

Stephane Lafortune (USA)

Keynote Speakers

David Ambrose: "Vortex sheets, Boussinesq equations,

and other problems in the Wiener algebra"

Alex Himonas:"Initial and boundary value problems for

evolution equations”

Stefano Trillo:"Nonlinear PDEs describing real

experiments: recurrences, solitons, and shock waves"

Yuri Latushkin (USA)

Jonatan Lenells (USA)

Changpin Li (China)

Andrei Ludu (USA)

Pavel Lushnikov (USA)

Dionyssis Mantzavions (USA)

Peter Miller (USA)

Dimitrios Mitsotakis (USA)

Nobutaka Nakazono (Japan)

Alan Newell (USA)

Katie Newhall (USA)

Beatrice Pelloni (UK)

Virgil Pierce (USA)

Barbara Prinari (USA)

Pamela Pyzza (USA)

Zhijun (George) Qiao (USA)

Vassilios Rothos (Greece)

Xu Runzhang (China)

Constance Schober (USA)

Brad Shadwick (USA)

Israel Michael Sigal (Canada)

Avraham Soffer (USA)

Martin Ostoja Starzewski (USA)

Thiab Taha (USA)

Michail Todorov (Bulgaria)

Muhammad Usman (USA)

Samuel Walsh (USA)

Jianke Yang (USA)

Vladimir Zakharov (USA)

Organized sessions

1. Jerry Bona, Min Chen,Shuming Sun, Bingyu Zhang, "Nonlinear waves"

2. Barbara Prinari, Alyssa K. Ortiz "Novel challenges in nonlinear waves and integrable systems"

3. John Carter, "Recent developments in mathematical models of water waves "

4. Andrei Ludu, Changpin Li, Thiab Taha, "Fractional diferential equations"

5. Alex Himonas, Curtis Holliman, Dionyssis Mantzavinos:"Evolution equations and integrable systems"

6. Vladimir Dragovic, Anton Dzhamay, Virgil Pierce: "Random matrices, Painleve equations, and integrable systems"

7. Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal, Samuel Walsh: "Stability and traveling waves"

8.Avraham Soffer, Gang Zhao, S. Gustafson: "Dispersive wave equations and their soliton interactions: Theory and applications"

9. Efstathios Charalampidis, Fotini Tsitoura: "Nonlinear evolutionary equations: Theory, numerics and experiments"

10. Robin Ming Chen, Runzhang Xu: "Recent advances in PDEs from fluid dynamics and other dynamical models"

11. Cancelled

12. Gino Biondini: “Dispersive shocks, semiclassical limits and applications"

13. Qi Wang and Xueping Zhao:"Recent advances in numerical methods of PDEs and applications in life science, material science"

14. Bedros Afeyan, Brad Shadwicn, Jon Wilkening: "Nonlinear kinetic self-organized plasma dynamics driven bycoherent, intense electromagnetic fields session"

15. Yi Zhu, Xu Yang, Hailong Guo: "Waves in topological materials"

16. Dmitry Pelinovsky and Anna Geyer: "Existence and stability of peaked waves in nonlinear evolution equations"

17. Pamela B. Pyzza: "Nonlinear dynamics of mathematical models in neuroscience"

18. Stephen Anco, Stephane Lafortune, Zhijun (George) Qiao: “Negative flows, peakons, integrable systems, and theirapplications"

19. Thomas Carty: "Network dynamics"

20. Nalini Joshi, Giorgio Gubbiotti, Nobutaka Nakazono, Milena Radnovic, Yang Shi, Dinh Tran: "Dynamical systemsand integrability"

21. Katie Newhall: "Stochastic dynamics in nonlinear systems"

22. Robert Buckkingham, Peter Miller: "Modern methods fordispersive wave equations"

23. Sergey Dyachenko, Katelyn Leisman, Denis Silantyev: "Nonlinear waves in optics, fluids and plasma"

24. Michael Sigal, Jianfeng Lu: "Mathematical perspectives in quantum mechanics and quantum chemistry"

25. Alexander O. Korotkevich and Pavel Lushnikov: "Nonlinear waves, singularities,vortices, and turbulence in hydrodynamics, physcal, and biological systems"

26. Ziad Musslimani, Matthew Russo: "Physical applied mathematics"

27. Cancelled

28. Chaudry Masood Khalique, Muhammad Usman: "Recentadvances in analytical and computational methods for nonlinear partial differential equations"

PROGRAM AT A GLANCE

Wednesday, April 17, 2019

Mahler auditorium

Room F/G

Room Y/Z

Room E

Room J

Room V/W

Room B

Room C

Room D

8.00–8.30am

Welcome

8.30–9.30am

Keynote lecture I: David Ambrose

9.30–10.00am

Coffee break

10.00–10.50am

S7 - I/IX

S3 - I/III

S24 - I/III

S25 - I/VII

S20 - I/II

S21 - I/II

S15 - I/III

S19 - I/II

10.55am–12.10pm

S7 - II/IX

S3 - II/III

S24 - II/III

S18 - I/IV

S6 - I/III

S5 - I/V

S15 - II/III

S19 - II/II

12.10–1.40pm

Lunch (attendees on their own)

1.40–3.20pm

S7 - III/IX

S5 - II/V

S9 - I/III

S18 - II/IV

S6 - II/III

S28 - I/I

S15 - III/III

Papers

3.20–3.50pm

Coffee break

3.50–5.55pm

S7 - IV/IX

S5 - III/V

S9 - II/III

S25 - II/VIIS6 - III/III

S16 - I/III

S21 - II/II

S24 - III/III

S10 - I/I

Thursday, April 18, 2019

Masters Hall

Room F/G

Room Y/Z

Room E

Room J

Room V/W

Room K

Room L

Room D

8:00–9:00am

9:10–10:00am

S4 - I/III

S8 - I/V

S25 - III/VIIS12 - I/III

S16 - II/III

Papers

S17 - I/II

10:00–10:30am

Coffee break

10:30–12:10pm

S18 - III/IV

S9 - III/III

S22 - I/II

S25 - IV/VIIS12 - II/III

S16 - III/III

S3 - III/III

S2 - I/II

S17 - II/II

12:10–1:40pm


1:40–3:20pm

S7 - V/IX

S5 - IV/V

S8 - II/V

S25 - V/VIIS1 - I/II

S12 - III/III

S26 - I/II

S2 - II/II

S20 - II/II

3:20–3:50pm

Coffee break

3:50–5:55pm

S7 - VI/IX

S5 - V/V

S8 - III/V

S18 - IV/IV

S1 - II/II

S22 - II/II

S26 - II/II

S4 - II/III

S23 - I/II

5:00–7:00pm

Posters, Hill Atrium (outside Mahler auditorium)

7:00–9:00pm

Conference banquet (including student papers award)

Friday, April 19, 2019 Masters Hall

Room F/G

Room Y/Z

Room K

Room V/W

8:00–9:00am

Keynote lecture 3: Stefano Trillo

9:10–10:00am

S7 - VII/IX

S13 - I/II

S14 - I/II

S25 - VI/VII

10:00–10:30am

Coffee break

10:30–12:10pm

S7 - VIII/IX

S13 - II/II

S8 - IV/V

S4 - III/III

S25 - VII/VII

12:10–1:40pm


1:40–3:20pm

S7 - IX/IX

S23 - II/II

S8 - V/V

S14 - II/II

3:20–3:50pm

Coffee break

: Alex Himonas

CONFERENCE PROGRAM

========================================================================================

TUESDAY, APRIL 16, 2019

5:00–6:00 REGISTRATION (in front of Mahler Hall)

5:00–7:00 RECEPTION

========================================================================================

WEDNESDAY, APRIL 17, 2019

7:30–9:30 REGISTRATION

8:00–8:30 WELCOME

Thiab Taha, Program Chair and Conference Coordinator

Alan Dorsey, Dean of the Franklin College of Arts and Sciences, UGA

8:30–9:30 KEYNOTE LECTURE I, Mahler Hall

David Ambrose: Vortex sheets, Boussinesq equations, and other problems in the Wiener algebra

Chair: Thiab Taha

9:30–10:00 COFFEE BREAK

10:00–10:50 SESSION 7, Mahler Hall: Stability and traveling waves – Part I/IX

Chairs: Bernard Deconinck, Anna Ghazaryan, Mat Johnson, Stephane Lafortune, Yuri Latushkin, Jeremy Upsal,

Samuel Walsh

10:00–10:25 Stephane Lafortune: Study of a model of a liquid in presence of a surfactant

10:25–10:50 Panayotis Kevrekidis: On some Select Klein-Gordon problems: internal modes, fat tails, wave collisions and

beyond

10:00–10:50 SESSION 3, F/G: Recent developments in mathematical studies of water waves – Part I/III

Chair: John Carter

10:00–10:25 John Carter: Particle paths and transport properties of NLS and its generalizations

10:25–10:50 Ben Akers: Asymptotics and numerics for modulational instabilities of traveling waves

10:00–10:50 SESSION 24, Room Y/Z: Mathematical perspectives in quantum mechanics and quantum chemistry – Part I/III

Chairs: Jianfeng Lu and Israel Michael Sigal

10:00–10:25 Christof Melcher: Spinning Landau-Lifschitz solitons - a quantum mechanical analogy

10:25–10:50 Benjamin Stamm: A perturbation-method-based post-processing of plane wave approximations for nonlinear

Schoedinger operators

10:00–10:50 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and

biological systems – Part I/VII

Chairs: Alexander O. Korotkevich and Pavel Lushnikov

10:00–10:25 David Kaup: Optical phase-modulated nonlinear waves in a graphene waveguide

10:25–10:50 Bo Yang and Jianke Yang: Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation

10:00 - 10:50 SESSION 20, Room J: Dynamical systems and integrability – Part I/II

Chairs: Nalini Joshi and Nobutaka Nakazono

10:00–10:25 Vladimir Dragovic and Milena Radnovic: Ellipsoidal Billiards and Chebyshev-type polynomials

10:25–10:50 Nalini Joshi, Christopher Lustri and Steven Luu: Hidden solutions of discrete systems

10:00–10:50 SESSION 21, Room V/W: Stochastic Dynamics in Nonlinear Systems – Part I/II

Chair: Katie Newhall

10:00–10:25 Katie Newhall: A network of transition pathways in a model granular system

10:25–10:50 Jay Newby: The effect of moderate noise on a limit cycle oscillator: counterrotation and bistability

10:00–10:50 SESSION 15, Room B: Waves in topological materials – Part I/III

Chairs: Yi Zhu, Xu Yang, Hailong Guo

10:00–10:25 Alexander Watson: Computing edge spectrum in the presence of disorder without spectral pollution

10:25–10:50 Justin Cole: Topologically Protected Edge Modes in Longitudinally Driven Waveguides

10:00–10:50 SESSION 19, Room C: Network Dynamics – Part I/II

Chair: Tom Carty

10:00–10:25 Mamoon Ahmed: The universal covariant representation and amenability

10:25–10:50 Dashiell Fryer: Adaptive zero determinant strategies in the iterated prisoner’s dilemma tournament

10:55–12:10 SESSION 7, Mahler Hall: Stability and traveling waves – Part II/IX


Samuel Walsh

10:55–11:20 Milena Stanislavova: Asymptotic stability for spectrally stable Lugiato-Lefever solutions in periodic waveguides

11:20–11:45 Efstathios Charalampidis: Formation of extreme events in NLS systems

11:45–12:10 Todd Kapitula: Viewing spectral problems through the lens of the Krein matrix

10:55–12:10 SESSION 3, F/G: Recent Developments in Mathematical Studies of Water Waves – Part II/III

Chair: John Carter

10:55–11:20 Chris Curtis: Nonlinear waves over patches of vorticity

11:20–11:45 Henrik Kalisch: Fully dispersive model equations for hydroelastic waves

11:45–12:10 Harvey Segur: Tsunami

10:55–12:10 SESSION 24, Room Y/Z: Mathematical perspectives in quantum mechanics and quantum chemistry – Part II/III

Chairs: Michael Sigal and Jianfeng Lu

10:55–11:20 Michael Weinstein: Edge states in honeycomb structures

11:20–11:45 Fabio Pusateri: Nonlinear Schroedinger equations with a potential in dimension 3

11:45–12:10 Artur Izmaylov: New developments in quantum chemistry on a quantum computer

10:55–12:10 SESSION 18, Room E: Negative flows, peakons, integrable systems, and their applications – Part I/IV

Chair: Zhijun (George) Qiao

10:55–11:20 Jing Kang: Liouville correspondences between multi-component integrable hierarchies

11:20–11:45 Huafei Di: Global well-posedness for a nonlocal semilinear pseudo-parabolic equation with conical degeneration

10:55–12:10 SESSION 6, Room J: Random matrices, Painleve equations, and integrable systems – Part I/III

Chair: Vladimir Dragovic

10:55–11:20 Anton Dzhamay: Discrete Painlevé equations in tiling problems

11:20–11:45 Tomoyuki Takenawa: The space of initial conditions for some 4D Painlevé systems

11:45–12:10 Nobutaka Nakazono: Classification of quad-equations on a cuboctahedron

10:55–12:10 SESSION 5, Room V/W: Evolution equations and integrable systems – Part I/IV

Chairs: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos

10:55–11:20 Gino Biondini: Riemann problems, solitons and dispersive shocks in modulationally unstable media

11:20–11:45 Barbara Prinari: Inverse scattering transform for the defocusing Ablowitz-Ladik equation with arbitrary nonzero

background

11:45–12:10 Satbir Malhi: Energy decay for the linear damped Klein Gordon equation on unbounded domain

10:55–12: 10 SESSION 15, Room B: Waves in topological materials – Part II/III

Chairs: Yi Zhu, Xu Yang, Hailong Guo

10:55–11:20 Junshan Lin: Embedded eigenvalues and Fano resonance for metallic structures with small holes

11:20–11:45 Alexis Drouo: Edge states in near-honeycomb structures

11:45–12:10 Hailong Guo: Unfitted Nitsche's method for computing edge modes in photonic graphene

10:55–12: 10 SESSION 19, Room C: Network dynamics – Part II/II

Chair: Tom Carty

10:55–11:20 Timothy Ferguson: Bistability in the Kuramoto model

11:20–11:45 Tom Carty: Configurational stability for the Kuramoto-Sakaguchi modelH

11:45–12:10 Sarah Simpson: A Matrix Valued Kuramoto Model

12:10–1:40 LUNCH (attendees on their own)

1:40–3:20 SESSION 7, Mahler Hall: Stability and traveling waves – Part III/IX


Samuel Walsh

1:40–2:05 Ross Parker: Spectral stability of multi-pulses via the Krein matrix

2:05–2:30 Anna Ghazaryan: Stability of planar fronts in a class of reaction-diffusion systems

2:30–2:55 Yuri Latushkin: Recent results on application of the Maslov index in spectral theory of differential operators

2:55–3:20 Alim Sukhtayev: Spectral stability of hydraulic shock profiles

1:40–3:20 SESSION 5, Room F/G: Evolution equations and integrable systems – Part II/V


1:40–2:05 David Nicholls: Well-posedness and analyticity of solutions to a water wave problem with viscosity

2:05–2:30 John Gemmer: Isometric immersions and self-similar buckling in non-Euclidean elastic sheets

2:30–2:55 Curtis Holliman: Non-uniqueness and norm-inflation for Camassa-Holm-type equations

2:55–3:20 Fredrik Hildrum: Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

1:40–3:20 SESSION 9, Room Y/Z: Nonlinear evolutionary equations: Theory, numerics and experiments – Part I/III

Chairs: Efstathios Charalampidis and Fotini Tsitoura

1:40–2:05 Roy Goodman: Bifurcations on a dumbbell quantum graph

2:05–2:30 Patrick Sprenger and Mark Hoefer: Traveling waves in the fifth order KdV equation and discontinuous shock solutions

of the Whitham modulation equations

2:30–2:55 Adilbek Kairzhan, Dmitry Pelinovsky & Roy Goodman: Nonlinear instability of spectrally stable shifted states on

star graphs

2:55–3:20 Yuan Chen and Keith Promislow: Curve Lengthening and shortening in Stong FCH

1:40–3:20 SESSION 18, Room E: Negative flows, peakons, integrable systems, and their applications – Part II/IV

Chair: Stephen Anco

1:40–2:05 Anna Geyer: Instability and uniqueness of the peaked periodic traveling wave in the reduced Ostrovsky equation

2:05–2:30 Huijun He: Some analysis results for the U(1)-invariant equation

2:30–2:55 Stephen Anco and Elena Recio: Accelerating dynamical peakons and their behaviour

2:55–3:20 Xiao-Jun Yang: A new perspective in anomalous viscoelasticity from the derivative with respect to another function

view point

1:40–3:20 SESSION 6, Room J: Random matrices, Painleve equations, and integrable systems – Part II/III

Chair: Virgil Pierce

1:40–2:05 Robert Buckingham: Representation of joint moments of CUE characteristic polynomials in terms of a Painlevé-V solution

2:05–2:30 Peter Miller: Rational solutions of Painlevé equations

2:30–2:55 Andrei Prokhorov: Asymptotic of solutions of three-component Painlevé-II equation

2:55–3:20 Sevak Mkrtchyan: Entropy of Beta Random Matrix Ensembles

1:40–3:20 SESSION 28, Room V/W: Recent advances in analytical and computational methods for nonlinear PDEs

Chairs: Chaudry Masood Khalique and Muhammad Usman

1:40–2:05 Muhammad Usman: A collocation method for a class of a nonlinear partial differential equations

2:05–2:30 Arshad Muhammad: Applications of fixed point theorems to integral and differential equations

2:30–2:55 Kinza Mumtaz & Mudassar Imran: The optimal control of HPV infection and cervical cancer with HPV vaccine

1:40–3:20 SESSION 15, Room B: Waves in topological materials – Part III/III

Chairs: Hailong Guo, Xu Yang, Yi Zhu

1:40–2:05 Lihui Chai: Frozen Gaussian Approximation for the Dirac equation in semi-classical regime

2:05–2:30 Yi Zhu: Linear and nonlinear waves in honeycomb photonic materials

2:30–2:55 Peng Xie and Yi Zhu: Wave-packet dynamics in slowly modulated photonic graphene

1:40–3:20 PAPERS, Room C

Chairs: Gennady El

1:40–2:05 Giacomo Roberti, Gennady El, Pierre Suret and Stéphane Randoux: Early stage of integrable turbulence in 1D

NLS equation: the semi-classical approach to statistics

2:05–2:30 Bryn Balls-Barker, Blake Barker & Olivier Lafitte: Spectral stability of ideal-gas shock layers in the strong shock limit

2:30–2:55 Camille R. Zaug and John D. Carter: Frequency Downshift in the Ocean

2:55–3:20 Ali Eshaghian Dorche, Ali Asghar Eftekhar and Ali Adibi: Advanced dispersion enginedgeeering for wideband on-chip

optical frequency comb generation


3:50–5:55 SESSION 7, Mahler Hall: Stability and traveling waves – Part IV/IX


Samuel Walsh

3:50–4:15 Blake Barker: Rigorous verification of wave stability

4:15–4:40 Alin Pogan: Nonlinear stability of layers in precipitation models

4:40–5:05 Vahagn Manukian: Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models

5:05–5:30 Zhiwu Lin: Turning point principle for the stability of stellar models

5:30–5:55 Robert Marangell: Stability of travelling waves in a haptotaxis model

3:50–5:55 SESSION 5, Room F/G: Evolution Equations and Integrable Systems – Part III/V


3:50–4:15 Sarah Raynor: Low regularity stability for the KdV equation

4:15–4:40 John Holmes: Existence of solutions for conservation laws

4:40–5:05 Ryan Thompson: On the evolution of dark matter

5:05–5:30 Yuexun Wang: Enhanced existence time of solutions to the fractional KdV equation

5:30–5:55 Jose Pastrana Chiclana: Non-uniform continuous dependence for Euler equations in Besov spaces

3:50–5:55 SESSION 9, Room Y/Z: Nonlinear evolutionary equations: Theory, numerics and experiments – Part II/III

Chairs: Efstathios Charalampidis and Fotini Tsitoura

3:50–4:15 Foteini Tsitoura: Observation of phase domain walls in deep water surface gravity waves

4:15–4:40 Hang Yang: Models for 3D Euler Equations

4:40–5:05 Igor Barashenkov: New PT-symmetric systems with solitons: nonlinear Dirac and Landau-Lifshitz equations

5:05–5:30 Demetrios Christodoulides: Parity-Time and other symmetries in optics and photonics

5:30–55:5 Guo Deng, Gino Biondini and Surajit Sen: Generation, propagation and interaction of solitary waves in integrable versus

non-integrable lattices


biological systems – Part II/VII


3:50–4:15 Fabio Pusateri, Massimiliano Berti, and Roberto Feola: The Zakharov-Dyachenko conjecture on the integrability of

gravity water waves

4:15–4:40 Joseph Zaleski, Miguel Onorato and Yuri Lvov: Anomalous correlators, “ghost” waves and nonlinear standing waves in

the beta-FPUT system

4:40–5:05 Denis Silantyev and Pavel Lushnikov: Powerful conformal maps for adaptive resolving of the complex singularities of the

Stokes wave

5:05–5:30 Amir Sagiv, Adi Ditkowski and Gadi Fibich: Efficient numerical methods for nonlinear dynamics with random parameters

3:50–5:55 SESSION 6, Room J: Random Matrices, Painleve Equations, and Integrable Systems – Part III/III

Chair: Anton Dzhamay

3:50–4:15 Vasilisa Shramchenko: Algebro-geometric solutions to Schlesinger and Painlevé-VI equations

4:15–4:40 Leonid Chekhov: SLk character varieties and quantum cluster algebras

4:40–5:05 Alessandro Arsie: A survey of bi-flat F-manifolds

5:05–5:30 Nicholas Ercolani: Integrable mappings and random walks in random environments

5:30–5:55 Virgil Pierce: Skew-orthogonal polynomials and continuum limits of the Pfaff lattice

3:50–5:55 SESSION 16, Room V/W: Existence and stability of peaked waves in nonlinear evolution equations – Part I/III

Chair: Anna Geyer

3:50–4:15 Mariana Haragus: Regular patterns and defects for the Rayleigh-Bénard convection.

4:15–4:40 Richard Kollar: Krein signature without eigenfunctions and without eigenvalues. What is Krein signature and

what does it measure?

4:40–5:05 Fabio Natali: Periodic Traveling-wave solutions for regularized dispersive equations: Sufficient conditions for

orbital stability with applications

5:05–5:30 Uyen Le: Convergence of Petviashvili's method near periodic waves in the fractional KdV equation

5:30–5:55 Elek Csobo: Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials

3:50–5:55 SESSION 21, Room B: Stochastic dynamics in nonlinear systems - PART II/II

Chair: Katie Newhall

3:50–4:15 Joe Klobusicky: Averaging for systems of nonidentical molecular motors

4:15–4:40 Ilya Timofeyev: Stochastic parameterization of subgrid-scales in one-dimensional shallow water equations

4:40–5:05 Nawaf Bou-Rabee: Coupling for Hamiltonian Monte Carlo

5:05–5:30 Yuan Gao: Limiting behaviors of high dimensional stochastic spin ensemble

5:30–5:55 Molei Tao: Improving sampling accuracy of SG-MCMC methods via non-uniform subsampling of gradients

3:50–5:55 SESSION 24, Room C: Mathematical perspectives in quantum mechanics and quantum chemistry – Part III/III

Chairs: Jianfeng Lu and Israel Michael Sigal

3:50–4:15 Dionisios Margetis: On the excited state of the interacting boson system: a non-Hermitian view

4:15–4:40 Akos Nagy: Concentration properties of Majorana spinors in the Jackiw-Rossi theory

4:40–5:05 Marius Lemm: A central limit theorem for integrals of random waves

5:05–5:30 Christof Sparber: Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures

5:30–5:55 Thomas Chen: Boltzmann equations via Wigner transform and dispersive methods

3:50–5:55 SESSION 10, Room D: Recent advances in PDEs from fluid dynamics and other dynamical models – Part I/I

Chairs: Robin Ming Chen, Runzhang Xu

3:50–4:15 Gary Webb, Qiang Hu, Avijeet Prasad and Stephen Anco: Godbillon-Vey helicity in magnetohydrodynamics and fluid

dynamics

4:15–4:40 Hua Chen, Robert Gilbert and Philippe Guyenne: Dispersion and attenuation in a poroelastic model for gravity waves on

an ice-covered ocean

4:40–5:05 Qingtian Zhang: Global solution of SQG front equation

5:05–5:30 Dongfen Bian and Jinkai Li: Finite time blow up of compressible Navier-Stokes equations on half space or outside a

fixed ball

5:30–5:55 Wei Lian, Runzhang Xu and Yi Niu: Global well-posedness of coupled parabolic systems

========================================================================================

THURSDAY, APRIL 18, 2019


8:00–9:00 KEYNOTE LECTURE 2, Masters Hall

Alex Himonas: Initial and boundary value problems for evolution equations

Chair: Jerry Bona

9:10–10:00 SESSION 4, Room F/G: Fractional Diferential Equations – Part I/III

Chair: Harihar Khanal

9:10–9:35 Dumitru Baleanu: On fractional calculus and nonlinear wave phenomena

9:35–1:00 Andrei Ludu: Time dependent order differential equations

9:10–10:00 SESSION 8, Room Y/Z: Dispersive Wave Equations and their Soliton Interactions: Theory and Applications – Part I/V

Chairs: Avraham Soffer, Gang Zhao, S. Gustafson

9:10–9:35 Peter Pickl: Higher Order Corrections to Mean Field Dynamics of Bose Cold Gases

9:35–10:00 Thomas Chen and Avy Soffer: Dynamics of a heavy quantum tracer particle in a Bose gas

9:10–10:00 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical,

and biological systems – Part III/VII


9:10–9:35 Svetlana Roudenko, Kai Yang and Yanxiang Zhao: Stable blow-up dynamics in the critical and supercritical NLS and

Hartree equations

9:35–1:00 Anastassiya Semenova, Alexander Korotkevich, and Pavel Lushnikov: Appearance of stokes waves in deep water

9:10–10:00 SESSION 12, Room J: Dispersive shocks, semiclassical limits and applications – Part I/III

Chair: Gino Biondini

9:10–9:35 Stephane Randoux: Modulational instability of a plane wave in the presence of localized perturbations: some

experimental results in nonlinear fiber optics

9:35–1:00 Gennady El: Wave-mean flow interactions in dispersive hydrodynamics

9:10–10:00 SESSION 16, Room V/W: Existence and stability of peaked waves in nonlinear evolution equations – Part II/III

Chair: Dmitry Pelinovsky

9:10–9:35 Stephen Anco: Evolution equations with distinct sectors of peakon-type solutions

9:35–10:00 Zhijun Qiao: High order peakon models

9:10–10:00 PAPERS, Room K

Chairs: Otis wright

9:10–9:35 Alessandro Barone, Alessandro Veneziani, Flavio Fenton and Alessio Gizzi: Cardiac conductivity estimation by a

variational data assimilation procedure: analysis and validation

9:35–10:00 Otis Wright: Effective Integration of Some Integrable NLS Equations

9:10–10:00 SESSION 17, Room L: Nonlinear dynamics of mathematical models in neuroscience – Part I/II

Chair: Pamela Pyzza

9:10–9:35 Shelby Wilson: On the dynamics of coupled Morris-Lecar neurons


10:30–12:10 SESSION 18, Masters Hall: Negative flows, peakons, integrable systems, and their applications – Part III/IV

Chair: Stephane Lafortune

10:30–10:55 Qilao Zha, Qiaoyi Hu and Zhijun Qiao: Short pulse systems produced through the negative WKI hierarchy

10:55–11:20 Evans Boadi, Sicheng Zhao and Stephen Anco: New integrable peakon equations from a modified AKNS scheme

11:20–11:45 Shuxia Li and Zhijun Qiao: Lax algebraic representation for an integrable hierarchy

10:30–12:10 SESSION 9, Room F/G: Nonlinear Evolutionary Equations: Theory, Numerics and Experiments – Part III/III

Chair: Efstathios Charalampidis and Fotini Tsitoura

10:30–10:55 Jason Bramburger: Snakes and lattices: Understanding the bifurcation structure of localized solutions to lattice

dynamical systems

10:55–11:20 Ryan Goh: Growing stripes, with and without wrinkles

11:20–11:45 Zoi Rapti, Jared Bronski & Andrea Barreiro: Nonlinear eigenvalue problems in biologically motivated PDEs

11:45–12:10 Joceline Lega: Grain boundaries of the Swift-Hohenberg equation: simulations and analysis

10:30–12:10 SESSION 22, Room Y/Z: Modern Methods for Dispersive Wave Equations – Part I/II

Chairs: Robert Buckingham and Peter Miller

10:30–10:55 Peter Perry: Soliton Resolution for the Derivative Nonlinear Schrödinger Equation

10:55–11:20 Aaron Saalmann: Long-time asymptotics for the massive Thirring model

11:20–11:45 Elliot Blackstone: Singular limits of certain Hilbert-Schmidt integral operators and applications to tomography

11:45–12:10 Tom Trogdon: The computation of linear and nonlinear dispersive shocks

10:30–12:10 SESSION 25, Room E: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical,

and biological systems – Part IV/VII

Chair: Alexander O. Korotkevich and Pavel Lushnikov

10:30–10:55 Jerry Bona: Dynamical problems arising in blood flow: nonlinear waves on trees

10:55–11:20 Curtis Menyuk, Zhen Qi, Shaokang Wang: Stability and noise in frequency combs: harnessing the music of the spheres

11:20–11:45 Tobias Schaefer: Instantons and fluctuations in complex systems

11:45–12:10 Katelyn Plaisier Leisman and Gregor Kovacic: Nonlinear waves acting like linear waves in NLS

10:30–12:10 SESSION 12, Room J: Dispersive shocks, semiclassical limits and applications – Part II/III


10:30–10:55 Alexander Tovbis: Towards kinetic equation for soliton and breather gases for the focusing NLS equation

10:55–11:20 Sitai Li: Universal behavior of modulationally unstable media with non-zero boundary conditions

1:20 –11:45 Jonathan Lottes: Nonlinear interactions between solitons and dispersive shocks in focusing media

11:45–12:10 Thibault Congy: Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

10:30–12:10 SESSION 16, Room V/W: Existence and stability of peaked waves in nonlinear evolution equations – Part III/III

Chair: Dmitry Pelinovsky

10:30–10:55 Mathias Arnesen: A nonlocal approach to waves of maximal height to the Degasperis-Procesi equation

10:55–11:20 Raj Dhara: Waves of maximal height for a class nonlocal equations with homogeneous symbol

11:20–11:45 Tien Truong: Large-amplitude solitary water waves for the Whitham equation

11:45–12:10 Bruno Vergara: Convexity of Whitham's highest cusped wave

10:30–12:10 SESSION 3, K: Recent Developments in Mathematical Studies of Water Waves – Part III/III

Chair: John Carter

10:30–10:55 Diane Henderson: Faraday waves with bathymetry

10:55–11:20 Olga Trichtchenko: Water waves under ice

11:20–11:45 Bernard Deconinck: The stability of stationary solutions of the focusing NLS equation

11:45–12:10 Debbie Eeltink: Effect of viscosity and sharp wind increase on ocean wave statistics

10:30–12:10 SESSION 2, Room L: Novel challenges in nonlinear waves and integrable systems – Part I/II

Chairs: Barbara Prinari, Alyssa K. Ortiz

10:30–10:55 Martin Klaus: Spectral properties of matrix-valued AKNS systems with steplike potentials

10:55–11:20 Alexei Rybkin: The effect of a positive bound state on the KdV solution. A case study

11:20–11:45 C van der Mee: Exact solutions of the focusing NLS equation with symmetric nonvanishing boundary conditions

11:45–12:10 Jeremy Upsal: Real Lax spectrum implies stability

10:30–12:10 SESSION 17, Room D: Nonlinear dynamics of mathematical models in neuroscience – Part II/II

Chair: Pamela B. Pyzza

10:30–10:55 Paulina Volosov and Gregor Kovacic: Network reconstruction: architectural and functional connectivity in the

cerebral cortex

10:55–11:20 Duane Nykamp and Brittany Baker: Network microstructure dominates global network connectivity in synchronous

event initiation

11:20–11:45 Pamela Pyzza, Katie Newhall, Douglas Zhou, Gregor Kovacic and David Cai: Idealized models of insect olfaction

11:45–12:10 Alexei Cheviakov and Jason Gilbert: The narrow-capture problem in a unit sphere: global optimization of volume trap

arrangements


1:40–3:20 SESSION 7, Masters Hall: Stability and traveling waves – Part V/IX


Samuel Walsh

1:40–2:05 Graham Cox: A Maslov index for non-Hamiltonian systems

2:05–2:30 Claire Kiers: A bifurcation analysis of standing pulses and the Maslov index

2:30–2:55 Selim Sukhtaie: Localization for Anderson models on tree graphs

2:55–3:20 Mariana Haragus: Dynamics of frequency combs modeled by the Lugiato-Lefever equation

1:40–3:20 SESSION 5, Room F/G: Evolution Equations and Integrable Systems – Part IV/V

Chair: Alex Himonas, Curtis Holliman & Dionyssis Mantzavinos

1:40–2:05 Natalie Sheils: Revivals and fractalisation in the linear free space Schrodinger equation

2:05–2:30 David Smith: Unified transform method with moving interfaces

2:30–2:55 Fangchi Yan: Well-posedness of initial-boundary value problems for dispersive equations via the Fokas method

2:55–3:20 Maria Christina van der Weele: Integrable systems in 4+2 dimensions and their reduction to 3+1 dimensions

1:40–3:20 SESSION 8, Room Y/Z: Dispersive wave equations and their soliton interactions: Theory and applications – Part II/V


1:40–2:05 Marius Beceanu, Juerg Froehlich and Avy Soffer: Semi-linear Schroedinger's equation with random time-dependent

potentials

2:05–2:30 Minh Binh & Avy Soffer: On the energy cascade of acoustic wave turbulence: Beyond Kolmogorov-Zakharov solutions

2:30–2:55 Matthew Rosenzweig: Global well-posedness and scattering for the Davey-Stewartson system at critical regularity


biological systems – Part V/VII


1:40–2:05 Evgeny Kuznetsov, Maxim Kagan and Andrey Turlapov: Expansion of the strongly interacting superfluid Fermi gas:

symmetry and self-similar regimes

2:05–2:30 Joseph Zaleski, Philip Zaleski, and Yuri Lvov: Excitation of interfacial waves via near-resonant surface-interfacial

wave interactions

2:30–2:55 Sergey Dyachenko, Alexander Dyachenko, Pavel Lushnikov & Vladimir Zakharov: Singularities in 2D fluids with free

surface

2:55–3:20 Israel Michael Sigal: On density functional theory

1:40–3:20 SESSION 1, Room J: Nonlinear Waves – Part I/II

Chair: Jerry Bona

1:40–2:05 Guillaume Fenger: Strong error order of time-discretization of the stochastic gBBM equation

2:05–2:30 Min Chen: Mathematical analysis of Bump to Bucket problem

2:30–2:55 Olivier Goubet: Wave equations with infinite memory

2:55–3:20 Bongsuk Kwon: Small Debye length limit for Euler-Poisson system

1:40–3:20 SESSION 12, Room V/W: Dispersive shocks, semiclassical limits and applications – Part III/III


1:40–2:05 Mark Hoefer: Evolution of broad initial profiles—solitary wave fission and solitary wave phase shift

2:05–2:30 Antonio Moro: Dispersive shocks dynamics of phase diagrams

2:30–2:55 Jeffrey Oregero: Semiclassical Lax spectrum of Zakharov-Shabat systems with periodic potentials

2:55–3:20 Bingying Lu: The universality of the semi-classical sine-Gordon equation at the gradient catastrophe

1:40–3:20 SESSION 26, Room K: Physical Applied Mathematics – Part I/II

Chairs: Ziad Musslimani, Matthew Russo

1:40–2:05 Nick Moore: Anomalous waves induced by abrupt changes in topography

2:05–2:30 Adam Binswanger: Oblique dispersive shock waves in steady shallow water flows

2:30–2:55 Justin Cole: Solitons and Psuedo-solitons in the Korteweg-de-Vries equation with step-up boundary conditions

2:55–3:20 Sathyanarayanan Chandramouli: Spectral Renormalization algorithm applied to solving initial-boundary value problems

1:40–2:05 SESSION 2, Room L: Novel Challenges in Nonlinear Waves and Integrable Systems – Part II/II

Chairs: Barbara Prinari, Alyssa K. Ortiz

1:40–2:05 Annalisa Calini: Integrable evolutions of twisted polygons in centro-affine space

2:05–2:30 Brenton LeMesurier: Studying DNA transcription pulses with refinements of a [discrete] sine-Gordon approximation

2:30–2:55 Deniz Bilman: Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy

2:55–3:20 Alyssa K. Ortiz: Soliton solutions of certain reductions of the matrix NLS equation with non-zero boundary conditions

1:40–3:20 SESSION 20, V/W: Dynamical Systems and integrability – Part II/II

Chairs: Nalini Joshi and Nobutaka Nakazono

1:40–2:05 Y. Ohta: Two dimensional stationary vorticity distribution and integrable system

2:05–2:30 Claire Gilson: Quasi-Pfaffians and noncommutative integrable systems

2:30–2:55 Masato Shinjo and Koichi Kondo: A discrete analogue of the Toda hierarchy and its some properties

2:55–3:20 Giorgio Gubbiotti: On the inverse problem of the discrete calculus of variations


3:50–5:55 SESSION 7, Masters Hall: Stability and traveling waves – Part VI/IX


Samuel Walsh

3:50–4:15 Mat Johnson: Modulational dynamics of spectrally stable Lugiato-Lefever periodic waves

4:15–4:40 Chongchun Zeng: Steady concentrated vorticity and its stability of the 2-dim Euler equation on bounded domains

4:40–5:05 Dmitry Pelinovsky: Double-periodic waves of the focusing NLS equation and rogue waves on the periodic background

5:05–5:30 Keith Promislow: Bulk verses Surface Diffusion in Highly Amphiphilic Polymer Networks

5:30–5:55 Doug Wright: Generalized solitary wave solutions of the capillary-gravity Whitham equation

3:50–5:55 SESSION 5, Room F/G: Evolution equations and integrable systems – Part V/V


3:50–4:15 Dionyssios Mantzavinos: Analysis of nonlinear evolution equations in domains with a boundary

4:15–4:40 Feride Tiglay: Non-uniform dependence of the data-to-solution map for the Hunter--Saxton equation in Besov spaces

4:40–5:05 Rafael Barostichi: The Cauchy problem for the "good" Boussinesq equation with analytic and Gevrey initial data

5:05–5:30 Renata Figueira: Gevrey regularity in time variable for solutions to the "good" Boussinesq equation.

5:30–5:55 Alex Himonas: The Cauchy problems for evolution equations with analytic data

3:50–5:55 SESSION 8, Room Y/Z: Dispersive wave equations and their soliton interactions: Theory and applications – Part III/V


3:50–4:15 Stefanos Aretakis: Conservation laws and asymptotics for the wave equation

4:15–4:40 Jonas Luhrmann: Local smoothing estimates for Schrodinger equations on hyperbolic space and applications

4:40–5:05 Hao Jia: Quantization of energy of blow up for wave maps

5:05–5:30 Baoping Liu: Long time dynamics for nonlinear dispersive equations

5:30–5:55 Qingquan Deng: Soliton Potential Interaction of NLS in R3

3:50–5:55 SESSION 18, Room E: Negative flows, peakons, integrable systems, and their applications – Part IV/IV

Chairs: Stephen Anco, Stephane Lafortune

3:50–4:15 Daniel Kraus: Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

4:15–4:40 Wenhao Liu: Some new exact solutions for the extended (3+1)-dimensional Jimbo-Miwa equation

4:40–5:05 Vesselin Vatchev: Some Properties of Wronskian Solutions of Nonlinear Differential Equations

3:50–5:55 SESSION 1, Room J: Nonlinear waves – Part II/II

Chairs: Min Chen

3:50–4:15 Douglas Svensson Seth: Three-dimensional steady water waves with vorticity

4:15–4:40 Shenghao Li: Lower regularity solutions of non-homogeneous boundary value problems of the sixth order Boussinesq

equation in a quarter plane

4:40–5:05 Hongqiu Chen: Well-posedness for a higher-order, nonlinear, dispersive equation: new approach

5:05–5:30 Shu-Ming Sun: Solitary-wave solutions for some BBM-type of equations with inhomogeneous nonlinearity

3:50–5:55 SESSION 22, Room V/W: Modern methods for dispersive wave equations – Part II/II

Chairs: Robert Buckingham and Peter Miller

3:50–4:15 Rowan Killip: KdV is well-posed in H-1

4:15–4:40 Jiaqi Liu: Long time asymptotics of the defocussing Manakov system in weighted Sobolev space

4:40–5:05 Donatius DeMarco: Asymptotics of rational solutions of the defocusing nonlinear Schrodinger equation

5:05–5:30 Bob Jenkins: Semiclassical soliton ensembles and the three-wave resonant interaction (TWRI) equations

3:50–5:55 SESSION 26, Room K: Physical applied mathematics – Part II/II

Chairs: Ziad Musslimani, Matthew Russo

3:50–4:15 Abdullah Aurko: Time-dependent spectral renormalization method applied to conservative PDEs

4:15–4:40 Constance Schober: Linear instability of the Peregrine breather: Numerical and analytical investigation

4:40–5:05 Ryan Roopnarain: Various dynamical regimes, and transitions from homogeneous to inhomogeneous steady states in

oscillators with delays and diverse couplings

5:05–5:30 Michail Todorov and Vladimir Gerdjikov: On N-soliton interactions: Effects of local and non-local potentials

3:50–4:15 SESSION 4, Room L: Fractional Diferential Equations – Part II/III

Chair: Andrei Ludu

3:50–4:15 Gavriil Shchedrin, Nathanael Smith, Anastasia Gladkina and Lincoln Carr: Generalized Euler's integral transform

4:15–4:40 Aghalaya Vatsala: One dimensional sub-hyperbolic equation via sequential Caputo fractional derivative

4:40–5:05 Christina Nevshehir: The gravity of light travel: riding the fractional wave of a visible universe from h to c-squared

5:05–5:30 Haret Rosu and Stefan Mancas: The factorization method for fractional quantum oscillators

5:30–5:55 Timothy Burns and Bert Rust: Closed-form projection method for regularizing a function defined by a discrete set of noisy

data and for estimating its derivative and fractional derivative

3:50–5:55 SESSION 23, Room D: Nonlinear waves in optics, fluids and plasma – Part I/II

Chairs: Sergey Dyachenko, Katelyn Leisman, Denis Silantyev

3:50–4:15 Jeffrey Banks & Andre Gianesini Odu: High-order accurate conservative finite differences for Vlasov equations in 2D+2V

4:15–4:40 Pavel M Lushnikov, Vladimir E Zakharov and Nikolay M. Zubarev: Non-canonical Hamiltonian structure and integrability

for 2D fluid surface dynamics

4:40–5:05 Jolene Britton and Yulong Xing: Well-balanced discontinuous Galerkin methods for blood flow simulation

with moving equilibrium

5:05–5:30 Yulong Xing: Invariant conserving local discontinuous Galerkin methods for the modified Camassa-Holm equation

5:00–7:00 POSTERS: Pecan Tree Galleria

Lucas Schauer and Geng Chen: Shock formation in finite time for the 1-d compressible Euler equations

Taylor Paskett and Blake Barker: Stability of traveling waves in compressible Navier-Stokes

Alexei Cheviakov and Caylin Lee: Nonlinear wave equations of shear radial wave propagation in fiber-reinforced

cylindrically symmetric media

Ryan Marizza, Jessica Harris, Michelle Maiden and Mark Hoefer: Theory and observation of interacting linear waves and

nonlinear mean flows in a viscous fluid conduit

7:00- 9:00 BANQUET

Speaker: TBA

Thiab Taha: Presentation of best Student Paper Awards

========================================================================================

FRIDAY, APRIL 19, 2019


8:00–9:00 KEYNOTE LECTURE 3, Masters Hall

Stefano Trillo: Nonlinear PDEs describing real experiments: recurrences, solitons, and shock waves


9:10–10:00 SESSION 7, Masters Hall: Stability and traveling waves – Part VII/IX


Samuel Walsh

9:10–9:35 Dag Nilsson: Solitary wave solutions of a Whitham-Bousinessq system

9:35–10:00 Ola Maehlen: Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities

9:10–10:00 SESSION 13, Room F/G: Recent advances in numerical methods of PDEs and applications in life science,

material science – Part I/II

Chairs: Qi Wang and Xueping Zhao

9:10–9:35 Thomas Lewis: Approximating nonlinear reaction-diffusion problems with multiple solutions

9:35–10:00 Shuang Liu and Xinfeng Liu: Efficient and stable numerical methods for a class of stiff reaction-diffusion systems with

free boundaries

9:10–10:00 SESSION 14, Room K: Nonlinear kinetic self-organized plasma dynamics driven by coherent, intense electromagnetic

fields – Part I/II

Chairs: Bedros Afeyan, Shadwick Brad,Wilkening Jon

9:10–9:35 Jon Wilkening, Bedros Afeyan and Rockford Sison: Spectrally accurate methods for kinetic electron plasma wave

dynamics

9:35–10:00 Bedros Afeyan and Richard Sydora: Improving the performance of plasma kinetic simulations by iteratively learned

phase space tiling: Variational constrained optimization meet machine learning

9:10–10:00 SESSION 25, Room V/W: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and

biological systems – Part VI/VII


9:10–9:35 Avadh Saxena and Avinash Khare: Family of potentials with power-law kink tails

9:35–10:00 Taras Lakoba and Jeffrey Jewell: Higher-order Runge--Kutta-type schemes based on the method of characteristics for

hyperbolic equations with crossing characteristics

10:00 - 10:30 COFFEE BREAK

10:30–12:10 SESSION 7, Masters Hall: Stability and traveling waves – Part VIII/IX


Samuel Walsh

10:30–10:55 Miles Wheeler: Coriolis forces and particle trajectories for waves with stratification and vorticity

10:55–11:20 Kristoffer Varholm: On the stability of solitary water waves with a point vortex

11:20–11:45 David Ambrose: Periodic traveling hydroelastic waves

11:45–12:10 Robin Ming Chen: Asymptotic stability of the Novikov peakons

10:30–12:10 SESSION 13, Room F/G: Recent advances in numerical methods of pdes and applications in life science,

material science – Part II/II

Chairs: Qi Wang and Xueping Zhao

10:30–10:55 Yi Sun and Qi Wang: A hybrid model for simulating sprouting angiogenesis in biofabrication.

10:55–11:20 Paula Vasquez and Erik Palmer: A parallel approach to kinetic viscoelastic modelling

11:20–11:45 Xiaofeng Yang: Efficient schemes with unconditionally energy stabilities for anisotropic phase field models: S-IEQ and

S-SAV

11:45–12:10 Qi Wang and Xueping Zhao: A second order fully-discrete linear energy stable scheme for a binary compressible

viscous fluid model

10:30–12:10 SESSION 8, Room Y/Z: Dispersive wave equations and their soliton interactions: Theory and applications – Part IV/V


10:30–10:55 Leonid Chaichenets: Dirk Hundertmark, Peer Kunstmann and Nikolaos Pattakos: Knocking out teeth in

one-dimensional periodic NLS: Local and Global wellposedness results

10:55–11:20 Nikolai Leopold and Soeren Petrat: Derivation of the Schroedinger-Klein-Gordon equations

11:20–11:45 Yifei Wu: Global well-posedness for mass-subcritical NLS in critical Sobolev space

10:30–12:10 SESSION 4, Room K: Fractional diferential equations – Part III/III

Chairs: Dumitru Baleanu

10:30–10:55 Harihar Khanal: Variable Order Differential Equations, Solutions and Applications

10:55–11:20 Thiab Taha: IST Numerical Schemes for Solving Nonlinear Evolution Equations and their possible applications

for solving time Fractional Differential Equations

10:30–12:10 SESSION 25, Room V/W: Nonlinear waves, singularities, vortices, and turbulence in hydrodynamics, physical, and

biological systems – Part VII/VII


10:30–10:55 Stephen Gustafson: Chiral magnetic skyrmions for 2D Landau-Lifshitz equations

10:55–11:20 Benno Rumpf: Clebsch variables for stratified compressible fluids


1:40–3:20 SESSION 7, Masters Hall: Stability and traveling waves – Part IX/IX


Samuel Walsh

1:40–2:05 Wesley Perkins: Modulational instability of viscous fluid conduit periodic waves

2:05–2:30 Hung Le: On the existence and instability of solitary water waves with a finite dipole

2:30–2:55 Peter Howard: Renormalized oscillation theory for linear Hamiltonian systems via the Maslov index

2:55–3:20 Jiayin Jin: Invariant manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space

1:40–3:20 SESSION 23, Room F/G: Nonlinear waves in optics, fluids and plasma – Part II/II

Chairs: Sergey Dyachenko, Katelyn Leisman, Denis Silantyev

1:40–2:05 Mimi Dai: Non-uniqueness of Leray-Hopf weak solutions for 3D Hall-MHD system

2:05–2:30 Ezio Iacocca: A hydrodynamic formulation for solid-state ferromagnetism

2:30–2:55 Alexander Korotkevich: Inverse cascade of gravity waves in the presence of condensate: numerical results and

analytical explanation

2:55–3:20 Alexey Cheskidov and Xiaoyutao Luo: Weak solutions for the 3D Navier-Stokes equations with discontinuous energy

1:40–3:20 SESSION 8, Room Y/Z: Dispersive Wave Equations and their Soliton Interactions: Theory and Applications – Part V/V


1:40–2:05 Scott Strong and Lincoln Carr: Nonlinear waves on vortex filaments in quantum liquids: A geometric perspective

2:05–2:30 Svetlana Roudenko, Anudeep Kumar Arora and Kai Yang: Stable blow-up dynamics in the generalized L2-critical

Hartree equation

2:30–2:55 M. Burak Erdogan, William R. Green and Ebru Toprak: The effect of threshold energy obstructions on the L1 to

L-infinity dispersive estimates for some Schrodinger type equations

2:55–3:20 Yanqiu Guo and Edriss Titi: Backward behavior of a dissipative KdV equation

1:40–3:20 SESSION 14, Room K: Nonlinear kinetic self-organized plasma dynamics driven by coherent, intense electromagnetic

fields – Part II/II

Chairs: Bedros Afeyan,Shadwick Brad,Wilkening Jon

1:40–2:05 B. A. Shadwick, Alexander Stamm and Bedros Afeyan: Nonlinear instabilities due to drifting species and magnetic fields in

high energy density plasmas

2:05–2:30 Richard Sydora, Bedros Afeyan and Bradley A. Shadwick: Impact of cyclotron harmonic wave instabilities on stability of

self-organized nonlinear kinetic plasma structures

2:30–2:55 Frank Lee, Michael Allshouse, Harry Swinney and Philip Morrison: Internal wave energy flux from density perturbations


KEYNOTE ABSTRACTS

Vortex sheets, Boussinesq equations, and otherproblems in the Wiener algebra

David M. Ambrose

Drexel University, Department of Mathematics

Philadelphia, PA 19104 USA

[email protected]

There are several approaches to proving the ill-posedness of vor-

tex sheets; we will explore the version due to Duchon and Robert.

Interestingly, the Duchon and Robert result is really about global

existence of small solutions. The functional setting is a space-time

version of the Wiener algebra with exponential weights, and func-

tions in this space, at any time after the initial time, are spatially

analytic. This existence theorem becomes an ill-posedness result

upon reversing time, finding small analytic solutions which lose

analyticity arbitrarily quickly. Both aspects of this proof – exis-

tence of solutions and ill-posedness – are of interest for other prob-

lems, and we will describe several applications. These applications

may include nonlinear ill-posedness of linearly ill-posed Boussi-

nesq equations, some small global solutions of the 2D Kuramoto-

Sivashinsky equation, small global solutions for a problem in epi-

taxial growth, and existence of solutions for non-separable mean

field games. This includes joint work with Jerry Bona, Anna Maz-

zucato, and Timur Milgrom.

Initial and boundary value problems for evolutionequations

Alex Himonas

Department of Mathematics, University of Notre Dame

Notre Dame, IN 46556

[email protected]

In the first part of the talk we shall discuss questions of existence,

uniqueness, dependence on initial data, and regularity of solutions

to the initial value problem of Camassa-Holm and related equa-

tions in a variety of function spaces. Some of these equations can

be thought as “toy” models for the Euler equations governing the

motion of an incompressible fluid, and the analytic techniques de-

veloped for these equations have been in some cases transferable

to the Euler equations. In the second part of the talk we shall fo-

cus on the advancement of the Unified Transform Method of Fokas

for solving the initial-boundary value problem (ibvp) of nonlinear

evolution equations in one and two space dimensions. Although

introduced as the ibvp analogue of the renowned Inverse Scatter-

ing Transform method for integrable nonlinear evolution equations,

Fokas’ approach can also be used to produce novel solution formu-

las for the linear versions of such equations. Replacing in Fokas’

solution formulas the forcing with the nonlinearity provides a new

framework for the analysis of nonlinear equations with a variety of

boundary conditions in appropriate solution spaces.

The talk is based on work in collaboration with R. Barostichi,

R.O. Figueira and G. Petronilho (Federal University of Sao Car-

los, Brazil) A. Fokas (University of Cambridge, UK), J. Gorsky

(University od San Diego), C. Holliman (Catholic University of

America), J. Holmes (Ohio State), H. Kalisch and S. Selberg (Uni-

versity of Bergen, Norway), C. Kenig (University of Chicago), G.

Misiołek and F. Yan (University of Notre Dame), D. Mantzavinos

(University of Kansas), G. Ponce (University of California, Santa

Barbara), R. Thompson (University of North Georgia), F. Tiglay

(Ohio State).

Nonlinear PDEs describing real experiments:recurrences, solitons, and shock waves

Stefano Trillo

Department of Engineering, University of Ferrara, Via Saragat 1, 44122

Ferrara, Italy

[email protected]

The Fermi-Pasta-Ulam-Tsingou (FPUT) paradox discovered in the

50s, i.e., the fact that a nonlinear system with many or even in-

finite degrees of freedom might exhibit near or exact recurrences

to the initial condition instead of a transition to equipartition of

energy between the modes, is still the driving force of many re-

search avenues in nonlinear physics. Historically, integrability of

the underlying models, their soliton solutions, and the generating

mechanisms such as shock formation have commonly believed to

play a key role.

In this paper, we will review the recent achievements obtained in

understanding FPUT recurrence phenomena with special emphasis

on the theoretical results that explain real experimental observa-

tions. Two main mechanisms will be discussed. The first entails

the fission of solitons from periodic initial data [1] akin to the fa-

mous numerical experiment performed by Zabusky and Kruskal

(1965) for the KdV equation. The case of the KdV and the defo-

cusing NLS equations will be contrasted to illustrate similarities

and differences.

A second scenario involves modulational instability in the focus-

ing NLS equation where recurrences are mediated by the inter-

action with a background according to a complicated homoclinic

structure where breather play a key role [2]. Latest developments

of such scenario and related open problems will be discussed also

in connection to parametric resonance governed by non-integrable

models [3].

1. S. Trillo, G. Deng, G. Biondini, M. Klein, G. Clauss, A. Chabchoub,

and M. Onorato, Experimental observation and theoretical description

of multisoliton fission in shallow water, Phys. Rev. Lett., 117 (2016),

144102.

2. A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szrift-

giser, and S. Trillo, Fibre multiwave-mixing combs reveal the broken

symmetry of Fermi-Pasta-Ulam recurrence, Nat. Photonics, 10 (2018),

303–308.

3. A. Mussot, M. Conforti, S. Trillo, F. Copie, and A. Kudlinski, Modu-

lation instability in dispersion oscillating fibers, Adv. Opt. Photon., 10

(2018), 1–42.

1

SESSION ABSTRACTS

SESSION 1

Nonlinear waves

Jerry Bona

Department of Mathematics, Statistics, and Computer Science

University of Illinois at Chicago, Chicago, IL 60607

[email protected]

Min Chen

Department of Mathematics, Purdue University, West Lafayette, IN 47907

[email protected]

Shu-Ming Sun

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061

[email protected]

Bing-yu Zhang

Department of Mathematical Sciences, University of Cincinnati

Cincinnati, OH 45221

[email protected]

This session will focus on the propagation of waves in water and

other media where nonlinearity, dispersion and sometimes dissipa-

tion and capillarity are all acting. Featured in the session will be

theoretical work, such as existence of solitary wave solutions and

existence of two dimensional standing waves, well-posedness of

dispersion-managed nonlinear systems, higher order model equa-

tions and equations with dispersive terms, and numerical investi-

gation on the solutions of various systems and equations.

SESSION 2

Novel challenges in nonlinear waves and integrablesystems

Barbara Prinari1,2,∗ and Alyssa K. Ortiz2

1 Department of Mathematics, University at Buffalo2 Department of Mathematics, University of Colorado Colorado Springs

[email protected], [email protected]

The study of physical phenomena by means of mathematical mod-

els leads in many cases to nonlinear wave equations. A special

class of nonlinear wave equations is represented by the so-called

soliton equations, which are infinite-dimensional completely inte-

grable Hamiltonian systems that admit an infinite number of con-

served quantities, and whose initial-value problem can be linearized

via a method called the inverse scattering transform. At the same

time, in realistic settings most nonlinear wave equations of phys-

ical interest are non-integrable, and integrability can only offer a

partial picture. Understanding integrable as well as non-integrable

nonlinear wave equations and their solutions and investigating their

remarkably rich mathematical structure requires a variety of tech-

niques from different branches of mathematics, such as exact meth-

ods, approximations, asymptotics and perturbation techniques, sym-

metry analysis, numerical simulations, etc.

Over the past fifty years, a large body of knowledge has been ac-

cumulated on nonlinear waves and integrable systems, which con-

tinue to be extensively studied worldwide and to offer interesting

research problems and new venues for applications. Among the

many current research topics in this area are multi-dimensional

systems, boundary value problems, discretization issues, connec-

tions with algebraic and differential geometry, number theory and

different areas of mathematics, etc.

This session aims at bringing together leading researchers in the

fields of nonlinear wave equations and integrable systems, and at

offering a broad overview of some of the current research activities

at the frontier of pure and applied mathematics.

SESSION 3

Recent developments in mathematical studies of wa-ter waves

John D. Carter

Mathematics Department, Seattle University

[email protected]

This minisymposium brings together mathematicians, engineers,

and oceanographers. With a focus on nonlinear water waves, the

speakers in this session will present experimental, analytical, and

numerical results from mathematical models of waves on shallow

and/or deep water.

SESSION 4

Fractional differential equations

Andrei Ludu

Embry-Riddle Aeronautical University

Dept. Mathematics & Wave Lab, Daytona Beach, FL, USA

[email protected]

Changpin Li

Shanghai University, Department of Mathematics, Shanghai, China

[email protected]

Thiab Taha

University of Georgia, Computer Science Department, Athens, GA, USA

[email protected]

Fractional Calculus is a field of pure and applied mathematics that

deals with derivatives and integrals of arbitrary orders and their ap-

plications in science, engineering, mathematics, and other fields.

In recent years considerable interest in fractional calculus has been

stimulated by the applications that this calculus finds in numerical

analysis and different areas of physics and engineering, possibly

including fractal phenomena. In this Special Session the talks will

cover areas from pure mathematical fractional calculus and theo-

rems for existence, uniqueness and stability of fractional differen-

tial equations, to time-dependent fractional order differential equa-

tions, and to several various fields of science including data sci-

ence, viscoelasticity, rheology, electrical engineering, biophysics,

signal and image processing, quantum physics, and control theory.

SESSION 5

Evolution equations and integrable systems

Alex A. Himonas

Department of Mathematics, University of Notre Dame, Notre Dame, IN

46556

[email protected]

2

Curtis Holliman

Department of Mathematics, The Catholic University of America

Aquinas Hall 116, Washington, DC 20064

[email protected]

Dionyssis Mantzavinos

Department of Mathematics, University of Kansas

Lawrence, KS 66045

[email protected]

Linear and nonlinear evolution equations have been at the fore-

front of advances in partial differential equations for a long time.

They are involved in beautiful, yet extremely challenging prob-

lems, with a strong physical background, for which progress is

achieved through a mixture of techniques lying at the interface be-

tween analysis and integrable systems. Topics studied for these

equations include, among others, traveling waves, initial-boundary

value problems, local and global well-posedness, inverse scatter-

ing, stability, and integrability.

SESSION 6

Random matrices, Painleve equations, andintegrable systems

Vladimir Dragovic

Department of Mathematical Sciences,

The University of Texas at Dallas, Richardson, TX 75080, USA

[email protected]

Anton Dzhamay and Virgil U. Pierce

School of Mathematical Sciences,

University of Northern Colorado, Greeley, CO 80639, USA


The theory of Integrable Systems is well-known for employing

tools from many different branches of mathematics and mathemat-

ical physics to perform qualitative and quantitative analysis of a

wide range of important natural phenomena. In this special ses-

sion we plan to primarily focus on the theory of discrete and con-

tinuous Painleve equations, and the interaction between Random

Matrices and Integrable Systems. Painleve equations are a special

class of nonlinear ordinary differential equations whose solutions

satisfy the Painleve property that their only movable singularities

are poles. Thus, solutions of Painleve equations have good an-

alytic properties and form a class of genuinely nonlinear special

functions known as the Painleve transcendents. The importance

of these special functions has been steadily growing and we now

know that many interesting models can be described in terms of

such Painleve transcendents. In recent years there had been many

interesting developments in the theory of discrete Painleve equa-

tions, whose theory is founded on deep ideas from the algebraic

geometry. A large class of such examples occurs in the theory of

Random Matrices as well as discrete dynamical systems of Ran-

dom Matrix type, such as determinantal point processes. Among

powerful tools for studying such problems, as well as for studying

the asymptotics of Painleve transcendents, is the Riemann-Hilbert

Problem approach. Talks in our session will give a broad overview

of this research area, highlight important recent developments, and

outline possible new research directions.

SESSION 7

Stability and traveling waves

Bernard Deconinck

Department of Applied Mathematics, University of Washington, Seattle,

WA 98195

[email protected]

Anna Ghazaryan

Department of Mathematics, Miami University, Oxford, OH 45056

[email protected]

Mat Johnson

Department of Mathematics, University of Kansas, Lawrence, KS 66045

[email protected]

Stephane Lafortune

Department of Mathematics, College of Charleston, Charleston, SC 29401

[email protected]

Yuri Latushkin

Mathematics Department, University of Missouri, Columbia, MO 65211

[email protected]

Jeremy Upsal

Department of Applied Mathematics, University of Washington, Seattle,

WA 98195

[email protected]

Samuel Walsh

Mathematics Department, University of Missouri, Columbia, MO 65211

[email protected]

This session will bring together researchers who study fronts, pulses,

wave trains and patterns of more complex structure which are re-

alized as special solutions of nonlinear partial differential equa-

tions. Existence, stability, dynamic properties, and bifurcations of

those solutions will be discussed, from both analytical and numer-

ical point of views.

SESSION 8

Dispersive wave equations and their soliton interac-tions: Theory and applications

Avraham Soffer, Gang Zhao, S. Gustafson

In this session we focus on modern results and approaches to the

large time complex dynamics of dispersive equations. In particular,

large time global existence and scattering problems will be consid-

ered, for rough initial data, nonlinear dynamics in the presence of

noise, dispersive estimates in the presence of threshold singular-

ities, soliton-potential collision dynamics and more. The mathe-

matical tools presented come from analytic, computational and nu-

merical approaches. On the more technical level, the emphasis will

be on dynamics which at large times is not asymptotically stable:

the solutions wander far away from the initial coherent structures,

be it solitons or other objects.

3

SESSION 9

Nonlinear evolutionary equations: Theory, numer-ics and experiments

Efstathios G. Charalampidis and Foteini Tsitoura

Department of Mathematics and Statistics, University of Massachusetts

Amherst, MA 01003-9305

[email protected] and [email protected]

This session will focus on the study of nonlinear waves in a broad

array of fields. It will bring together caliber experts studying the-

oretically, numerically as well as experimentally nonlinear waves

in novel physical settings. Furthermore, this session will highlight

some of the newest findings in these settings as well as introduce

novel theoretical and computational techniques associated with the

underlying equations. Among the topics to be discussed in the ses-

sion include, bifurcation analysis, stability problems motivated by

biological settings, discontinuous shock wave solutions, flows in

Boussinesq models, and experimentally observed patterns on deep

water, among others.

SESSION 10

Recent advances in PDEs from fluid dynamics andother dynamical models

Robin Ming Chen

Department of Mathematics

University of Pittsburgh,

USA

[email protected]

Runzhang Xu

College of Science

Harbin Engineering University,

Harbin, P R China

[email protected]

This session will mainly focus on a number of recent developments

in the very active areas of fluid mechanics, integrable systems, clas-

sical physical models and other dynamical systems. The topics in-

clude Hamiltonian structures; derivation of physical model equa-

tions; well-posedness; formation of singularities; stability analy-

sis and geometric aspects, etc.. Directions of work related to new

methods and their applications to nonlinear PDEs will be empha-

sized, with the aim of bringing together a number of researchers at

all career stages working in on these topics.

SESSION 12

Dispersive shocks, semiclassical limits and applica-tions

Gino Biondini

State University of New York at Buffalo, Department of Mathematics

[email protected]

The combination of nonlinearity and weak dispersion can often re-

sult in the formation of dispersive shock waves (DSW). The study

of the formation and interaction of dispersive shocks has received

renewed interest in recent years, both from a theoretical and from a

practical point of view. On theoretical side, the study of DSW can

involve a variety of techniques such as Whitham modulation the-

ory, the inverse scattering transform, oscillatory Riemann-Hilbert

problems, the algebro-geometric approach, and the Deift-Zhou non-

linear steepest descent method. On the practical side, DSWs have

been experimentally observed in many physical contexts such wa-

ter waves, nonlinear optics, plasmas and Bose-Einstein conden-

sates among others. This session aims at presenting several recent

results on the subject.

SESSION 13

Recent advances in numerical methods of PDEs andapplications in life science, material science

Qi Wang

1523 Greene Street, Office 313C,

Columbia, South Carolina, 29208

[email protected]

Xueping Zhao

1523 Greene Street, Office 313A,

Columbia, South Carolina, 29208

[email protected]

This session will focus on the recent advances in numerical ap-

proaches of PDEs and their applications in life science and material

science.

Partial differential equation is a powerful tool to investigate the

underlying mechanisms of various phenomena in nature. Due to

the lack of analytical solutions, accurate and efficient numerical

methods of PDEs and their applications are required to promote

the development of various fields. This session includes multiple

numerical methods developed very recently and their applications,

such as group behavior, dendritic solidification. The following is a

list of titles of the talks and information of the authors.

SESSION 14

Nonlinear kinetic self-organized plasma dynamicsdriven by coherent, intense electromagnetic fields

Bedros Afeyan and Jon Wilkening

Polymath Research Inc.

University of California, Berkeley


Brad Shadwick∗

University of Nebraska, Lincoln

[email protected]

Session 14 is dedicated to the study of nonlinear kinetic plasma

wave structures in phase space and analogues in fluids. The pre-

sentations focus strogly on numerical methods that either extend

current state of the art towards spectral accuracy, or adaptive refine-

ment tied to machine learning, or variational techniques or tradi-

tional particle-in-cell codes but in new regimes. Emphasis is given

to strong, coherent (laser) field interactions with high energy den-

sity plasmas. Here, both one dimensional and higher dimensional

models are treated in both the electrostatic limit and for fully elec-

tromagnetic cases. Self-consistent and externally imposed mag-

netic fields play an important role in most of the presentations.

4

Two fluid analogues are also included so as to show the breadth

and potential impact of these works on nonlinear wave science in

general.

SESSION 15

Waves in topological materials

Hailong Guo

School of Mathematics and Statistics, The University of Melbourne, Parkville,

VIC 3010, Australia

[email protected]

Xu Yang

Department of Mathematics, University of California, Santa Barbara, CA,

93106, USA

[email protected]

Yi Zhu

Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Bei-

jing, 100084, China

[email protected]

In recent several years, there have been intense efforts toward ex-

ploiting the topological protected wave propagation–immune from

scattering by defects and disorder. These novel and subtle wave

patterns are investigated in different physical systems, which in-

clude, but not limited to, matter waves in quantum systems, acous-

tic waves in nano-systems, electromagnetic waves in photonic sys-

tems. A vast of new experiments and theories come out to describe

the wave phenomena in topological materials. The goal of this spe-

cial session is to bring together theoretical and applied researchers

in these areas to discuss some recent advances in the mathematical

theories and physical applications. Topics include, but not limited

to, the analysis of the underlying governing equations, numerical

methods on computing the edge states, experimental realizations.

SESSION 16

Existence and stability of peaked waves in nonlinearevolution equations

Anna Geyer

Address: Delft Institute of Applied Mathematics,

Delft University of Technology,

Mekelweg 4, 2628 CD Delft, The Netherlands

[email protected]

Dmitry E. Pelinovsky

Department of Mathematics and Statistics,

McMaster University,

Hamilton, Ontario, Canada, L8S 4K1

[email protected]

This session will focus on peaked waves in nonlinear evolution

equations.

Several well-known nonlinear dispersive equations arising for in-

stance from models for water waves allow for solutions which ex-

hibit singularities in the derivatives. Examples of such equations

are the Camassa-Holm equation, the Whitham equation and other

members of the fractional KdV equation, the reduced Ostrovsky

equation and the short pulse equation. In this session, different as-

pects regarding the existence and stability of such peaked or cusped

solutions will be discussed using analytical as well as numerical

methods.

SESSION 17

Nonlinear dynamics of mathematical models inneuroscience

Pamela B. Pyzza

Ohio Wesleyan University

61 S. Sandusky Street

Delaware, OH 43015

[email protected]

Nonlinear dynamics appear often in neuroscience and thus math-

ematical modeling lends itself as an effective approach to investi-

gating neuronal phenomena. This session will feature recent con-

tributions of mathematics to neuroscience and neuronal networks,

including innovations in modeling and in the analysis of models.

The speakers in this session will present work applying compu-

tational, analytical, and experimental tools to address a variety of

problems in mathematical neuroscience.

SESSION 18

Negative flows, peakons, integrable systems and theirapplications

Stephen Anco

Brock University

St. Catharines, ON, L2S 3A1, Canada

[email protected]

Zhijun (George) Qiao

University of Texas Rio Grande Valley

Edinburg, TX 78539

[email protected]

Stephane Lafortune

College of Charleston

Charleston, South Carolina 29424

[email protected]

Nonlinear dispersive wave equations appear in many fields, includ-

ing fluid mechanics, plasma physics, optics, and differential geom-

etry. There has been much recent work on the study of these equa-

tions, especially ones that describe negative flows and peakons, yet

many interesting questions and problems remain to be solved. This

session will bring together researchers at all career stages to share

their recent results on various topics related to integrable systems,

negative flows, peakons, and nonlinear soliton models. Specific

topics will focus on (but not be restricted to) peakons and other

soliton solutions, negative flows and their integrability structure,

reciprocal/Liouville transformations, Hamiltonian structures, con-

servation laws relating negative flows and peakon equations, as

well as other developments connected with these types of equa-

tions and their solutions.

5

SESSION 19

Network dynamics

Thomas Carty

Department of Mathematics, Bradley University, Peoria, IL, 61625

[email protected]

This special section will concentrate on network dynamics. Recent

analytical advances have led to an explosion of the use of dynam-

ical systems on graphs in the modeling of natural phenomena. In

the last ten years alone, we have seen network models applied to

neurochemisty in the study of brain neurons, to social science in

models of group decision making and group participation, to eco-

nomics as a model of electrical power distribution on an energy

grid, and more. One focus of this special session is on Kuramoto-

type models for finite networks. The Kuramoto variants have been

indispensable in modeling dynamics where synchrony of oscilla-

tory behavior arise. Interesting and difficult problems arise as the

complexity of the oscillatory behavior of the individual actors in-

creases.

SESSION 20

Dynamical systems and integrability

Nalini Joshi

School of Mathematics and Statistics, The University of Sydney, New

South Wales 2006, Australia.

[email protected]

Nobutaka Nakazono and Giorgio Gubbiotti

Department of Physics and Mathematics, Aoyama Gakuin University

Sagamihara, Kanagawa 252-5258, Japan.

School of Mathematics and Statistics, The University of Sydney

New South Wales 2006, Australia.


Milena Radnovic and Yang Shi

School of Mathematics and Statistics, The University of Sydney

New South Wales 2006, Australia.

College of Science and Engineering, Flinders University

Adelaide 5042, Australia.


The study of integrability and integrable systems addresses impor-

tant questions from mathematics and physics. Many of these ques-

tions arise from the study of models involving finite operations,

and require the analysis of discrete integrable systems in order to

be answered. In this session we are bringing together researchers

working with various aspects of integrable systems with purpose

of intensifying the exchange of experience, methods and ideas.

SESSION 21

Stochastic dynamics in nonlinear systems

Katie Newhall

UNC Chapel Hill

[email protected]

Stochastic dynamics arise in the modeling of biological and phys-

ical systems but also from optimization algorithms. Examples in-

clude transport within a cell by molecular motors, dynamics of a

system of magnetization vectors, random arrangements of gran-

ular material and stochastic gradient methods for training neural

networks or Bayesian inferences with big data. This session will

address common challenges in stochastic systems across broadly

different applications, for example, high dimensionality, multiple

time-scales, and nonlinear multiplicative noise.

SESSION 22

Modern methods for dispersive wave equations

Robert J. Buckingham

Department of Mathematical Sciences, University of Cincinnati

P.O. Box 210025, Cincinnati, OH 45221-0025

[email protected]

Peter D. Miller

Departmentof Mathematics, University of Michigan

530 Church St., Ann Arbor, MI 48109

[email protected]

This special session will bring together researchers developing the

latest analytic, asymptotic, and numerical techniques for under-

standing dispersive wave equations.

SESSION 23

Nonlinear waves in optics, fluids and plasma

Sergey A. Dyachenko and Katelyn Leisman

Department of Mathematics,

273 Altgeld Hall, 1409 W. Green Street (MC-382),

Urbana, IL 61801 USA


Denis Silantyev


Courant Institute of Mathematical Sciences,

251 Mercer Street,

New York, NY 10012 USA

[email protected]

We present some of the recent advancements in numerical meth-

ods, and theoretical results in the field of nonlinear science. We

focus mainly on plasma, fluids, and nonlinear optics. We demon-

strate recent advances in turbulence theory of water waves, in par-

ticular the corrections to Kolmogorov spectra due to interaction

with condensate. We present recent advances on motion constants

that give hint of 2D water waves may be integrable system after all.

SESSION 24

Mathematical perspectives in quantum mechanicsand quantum chemistry

Jianfeng Lu

Mathematics Department, Duke University, Durham, NC, USA

[email protected]

Israel Michael Sigal

Department of Mathematics, University of Toronto, Toronto, Canada

[email protected]

6

This session is aimed at a review of the current progress and a dis-

cussion of outstanding issues in quantum mechanics and quantum

chemistry. The emphasis is on understanding behaviour of large

systems of quantum particles, such as atoms, molecules, solids,

etc.

To give an account of complex quantum systems one uses effective

theories (in which large number of degrees of freedom are swapped

for the nonlinearity) and the main problems here are the derivation,

mathematical analysis and application of such theories.

One of the most prominent examples of above is the density func-

tional theory (DFT). Despite the intensive use of this theory in

physics, chemistry, materials science and biology, with thousands

of papers published every month, and considerable progress

achieved, the time-dependent DFT is still in an initial stage of de-

velopment. This gap is even more daunting since the theory is

being recently applied for understanding energy transfer and de-

signing energy storage.

In this session we concentrate on mathematical underpinning of

the key effective theories, their justifications, applications and the

computational techniques used. The topics to be discussed will

include the rigorous analysis of the density functional theory, the

meso/macroscopic theories, described by the Ginzburg-Landau,

Landau-Lifshitz equations, two-dimensional quantum systems and

their geometrical and topological properties, measurement and de-

coherence and the interaction of radiation and matter.

SESSION 25

Nonlinear waves, singularities, vortices, and turbu-lence in hydrodynamics, physical, and biological sys-tems

Alexander O. Korotkevich and Pavel M. Lushnikov


MSC01 1115, 1 University of New Mexico,

Albuquerque, NM 87131-0001 USA


Waves dynamics is one of the most interesting phenomena in ap-

plied mathematics and physics. We encounter waves in all areas of

our everyday lives, from ripples on the surface of a cup of coffee

and sound waves to the extremely powerful laser pulses propaga-

tion in controlled fusion and plasma excitations in super novas.

In wast majority of interesting cases the problem of wave propa-

gation should be solved not only in the linear approximation but

also with nonlinear effects taken into account. Powerful tools of

modern applied mathematics and theoretical physics together with

rapidly emerging computational power leads to new amazing ad-

vances in the study of waves dynamics in different media. Com-

mon approaches stimulate intensive cross fertilization of ideas in

the field which accelerates the development of the wave dynamics

even further. Our session is devoted to new advances in the theory

of waves and demonstrates vividly the similarity of approaches in

a broad spectrum of important applications.

SESSION 26

Physical applied mathematics

Ziad Musslimani and Matthew Russo


Florida State University

Tallahassee, FL 32306, USA


Nonlinear waves pervade nature over a wide range of scales and

across many disciplines. In many cases their properties, includ-

ing their evolution, may be exactly or approximately determined

by one or more nonlinear evolution equations. This session will

include recent analytical and numerical work on nonlinear systems

such as the classical and PT-symmetric NLS, KdV equation, and

a generalized Heisenberg ferromagnet equation, with an emphasis

on their role in describing physical phenomena. Applications will

include hydrodynamics, nonlinear optics, aerodynamics, and other

areas.

SESSION 28

Recent advances in analytical and computationalmethods for nonlinear partial differential equations

Muhammad Usman

Department of Mathematics, University of Dayton, Dayton OH 45469-

2316, USA

[email protected]

Chaudry Masood Khalique


North-West University, Mafikeng Campus,

Private Bag X 2046,

Mmabatho 2735, South Africa

[email protected]

Nonlinear differential/partial differential equations (NDEs/NPDEs)

describe many physical phenomena arising in science and engi-

neering. Thus, finding their solutions play a vital role in provid-

ing information to understand and interpret the structure of such

physical phenomena. Researchers have developed many analytical

and numerical methods to solve these equations. Recent numeri-

cal methods include finite difference methods, collocation meth-

ods and finite element methods. While testing numerical tech-

niques, when exact solutions of initial and boundary value problem

of NPDEs are not available, conservation laws play an important

role. Well-known analytical tools include Lie symmetry method,

Backland transformation method, and inverse scattering transfor-

mation method.

This special session is dedicated to showcase recent progress in

finding analytical and numerical solutions to nonlinear differen-

tial/partial differential equations by various methods and to stimu-

late collaborative research activities.

7

PAPER ABSTRACTS

SESSION 1: “Nonlinear waves”

Three-dimensional steady water waves withvorticity

Evgeniy Lokharu, Douglas Svensson Seth∗, Erik Wahlen

Matematikcentrum, Box 118, 221 00 Lund, Sweden

[email protected], douglas.svensson [email protected],

[email protected]

We will consider the nonlinear problem of steady gravity-driven

waves on the free surface of a three-dimensional flow of an incom-

pressible fluid. In the talk we will discuss a recent progress on

three-dimensional waves with vorticity, which is a relatively new

subject. The rotational nature of the flow is modeled by the as-

sumption on the velocity field, that it is proportional to its curl.

Such vector fields are known in magnetohydrodynamics as Bel-

trami fields. We plan to provide a necessary background on the

topic and prove the existence of three-dimensional doubly periodic

waves with vorticity.

Strong error order of time-discretization of thestochastic gBBM equation

Guillaume Fenger∗, Olivier Goubet and Youcef Mammeri

LAMFA CNRS UMR 7352

Universite de Picardie Jules Verne,

33, rue Saint-Leu, 80039 Amiens, France.

[email protected], [email protected] and

[email protected]

We consider a Crank-Nicolson scheme to approximate analytical

solutions to the generalized Benjamin-Bona-Mahony equation

(gBBM) with white noise dispersion introduced in [2]. This equa-

tion reads, for p ≥ 1, x ∈ T the one-dimensional torus

du − duxx + ux dW + upuxdt = 0,

where (Wt)t≥0 is a standard real valued Brownian motion and is

the so-called Stratonovich product.

We choose a functional space in which the problem is well posed

and we study the strong error order of the time-discrete approxi-

mation. Due to the presence of a brownian motion we prove that

the strong error order of this Crank-Nicolson scheme is 1 instead

of 2 for the determinist equation.

1. R. Belaouar, A. de Bouard and A. Debussche, Numerical analysis of

the nonlinear Schrodinger equation with white noise dispersion, Stoch.

Partial Differ. Equ. Anal. Comput., 3 no.1 (2015), 103-132.

2. M. Chen, O. Goubet and Y. Mammeri, Generalized regularized long

waves equations with white noise dispersion, Stoch. Partial Differ. Equ.

Anal. Comput., 5 no. 3 (2017), 319-342.

3. G. Fenger, O. Goubet, Y. Mammeri, Numerical Analysis of the Crank-

Nicolson scheme for the Generalized Benjamin-Bona-Mahony Equa-

tion with White Noise Dispersion, (upcoming in 2019).

Small Debye length limit for the Euler-Poisson sys-tem

Chang-Yeol Jung and Bongsuk Kwon∗

Department of Mathematical Sciences, UNIST

Ulsan, 44919 Korea


Masahiro Suzuki

Department of Computer Science and Engineering, Nagoya Institute of

Technology

Nagoya, 466-8555 Japan

[email protected]

We discuss existence, time-asymptotic behavior, and quasi-neutral

limit for the Euler-Poisson equations. Specifically, under the Bohm

criterion, we construct the global-in-time solution near the sta-

tionary solution of plasma sheath, and also investigate its time-

asymptotic behavior and small Debye length limit. If time permits,

some key features of the proof and related problems will be dis-

cussed. This is joint work with C.-Y. Jung (UNIST) and M. Suzuki

(Nagoya Tech.).

Wave equations with infinite memory

Filippo Dell’Oro, Vittorino Pata

Politecnico di Milano - Dipartimento di Matematica

Via Bonardi 9, 20133 Milano, Italy


Olivier Goubet∗, Youcef Mammeri

Laboratoire Amienois de Mathematique Fondamentale et Appliquee

CNRS UMR 7352, Universite de Picardie Jules Verne, 80039 Amiens,

France


We introduce a new mathematical framework for the time discretiza-

tion of evolution equations with memory. As a model, we focus on

an abstract version of the equation

∂tu(t)−∫ ∞

0g(s)∆u(t − s) ds = 0

with Dirichlet boundary conditions, modeling hereditary heat con-

duction with Gurtin-Pipkin thermal law. Well-posedness and ex-

ponential stability of the discrete scheme are shown, as well as the

convergence to the solutions of the continuous problem when the

time-step parameter vanishes.

We consider also the nonlinear integrodifferential Benjamin-Bona-

Mahony equation

ut − utxx + ux −∫ ∞

0g(s)uxx(t − s)ds + uux = f

where the dissipation is entirely contributed by the memory term.

Under a suitable smallness assumption on the external force f , we

prove the convergence of trajectories to some global attractor.

1. F. Dell’Oro, O. Goubet, Y. Mammeri, V. Pata, Global attractor for the

Benjamin-Bona-Mahony equations with memory, Accepted to Indiana

University J. of Math.

2. F. Dell’Oro, O. Goubet, Y. Mammeri, V. Pata, A semidiscrete scheme

for evolution equations with memory, to appear

8

Mathematical analysis of Bump to Bucket problem

Min Chen∗

Department of Mathematics, Purdue University,

West Lafayette, IN 47907, USA

[email protected]

Olivier Goubet

LAMFA UMR 7352 CNRS, Universitede Picardie Jules Verne,

80039 Amiens CEDEX 1, France

[email protected]

Shenghao Li

Department of Mathematics, Purdue University, West Lafayette, IN 47907,

USA

[email protected]

In numerical simulations of surface water waves, when there is a

deformation on the bottom, it is a common practice to transform

form the boundary deformation data to the free surface. In this talk,

we investigate this procedure, by comparing the waves generated

by the moving bottom (Bump) and by the initial surface variation

(Bucket), using linear and nonlinear Boussinesq-type models.

Lower regularity solutions of non-homogeneousboundary value problems of the sixth orderBoussinesq equation in a quarter plane

Shenghao Li, and Min Chen

Department of Mathematics, Purdue University, West Lafayette, IN 47907


Bingyu Zhang

Department of Mathematics, University of Cincinnati, Cincinnati, OH 45221

[email protected]

We study an initial-boundary-value problem of the sixth-order

Boussinesq equation on a half line with nonhomogeneous bound-

ary conditions:

utt − uxx + βuxxxx − uxxxxxx + (u2)xx = 0, x > 0, t > 0,

u(x, 0) = ϕ(x), ut(x, 0) = ψ′′(x),

u(0, t) = h1(t), uxx(0, t) = h2(t), uxxxx(0, t) = h3(t),

where β = ±1. It is shown that the problem is locally well-posed

in Hs(R+) for − 12 < s ≤ 0 with initial condition (ϕ, ψ) ∈

Hs(R+)× Hs−1(R+) and boundary condition (h1, h2, h3) in the

product space Hs+1

3 (R+)× Hs−1

3 (R+)× Hs−3

3 (R+).

1. J. L. Bona, S. M. Sun, and B.-Y. Zhang. A non-homogeneous

boundary-value problem for the Korteweg-de Vries equation in a

quarter plane. Transactions of the American Mathematical Society,

354(2):427–490, 2002.

2. E. Compaan and N. Tzirakis. Well-posedness and nonlinear smoothing

for the good Boussinesq equation on the half-line. Journal of Differen-

tial Equations, 262(12):5824–5859, 2017.

3. A. Esfahani and L. G. Farah. Local well-posedness for the sixth-order

Boussinesq equation. Journal of Mathematical Analysis and Applica-

tions, 385(1):230–242, 2012.

4. S. Li, M. Chen, and B. Zhang. A non-homogeneous boundary value

problem of the sixth order Boussinesq equation in a quarter plane. Dis-

crete and Continuous Dynamical Systems - Series A, 38(5):2505–2525,

2018.

5. S. Li, M. Chen, and B. Zhang. Wellposedness of the sixth order Boussi-

nesq equation with non-homogeneous boundary value on a bounded

domain. Accepted by Physica D: Nonlinear Phenomena

Well-posedness for a higher-order, nonlinear, disper-sive equation: new approach

Jerry Bona and Hongqiu Chen

University of Illinois at Chicago and University of Memphis


Colette Guillope∗

University Paris-Est Creteil

[email protected]

A class of higher-order models for unidirectional water wave of the

form

ηt + ηx − γ1βηxxt + γ2βηxxx + δ1β2ηxxxxt + δ2β2ηxxxxx

+3

4α(η2)x + αβ

(γ(η2)xx −

7

48η2

x

)x− 1

8α2(η3)x = 0 (1)

was derived by Bona, Carvajal, Panthee and Scialom [1]. With ap-

propriate choices of the parameters γ1, γ2, δ1, δ2 and γ, this equa-

tion serves as a model for the propagation of small-amplitude, long-

crested surface waves moving to the direction of increasing values

of the spatial variable x. Here α is a typical ratio of wave am-

plitude to depth, β is a representative value of the square of the

depth to wavelength and t is proportional to elapsed time. The

dependent variable η = η(x, t) is a real-valued function of x ∈(−∞, ∞), t ≥ 0 representing the deviation of the free surface

from its undisturbed position at the point corresponding to x at

time t. This model subsists on the assumptions that α and β are

comparably-sized small quantities while η and its first few partial

derivatives are of order one. Moreover, γ1 and γ2 are restricted by

γ1 + γ2 = 16 .

The new result is that when γ = 748 , δ2 > δ1 > 0 and the initial

data

η(x, 0) = η0(x, 0) (2)

lies in H1 and not too big, then the initial-value problem of (1)-(2)

is globally well posed and the H1-norm of the solution is uniformly

bounded for t ≥ 0.

1. J. L. Bona, X. Carvajar, M. Panthee and M. Scialom, Higher-order

Hamiltonian model for unidirectional water waves, Journal of Nonlin-

ear Science, 28 (2018), no. 2, 543-577.

Solitary-wave solutions for some BBM-type of equa-tions with inhomogeneous nonlinearity

Shu-Ming Sun


Virginia Tech

Blacksburg, VA 24061

email: [email protected]

9

The talk considers the existence of solitary-wave solutions of some

higher-order Benjamin-Bona-Mahony (BBM) equations, whose lin-

ear parts are pseudo-differential operators and nonlinear parts in-

volve inhomogeneous polynomials of solutions and their deriva-

tives, which have not been studied before. One of such equations

can be derived from water-wave problems as the second-order ap-

proximate equation from fully nonlinear governing equations. Un-

der some conditions on the symbols of pseudo-differential oper-

ators and the nonlinear terms, it is shown that the equation has

solitary-wave solutions. Numerical study of the solitary-wave so-

lutions for some special fifth-order BBM equations will also be

discussed. (This is a joint work with J. Bona, H. Chen, and J.-M.

Yuan).

SESSION 2: “Novel challenges in nonlinear waves and integrable

systems”

We consider a slowly decaying oscillatory potential such that the

corresponding 1D Schrodinger operator has a positive eigenvalue

embedded into the absolutely continuous spectrum. This potential

does not fall into a known class of initial data for which the Cauchy

problem for the Korteweg-de Vries (KdV) equation can be solved

by the inverse scattering transform. We nevertheless show that the

KdV equation with our potential does admit a closed form classical

solution in terms of Hankel operators. Comparing with rapidly de-

caying initial data our solution gains a new term responsible for the

positive eigenvalue. To some extend this term resembles a positon

(singular) solution but remains bounded. Our approach is based

upon certain limiting arguments and techniques of Hankel opera-

tors.

Soliton solutions of certain reductions of the matrixnonlinear Schrodinger equation with non-zeroboundary conditions

Alyssa K. Ortiz1,∗ and Barbara Prinari1,2

1 Department of Mathematics, University of Colorado Colorado Springs2 Department of Mathematics, University at Buffalo


We will present soliton solutions for two novel reductions of the

matrix nonlinear Schrodinger equation (MNLS), introduced in [1],

which are integrable and are the analog of the modified Manakov

system with mixed signs of the nonlinear coefficients, i.e. a nonlin-

earity in the norm which is of Minkowski type instead of Euclidean

type.

In this presentation we will develop the Inverse Scattering Trans-

form (IST) for these novel reductions of MNLS in the case of non-

zero boundary conditions, using similar methods as those shown

in [2]. We will also discuss the resulting one-soliton solutions of

such equations under various conditions on the norming constant

matrices.

1. B. Prinari and A. Ortiz, Inverse Scattering Transform and Solitons

for Square Matrix Nonlinear Schrodinger Equations, Studies in Appl.

Math., 141 (2018), 308-352.

2. B. Prinari, F. Demontis, S. Li, and T. Horikis, Inverse scattering trans-

form and soliton solutions for square matrix nonlinear Schrodinger

equations with non-zero boundary conditions, Physica D, 368 (2017),

22-49.

Real Lax spectrum implies stability

Bernard Deconinck and Jeremy Upsal∗

Department of Applied Mathematics, University of Washington

Seattle, WA 98195-2420, USA


We consider the dynamical stability of elliptic solutions to inte-

grable equations that belong to the AKNS hierarchy. The spectrum

of the differential operator obtained through linearization is impor-

tant for determining the stability of solutions. When the spectrum

is on the imaginary axis, the solutions are spectrally stable. To de-

termine this stability spectrum, we use the integrability properties

of the underlying equation.

The spatial component of the Lax pair for members of the AKNS

hierarchy naturally defines an eigenvalue problem for the Lax pa-

rameter, ζ. The collection of these eigenvalues is called the Lax

spectrum, σL. When the eigenvalue problem is self adjoint, σL ⊂R. If it is not self adjoint, significantly more work is required to

resolve the Lax spectrum. We define a function whose solutions

determine the Lax spectrum as well as an explicit construction of

the eigenfunctions. Using the eigenfunctions, we present a method

for constructing the Floquet discriminant, a commonly used tool

for computing the Lax spectrum.

For stationary solutions of equations in the AKNS hierarchy, we

use knowledge of the Lax spectrum to determine spectral stability.

In particular, we find that (1) R ⊂ σL when the problem is not self

adjoint, and (2) for self-adjoint or non self-adjoint problems, real

Lax spectra gives rise to imaginary, and hence stable, eigenvalues.

Studying DNA transcription pulses with refinementsof a [discrete] Sine-Gordon approximation

Brenton LeMesurier* and Alex Kasman

College of Charleston, Charleston SC 29424

[email protected]/[email protected]

Transcription from DNA to RNA involves a traveling opening the

double helix, and with a great many approximations and assump-

tions, Englander et al [1] proposed modeling this by the equations

of the pendulum chain model, and thence via continuum limit by

the Sine-Gordon equation. Indeed, the kink solutions of the latter

are a fair qualitative approximation of the phenomenon, and these

kinks are “topological”, so they might be expected to be robust

under deviations from the exactly integrable and continuum form.

In this work in progress, starting with the work of Kasman [2],

we consider the effects of more accurate modeling, developing on

ideas of Salerno, Yakushevich, et al [3, 4]. In particular we account

for (a) the non-uniformity in the masses and sizes of the four nu-

cleobases and in the base pair bond strengths (A-T pairs have two

hydrogen bonds; C-T pairs have three), and (b) asymmetry in the

motions of the two nucleobases within each pair.

Questions include whether sustained propagation persists and

whether some of the many possible encodings of a given protein

10

are evolutionarily preferred due to easier propagation of this open-

ing “kink”; some preliminary observations will be made. A more

basic question is whether the symmetry assumed in the basic model

so as to get a single DOF per base pair is stable. In fact it is not,

but discrete Sine-Gordon style approximations are seen to return in

another more stable form!

1. S.W. Englander, N.R. Kallenbach, A.J. Heeger, J.A. Krumhansl, and

S. Litwin. Nature of the open state in long polynucleotide double he-

lices: possibility of soliton excitations. Proceedings of the National

Academy of Sciences, 77(12):7222–7226, 1980.

2. A. Kasman. DNA solitons and codon bias. In Mathematics of DNA

Structure, Function and Interactions. IMA conference, 2007.

3. M. Salerno. Discrete model for DNA-promoter dynamics. Phys. Rev.

A, 44(8):5292–5297, 1991.

4. L. V. Yakushevich. Nonlinear Physics of DNA. Wiley, 2004.

Integrable evolutions of twisted polygons in centro-affine space

Annalisa Calini* and Gloria Mari-Beffa

College of Charleston/University of Wisconsin-Madison

[email protected]/[email protected]

Many classical objects in differential geometry are described by

integrable systems: nonlinear partial differential equations (PDE)

with infinitely many conserved quantities that are (in some sense)

solvable. Beginning in the 1980s, studies of curve evolutions that

are invariant under the action of a geometric group of transfor-

mations have unveiled more connections between geometric curve

flows and well-known integrable PDE (among them are the KdV,

mKdV, sine-Gordon, and NLS equations). More recently, efforts

have been directed towards understanding geometric discretiza-

tions of surfaces and curves and associated evolutions.

This talk focuses on a natural geometric flow for polygons in centro-

affine geometry derived from discretizations of the Adler-Gel’fand-

Dikii flows for curves in projective space. Such discretizations, to-

gether with a pair of Hamiltonian structures, were introduced in

Mar227-Beffa and Wang [2]. We prove the compatibility of the

two Hamiltonian structures in arbitrary dimension by lifting them

to a pair of pre-symplectic forms on the moduli space of centro-

affine arc length parametrized polygons. We also describe their

kernels and discuss implications on the integrability of the polygo-

nal flows.

1. G. Mari Beffa and A. Calini, Integrable evolutions of twisted polygons

in centro-affine Rm, Preprint.

2. G. Mari Beffa and J.P. Wang, Hamiltonian structures and integrable

evolutions of twisted gons in cn, Nonlinearity 26 (2013) 2515-2551.

Exact solutions of the focusing nonlinear Schrodingerequation with symmetric nonvanishing boundaryconditions

Francesco Demontis and Cornelis van der Mee∗

Dip. Matematica e Informatica, Universita di Cagliari, Italy


Giovanni Ortenzi

Dip. Matematica e Applicazioni, Universita di Milano Bicocca, Italy

[email protected]

Barbara Prinari

Dept. of Mathematics, University of Buffalo, USA

[email protected]

After a quick review of the direct and inverse scattering theory of

the focusing Zakharov-Shabat system with symmetric nonvanish-

ing boundary conditions, we derive the exact expressions for its re-

flectionless solutions using Marchenko theory. Since the Marchenko

integral kernel has separated variables, the matrix triplet method

consisting of representing the Marchenko integral kernel in the

form

F(x + y, t) = Ce−(x+y)AetH B

is applied to express the multisoliton solutions of the focusing non-

linear Schrodinger equation with symmetric nonvanishing bound-

ary conditions in terms of the matrix (A, B, C). Since these exact

expressions contain matrix exponentials and matrix inverses, com-

puter algebra can be used to “unpack” and graph them. Here Ahas only eigenvalues with positive real part, H is a suitable func-

tion of A, and B and C are size compatible rectangular matrices.

The 2p × 2q matrices involved are p × q matrices with its entries

belonging to a division ring of 2 × 2 matrices that is isomorphic

with Hamilton’s quaternion algebra, thus supplying an application

of quaternion linear algebra [1]

1. L. Rodman, Topics in Quaternion Linear Algebra (Princeton Univer-

sity Press, 2014).

Extreme superposition: Rogue waves of infinite or-der and the Painleve-III hierarchy

Deniz Bilman and Peter D. Miller

Department of Mathematics, University of Michigan

530 Church St. Ann Arbor, MI, USA


Liming Ling

South China University of Technology

[email protected]

We study the fundamental rogue wave solutions of the focusing

nonlinear Schrodinger equation in the limit of large order. Using

a recently-proposed Riemann-Hilbert representation of the rogue

wave solution of arbitrary order k, we establish the existence of

a limiting profile of the rogue wave in the large-k limit when the

solution is viewed in appropriate rescaled variables capturing the

near-field region where the solution has the largest amplitude. The

limiting profile is a new particular solution of the focusing non-

linear Schrodinger equation in the rescaled variables — the rogue

wave of infinite order — which also satisfies ordinary differential

equations with respect to space and time. The spatial differential

equations are identified with certain members of the Painleve-III

hierarchy. We compute the far-field asymptotic behavior of the

near-field limit solution and compare the asymptotic formulæwith

the exact solution with the help of numerical methods for solving

Riemann-Hilbert problems. In a certain transitional region for the

asymptotics the near field limit function is described by a specific

globally-defined tritronquee solution of the Painleve-II equation.

11

These properties lead us to regard the rogue wave of infinite order

as a new special function.

1. D. Bilman, L. Ling, and P. D. Miller, Extreme Superposition: Rogue

Waves of Infinite Order and the Painleve-III Hierarchy, Preprint.,

Arxiv-id: 1806.00545, 2018

Spectral properties of matrix-valued AKNS systemswith steplike potentials

Martin Klaus

Department of Mathematics, Virginia Tech

Blacksburg, Virginia, USA

[email protected]

We consider AKNS systems of the form

v′ =(−iξ In Q

R iξ Im

)v, x ∈ R (1)

where Q and R are n × m and m × n complex valued matrix func-

tions and ξ is a complex-valued eigenvalue parameter (In, Im are

n × n, m × m identity matrices). We are particularly interested in

the case where Q and R tend to nonzero and possibly different lim-

its as x → ±∞. Our focus will be on the location and existence of

eigenvalues and spectral singularities. The motivation for studying

spectral singularities comes from the fact that they cause techni-

cal difficulties in the application of the inverse scattering transform

(IST) to the matrix nonlinear Schrodinger equation associated with

(1). We hope to provide some insights that will lead to a better un-

derstanding of the conditions under which the IST can be applied

to AKNS systems.

SESSION 3: “Recent developments in mathematical studies of wa-

ter waves”

Particle paths and transport properties of NLS andits generalizations

John D. Carter∗

Mathematics Department

Seattle University

[email protected]

Chris W. Curtis

Department of Mathematics & Statistics

San Diego State University

[email protected]

Henrik Kalisch


University of Bergen

[email protected]

The nonlinear Schrodinger equation (NLS) is well known as a uni-

versal equation in the study of wave motion. In the context of wave

motion at the free surface of an incompressible fluid, NLS accu-

rately predicts the evolution of modulated wave trains with low to

moderate wave steepness.

In this talk, we reconstruct the velocity potential and surface dis-

placement from NLS coordinates in order to compute particle tra-

jectories in physical coordinates. We use these particle trajectories

to compute the mean transport properties of modulated wave trains.

Additionally, we present particle trajectories and mean transport

properties for the Dysthe equation and two dissipative generaliza-

tions of NLS.

1. C.W. Curtis, J.D. Carter, and H. Kalisch, Deep water particle paths in

the presence of currents, Journal of Fluid Mechanics, 855 (2018), 322-

350.

2. J.D. Carter, C.W. Curtis, and H. Kalisch, Particle trajectories in nonlin-

ear Schrodinger models, arXiv:1809.08494 [physics.flu-dyn].

Asymptotics and numerics for modulational insta-bilities of traveling waves

Benjamin F. Akers

Air Force Institute of Technology

[email protected]

The spectral stability problem for periodic traveling waves for wa-

ter wave models is considered. The structure of the spectrum is dis-

cussed from the perspective of resonant interaction theory. Modu-

lational asymptotic expansions are used to predict the location of

instabilities in frequency-amplitude space. These predictions ex-

plain numerical results in [1]. Asymptotics results are presented in

the potential flow equations [2] as well as weakly nonlinear mod-

els [3]. The asymptotic predictions are compared to the results of

a direct numerical simulation of the modulational spectrum.

1. Nicholls, David P., Spectral data for travelling water waves: singulari-

ties and stability Journal of Fluid Mechanics, 624 (2009), 339-360.

2. Akers, Benjamin F., Modulational instabilities of periodic traveling

waves in deep water, Physica D: Nonlinear Phenomena, 300 (2015),

26-33.

3. Akers, Benjamin F. and Milewski, Paul A., A Model Equation for

Wavepacket Solitary Waves Arising from Capillary-Gravity Flows,

Studies in Applied Mathematics, 122 (2009), 249-274.

Fully dispersive model equations for hydroelasticwaves

Evgueni Dinvay and Henrik Kalisch∗

Dept. of Mathematics, University of Bergen, Norway

[email protected]

Emilian Parau

School of Mathematics, University of East Anglia, UK

[email protected]

In 1967, G. Whitham put forward a simple nonlinear nonlocal model

equation for the study of gravity waves at the free surface of an

inviscid fluid [9]. The advantage of this equation was that it de-

scribed the propagation of small amplitude waves nearly perfectly,

and in addition was able to feature some nonlinear effects.

In this lecture we review Whitham’s idea and present recent de-

velopments on formal asymptotics. We then present a model of

Whitham type for hydro-elastic waves [4, 7], which is similar to

12

the systems given in [1, 3, 5]. The model is tested in the case

of wave-sea-ice interactions and the response of an ice sheet to a

moving load [2, 6, 8].

1. P. Aceves-Sanchez, A.A. Minzoni and P. Panayotaros, Numerical study

of a nonlocal model for water-waves with variable depth. Wave Motion

50 (2013), 80-93.

2. J.W. Davys, R.J. Hosking and A.D. Sneyd, Waves due to a steadily

moving source on a floating ice plate, J. Fluid Mech. 158 (1985), 269-

287.

3. P. Guyenne and E.I. Parau, Finite-depth effects on solitary waves in a

floating ice sheet, J. Fluids and Structures 49, (2014), 242-262.

4. A.K. Liu and E. Mollo-Christensen, Wave propagation in a solid ice

pack, J. Phys. Oceanogr. 18 (1988), 1702-1712.

5. D. Moldabayev, H. Kalisch and D. Dutykh, The Whitham equation as

a model for surface water waves, Physica D, 309 (2015), 99–107.

6. E. Parau and F. Dias, Nonlinear effects in the response of a floating ice

plate to a moving load, J. Fluid Mech. 460 (2002), 281-305.

7. V.A. Squire, R.J. Hosking, A.D. Kerr and P.J. Langhorne, Moving

Loads on Ice Plates (Kluwer, Dordrecht, 1996).

8. T. Takizawa, Field studies on response of a floating sea ice sheet to

a steadily moving load, Contrib. Inst. Low Temp. Sci. A 36 (1987),

31-76.

9. G.B. Whitham, Variational methods and applications to water waves,

Proc. Roy. Soc. London A 299 (1967), 6-25.

The stability of stationary solutions of the focusingNLS equation

Bernard Deconinck∗ and Jeremy Upsal

Department of Applied Mathematics, University of Washington

Seattle, WA 98195-2420, USA


We examine the stability of the elliptic solutions of the focusing

nonlinear Schrdinger equation (NLS) with respect to subharmonic

perturbations. Using integrability properties of NLS, we discuss

the spectral stability of the solutions. We show that the spectrally

stable solutions are orbitally stable by constructing a Lyapunov

functional using higher-order conserved quantities of NLS.

This follows earlier work on the stability of elliptic solutions of

integrable equations, but in all these previous works, the Lax pair of

the integrable equation was self adjoint, significantly simplifying

the study.

Effect of viscosity and sharp wind increaseon ocean wave statistics

D. Eeltink*, A. Armaroli, Y.M. Ducimetiere, J. Kasparian and

M. Brunetti

GAP-Nonlinearity and Climate, University of Geneva,

Bd Carl-Vogt 66, CH1205 Geneva, Switzerland

[email protected]

The evolution of gravity waves is very sensitive to initial condi-

tions. That is, after a certain time, no information is left on the

initial conditions [1], even in the absence of irreversible processes

such as wave breaking. Therefore, a statistical approach is needed.

We study the statistical properties of narrow-banded waves propa-

gating in one direction, during and after a squall (a sudden episode

of wind). The model is initialized with a Gaussian shaped spec-

trum with random phases, and propagated using a forced-damped

higher-order Nonlinear Schrodinger (NLS) equation [2]. During

the squall the wave action increases, the spectrum broadens, the

spectral mean shifts up and the Benjamin-Feir index (BFI) and kur-

tosis increase. Conversely, after the squall, due to viscous dissipa-

tion, the opposite effect for each quantity occurs.

Kurtosis is considered the main parameter indicating if rogue waves

are likely to occur in a sea state. In turn, the BFI is often mentioned

as a means to predict the kurtosis. We confirm that there is indeed

a quadratic relation between these these two quantities. However,

this relation depends on the intensity of wind forcing and damping,

and is therefore not general. Instead, we find a simple and robust

exponential relation between the spectral mean and kurtosis, and

between the spectral width and kurtosis, which are independent of

any other quantity. Because of this simple relation, a single spec-

trum allows to assess the risk of rogue wave occurrence.

1. S. Y. Annenkov, V. I. Shrira, On the predictability of evolution of sur-

face401gravity and gravity-capillary waves, Physica D: Nonlinear Phe-

nomena 152-153 (2001) 665 - 675

2. D. Eeltink, A. Lemoine, H. Branger, O. Kimmoun, C. Kharif, J. D.

Carter, A. Chabchoub, M. Brunetti, J. Kasparian, Spectral up- and

downshifting of Akhmediev breathers under wind forcing, Physics of

Fluids 29 (10) (2017) 107103

Faraday waves with bathymetry

Diane Henderson & Austin Red Wing

William G. Pritchard Fluid Mechanics Laboratory


Penn State University University Park, PA 16803


Azar Eslam Panah∗

Department of Mechanical Engineering

Penn State Berks

Reading, PA 19610

[email protected]

We study, experimentally, parametrically excited surface waves in

water of finite depth with bathymetry. The bathymetry varies with

respect to the long dimension of the tank (1 ft) and is uniform with

respect to the width of the tank (1in). It is either symmetric or anti-

symmetric with respect to the centerline of the tank. Measurements

of neutral stability, wave amplitude evolution, and fluid particle

velocities are presented. A theoretical framework for the normal

modes of standing waves with such bathymetry is given by [1] .

1. J. Yu and L. N. Howard, Exact Floquet theory for waves over arbitrary

periodic topographies. J. Fluid Mech. 712, (2012), 451–470.

Water waves under ice

Olga Trichtchenko∗

The Department of Physics and Astronomy

The University of Western Ontario

[email protected]

Emilian I. Parau

School of Mathematics

University of East Anglia

13

[email protected]

Jean-Marc Vanden-Broeck


University College London

[email protected]

Paul Milewski

Department of Mathematical Sciences

University of Bath

[email protected]

In this talk, we present solutions for models of three-dimensional

nonlinear flexuralgravity waves, propagating at the interface be-

tween a fluid and an ice sheet. The fluid is assumed to be invis-

cid and incompressible, and the flow irrotational resulting in Euler

equations. We present the details of the numerical method based

on boundary integral equations used for computing both forced and

solitary wave solutions, show results in different regimes, and com-

pare different models for the ice sheet [1].

1. O. Trichtchenko, E. I. Parau, J.-M. Vanden-Broeck, and P. Milewski,

Solitary flexural–gravity waves in three dimensions, Philosophical

Transactions of the Royal Society A, 376 (2018), 20170345.

Tsunami

Harvey Segur∗

Department of Applied Mathematics

University of Colorado at Boulder

[email protected]

Diego Arcas

NOAA Center for Tsunami Research

Seattle, WA

[email protected]

Tsunami have received great deal of public interest in the last 20

years, because of two very destructive tsunami – one in the Indian

Ocean in December 2004 and the other off the eastern coast of

Japan in March 2011. Tsunami are often generated by undersea

earthquakes that occur at the common boundary of adjacent tec-

tonic plates. The moment magnitude is a measure of the energy

released by an earthquake, and that same measure is also used to

characterize the resulting tsunami, when the earthquake generates a

tsunami. The objective of this talk is to show two significant short-

comings of this procedure, both of which were demonstrated by the

tsunami of 2004. We are now trying to construct other measures of

tsunami, which we hope will provide more useful information.

Nonlinear waves over patches of vorticity

Christopher W. Curtis ∗

San Diego State University

[email protected]

Henrik Kalisch


[email protected]

In this talk, we present a method for numerically simulating freely

evolving surface waves over patches of vorticity. This is done via

point-vortex approximations and the use of fast-multipole methods

for updating point-vortex velocities. We then present results which

show the impact of varying types of vortex patches on nonlinear-

shallow-water wave propagation. A key result we find is that the

more nonlinear a surface wave, the more robust it is with respects

to the influence of submerged eddies. In contrast, nearly linear

waves can be strongly deformed, possibly to the point of breaking

by underwater vorticity patches.

SESSION 4: “Fractional differential equations”

Time dependent order differential equations

Andrei Ludu

Embry-Riddle Aeronautical University, Dept. Mathematics & Wave Lab

600 S. Clyde Morris Blvd. Daytona Beach, FL 32114 USA

[email protected]

Differential equations with space/time-dependent order of differ-

entiation recently emerge as valid predictive models for physical or

social systems with fast changing dynamics like population growth,

anomalous phase transitions, laws of evolution of technology, emer-

gency of novelty and the adjacent possible [1, 2]. Such new differ-

ential tool can provide answers to deeper mathematical-physical

questions like Hamiltonian flows on pseudo-manifolds, dimension

spectrum, fractional dimensions in homological algebra, or the re-

lation between fractional cohomology and fractal boundaries. The

natural frame for variable order equations is provided by fractional

differential equations. We present as an application the reduction

of such a variable order equation (from 1st order ODE to 3rd order

ODE) to a Volterra integral equation of second kind with singular

integrable kernel, and we solve the initial condition and the ex-

istence and uniqueness of solutions for such equation, or similar

types.

1. A. Ludu, Differential Equations of Time Dependent Order, Technical

and Natural Sciences-AMiTaNS15, AIP Publishing, 1684 (2015).

2. A. Ludu, and H. Khanal, Differential Equations of Dynamical Order,

Electronic J. Diff. Eqs., 24 (2017) 47-61.

On fractional calculus and nonlinear wave phenom-ena

Dumitru Baleanu

Department of Mathematics, Cankaya University, Anakara, Turkey

Institute of Space Sciences, Magurele-Bucharest, Romania

[email protected]

Fractional calculus deals with the study of fractional order integral

and derivative operators over real or complex domains [1, 2]. It is

an emerging field with important real world applications in various

areas of science and engineering [3, 4, 5]. To accurately describe

the non-local, frequency and history dependent properties of power

law phenomena, some modeling tools have to be introduced such

as fractional calculus.

In this paper, we show a new fractional extension of regularized

long-wave equation. Besides, the existence and uniqueness of the

14

solution of the regularized long-wave equation within fractional

derivative having Mittag-Leffler type kernel is analyzed. The re-

lated numerical results are also given. .

1. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and

derivatives: Theory and applications, Gordon and Breach, Yverdon,

(1993).

2. A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and applica-

tion of fractional differential equations, North Holland Mathematics

Studies 204, (2006).

3. I. Podlubny, Fractional differential equations, Academic Press: San

Diego CA, (1999).

4. D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional cal-

culus models and numerical methods, Series on Complexity, Nonlin-

earity and Chaos, World Scientific, (2012).

5. A. Atagana and D. Baleanu, New fractional derivative with non-local

and non-singular kernel, Thermal Sci. 20 (2006), 763-769.

Variable order differential equations, solutions andapplications

Harihar Khanal

Embry-Riddle Aeronautical University, Department of Mathematics

600 S. Clyde Morris Blvd., Daytona Beach, FL 32114 USA

[email protected]

Recently we introduced a special type of ordinary differential equa-

tions whose order of differentiation is a continuous function of the

independent variable [1, 2]. We show that such dynamical order

of differentiation equations can be approached by using the for-

malism for Volterra integral equations of second kind with singu-

lar integrable kernel. We present the numeric approach and so-

lutions for particular cases when order of differentiation changes

smoothly from 1 to 2 and backwards, and we discuss the asymp-

totic approach of the solutions towards the limiting classical ODE.

We study the numerical solutions by collocation method based on

a Taylor expansion of the solution and identification of the series

coefficients [3]. In this way the critical open problem of the initial

conditions is solved in an efficient way. The model is applied to

models for social systems with fast changing dynamics like popu-

lation growth, emergency of novelty and world computer networks.

1. A. Ludu, Differential Equations of Time Dependent Order, Technical

and Natural Sciences-AMiTaNS15, AIP Publishing, 1684 (2015).

2. A. Ludu, and H. Khanal, Differential Equations of Dynamical Order,

Electronic J. Diff. Eqs., 24 (2017) 47-61.

3. A. Zacharias, H. Khanal, and A. Ludu, Variable Order Differential

Equations and Applications, in print (2018).

Generalized Euler’s integral transform

G. Shchedrin, N. C. Smith, A. Gladkina∗, and L. D. Carr

Physics Department at Colorado School of Mines

1400 Illinois St, Golden, Colorado 80401, USA

[email protected], [email protected], [email protected],

[email protected]

Fractional calculus is recognized as a ubiquitous tool to character-

ize the dynamics of complex nonlocal systems, described by spa-

tial heterogeneity, non-Gaussian statistics, non-Fickian transport,

and scale-free distributions. The poster child for fractional deriva-

tives is anomalous diffusion, where in the superdiffusive regime

particles are allowed to jump farther than in a Gaussian-distributed

random walk [1].

One of the most useful fractional derivatives in modeling physical

systems is the Caputo fractional derivative, which ensures the con-

vergence of the fractional derivative at the origin. In this paper we

formulate the Caputo fractional derivative in terms of the general-

ized Euler’s integral transform, which allows us to take fractional

derivatives of a wide class of functions expressible as generalized

hypergeometric functions with a power law argument [2]. These

functions can take on the form of many common functions, such as

trigonometric, hyperbolic, and a family of Gaussian and Lorentzian

functions.

The conventional Euler’s integral transform (developed in 1778)

integrates a power law with a linear argument hypergeometric func-

tion, which is commensurate with taking a Caputo fractional deriva-

tive of the hypergeometric function. Here we develop a method to

take the Caputo fractional derivative of a power law argument hy-

pergeometric function by expanding the hypergeometric function

into its constituent hypergeometric series and utlizing the proper-

ties of the Pochhammer symbol. This allows us to extend the scope

of the transform.

Furthermore, the generalized Euler’s integral transform can be used

to solve linear fractional differential equations by assuming a gen-

eralized hypergeometric test function with a power law argument.

For example, we can use the framework of the generalized Eu-

ler’s integral transform to solve the fractional Schrodinger equa-

tion, which effectively models quantum transport in multiscale po-

tentials.

1. Y. Sagi, M. Brook, I. Almog, and N. Davidson, Observation of anoma-

lous diffusion and fractional self-similarity in one dimension, Phys.

Rev. Lett. 108, 093002 (2012).

2. G. Shchedrin, N. C. Smith, A. Gladkina, and L. D. Carr, Exact results

for a fractional derivative of elementary functions, SciPost Phys. 4, 029

(2018).

One dimensional sub-hyperbolic equation via sequen-tial Caputo fractional derivative

Aghalaya Vatsala

The representation form for sub hyperbolic one dimensional equa-

tion for Caputo derivative can be easily obtained by the usual stan-

dard procedure of eigen function expansion method. See reference

[1] below. However, this result does not yield the integer result as a

special case. In order to obtain the integer result as a special case,

in this work we assume that the Caputo derivative involved is se-

quential of order q, where 0.5 ≤ q ≤ 1. Here, we will assume that

the Caputo fractional partial derivative of order 2q, with respect to

t is sequential. This means that the Caputo derivative of order 2qcan be taken as the Caputo derivative of a function of order q fol-

lowed by the Caputo derivative of order q. The reason this helps

us to obtain the integer result as a special case is that the integer

derivative is sequential. In addition, the initial conditions need to

be modified also accordingly. In general, the initial and bound-

ary conditions involving Caputo derivative has the same initial and

boundary conditions as that of the integer derivative. In order to

15

obtain, the integer derivative results as a special case, our initial

conditions should be given at u(x, 0) and at the Caputo fractional

q derivative of u(x, t) at t = 0 should be known as functions of x.We will obtain a representation form for the sub hyperbolic equa-

tion in one dimensional space, using the sequential Caputo partial

derivative with respect to t. The representation form is obtained by

eigen function expansion method and followed by Laplace trans-

form method for Sequential Caputo derivatives. See reference [2]for

Laplace transform method for ordinary sequential fractional differ-

ential equations. If q = 1, our result yields the classical linear

hyperbolic equation as a special case. If q = .5, then we get the

linear parabolic result as a special case. The Green’s function will

involve the fractional trigonometric functions of Sin and cosine

functions. These trigonometric functions arise from some combi-

nation of Mittag-Leffler functions instead of the usual exponential

function. The main reason we want to obtain the integer result as a

special case is to establish that the fractional differential equations

represent as a better mathematical model and yield better results

compared with the integer derivative models.

1. Donna. S. Stutson, Aghalaya S. Vatsala, Sub Hyperbolic linear Partial

Fractional Differential Equation in One Dimensional Space with Nu-

merical Results, NONLINEAR STUDIES., V 20, No 4, (2013),483-

492.

2. Aghalaya S. Vatsala, Bhuvaneswari Sambandham, Laplace Trans-

form Method for Sequential Caputo Fractional Dofferential Equations,

Mathematics, in Engineering, Science and Aerospace V 7, No 2,

(2016), 339-347.

The gravity of light travel: Riding the fractional waveof a visible universe from h to c-squared

Christina Nevshehir

820 N. Center Ave

Gaylord, Michigan 49735, USA

[email protected]

As light travels fastest of all, all else must be relative to it. How

all else relates to light is explored first by establishing a mathe-

matical foundation on terra firma. Infinite regress objections of set

theory with Russell’s and Zeno’s paradoxes can be set aside. Re-

lying on the concept of Ulams spiral and Perelman’s solution of

the Poincare conjecture, this approach paves the way computation-

ally for fractional differential equations to solve the Riemann hy-

pothesis and other unsolved Millennium problems. This approach

proposes a Theory of Everything which unifies Einstein’s relativity

quanta with the overarching cosmos. This leads deterministically

rather than with uncertainty to the most optimal solutions for any

and all future systems and their applications like AI.

The factorization method for fractional quantum os-cillators

Haret C. Rosu∗ and Stefan C. Mancas

IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica,

Camino a la presa San Jose 2055, Col. Lomas 4a Seccion, 78216 San Luis

Potos038, S.L.P., Mexico

Department of Mathematics, Embry-Riddle Aeronautical University, Day-

tona Beach, FL 32114-3900, USA


We extend the factorization method to the case of fractional-differ-

ential Hamiltonians [1, 2]. Taking the quantum harmonic oscillator

as a primary example for this fractional-factorization framework,

we present two such factorizations, one with a single Levy index

[3] and the other with two Levy indices. Proceeding like in super-

symmetric quantum mechanics, we also revert the fractional factor-

ization brackets in order to introduce the fractional supersymmetric

partner problem. Nonlinear oscillators of the type xm, m ∈ N, are

also discussed in the same context.

1. K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press,

New York, 1974.

2. N. Laskin, Fractional quantum mechanics, Phys. Rev. E , 62 (2000),

3135; Fractional Schrodinger equation, Phys. Rev. E, 66 (2002),

056108.

3. F. Olivar-Romero, O. Rosas-Ortiz, Factorization of the quantum frac-

tional oscillator J. Phys: Conf. Series, 698 (2016), 012025.

Closed-form projection method for regularizing afunction defined by a discrete set of noisy data andfor estimating its derivative and fractional derivative

Timothy J. Burns and Bert W. Rust

Applied and Computational Mathematics Division

National Institute of Standards and Technology

100 Bureau Drive, Stop 8910

Gaithersburg, MD 20899-8910

[email protected]

We present a finite-dimensional projection method for regulariz-

ing a smooth function that has been defined by a discrete set of

measurement data, which have been contaminated by random, zero

mean errors. Our approach extends a statistical time series tech-

nique for separating signal from noise in the data, that was orig-

inally developed by Rust [1] for the study of Fredholm integral

equations of the first kind, with a smooth kernel. We then show

how to obtain closed-form estimates of the derivative and fractional

derivative of the data function, by finding approximate solutions of

the Volterra integral equations of the first kind which correspond

to integration and fractional integration, respectively. These esti-

mates are finite linear combinations of trigonometric or Legendre

polynomials of low degree.

1. B.W. Rust, Truncating the singular value decomposition for ill-posed

problems, NISTIR 6131, National Institute of Standards and Technol-

ogy, Gaithersburg, MD, July, 1998.

IST numerical schemes for solving nonlinear evo-lution equations and their possible applications forsolving time fractional differential equations

Thiab Taha

Computer Science Department

Univeristy of Georgia, Athens, GA

[email protected]

In this talk a survey and a method of derivation of certain class

of numerical schemes and an implementation of these schemes

16

will be presented. These schemes are constructed by methods re-

lated to the Inverse Scattering Transform (IST) and can be used as

numerical schemes for their associated nonlinear evolution equa-

tions. They maintain many of the important properties of their

original partial di?erential equations such as in?nite numbers of

conservation laws and solvability by IST. Numerical experiments

have shown that these schemes compare very favorably with other

known numerical methods.In addition,I will talk about their possi-

ble applications for solving time Fractional Differential Equations.

SESSION 5: “Evolution equations and integrable systems”

Integrable systems in 4+2 dimensionsand their reduction to 3+1 dimensions

M.C. van der Weele∗ and A.S. Fokas

Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge CB3 0WA, United Kingdom


Telephone: +447452866946

One of the main current topics in the field of integrable systems

concerns the existence of nonlinear integrable evolution equations

in more than two spatial dimensions. The fact that such equations

exist has been proven by one of the authors [1], who derived equa-

tions of this type in four spatial dimensions, which however had

the disadvantage of containing two time dimensions. The associ-

ated initial value problem for such equations, where the dependent

variables are specified for all space variables at t1 = t2 = 0, can

be solved by means of a nonlocal d-bar problem.

The next step in this program is to formulate and solve nonlinear

integrable systems in 3+1 dimensions (i.e., with three space vari-

ables and a single time variable) in agreement with physical reality.

The method we employ is to first construct a system in 4+2 dimen-

sions, which we then aim to reduce to 3+1 dimensions.

In this paper we focus on the Davey-Stewartson system [2] and

the 3-wave interaction equations. Both these integrable systems

have their origins in fluid dynamics where they describe the evolu-

tion and interaction, respectively, of wave packets on e.g. a water

surface. We start from these equations in their usual form in 2+1

dimensions (two space variables x, y and one time variable t) and

we bring them to 4+2 dimensions by complexifying each of these

variables. We solve the initial value problem of these equations in

4+2 dimensions. Subsequently, in the linear limit we reduce this

analysis to 3+1 dimensions to comply with the natural world. Fi-

nally, we discuss the construction of the 3+1 reduction of the full

nonlinear problem, which is currently under investigation.

1. A.S. Fokas, Integrable Nonlinear Evolution Partial Differential Equa-

tions in 4 + 2 and 3 + 1 Dimensions, Phys. Rev. Lett. 96 (2006),

190201.

2. A.S. Fokas and M.C. van der Weele, Complexification and integrability

in multidimensions, J. Math. Phys. 59 (2018), 091413.

Enhanced existence time of solutions to the fractionalKorteweg–de Vries equation

Mats Ehrnstrom and Yuexun Wang

Department of Mathematical Sciences, Norwegian University of

Science and Technology, 7491 Trondheim, Norway


We consider the fractional Korteweg–de Vries equation ut +uux −|D|αux = 0 in the range of −1 < α < 1, α 6= 0. Using basic

Fourier techniques in combination with the normal form transfor-

mation and modified energy method we extend the existence time

of classical solutions in Sobolev space with initial data of size εfrom 1/ε to a time scale of 1/ε2.

1. M. Ehrnstrom, and Y. Wang, Enhanced existence time of solutions

to the fractional Korteweg-de Vries equation, ARXIV:1804.06297,

2018.

Low regularity stablity for the KdV equation

Brian Pigott

Wofford College

[email protected]

Sarah Raynor∗

Wake Forest University

[email protected]

In this talk, we consider the stability of solitons for the KdV equa-

tion below the energy space, using spatially-exponentially-weighted

norms. We discuss known results including our own recent work

demonstrating arbitrarily long time stability in this setting, as well

as new progress towards full asymptotic stability.

Revivals and fractalisation in the linear free spaceSchrodinger equation

Peter J. Olver and Natalie E. Sheils∗

School of Mathematics,

University of Minnesota


David A. Smith

Division of Science,

Yale-NUS College,

[email protected]

We consider the one-dimensional linear free space Schrodinger

equation on a bounded interval subject to homogeneous linear

boundary conditions. We prove that, in the case of pseudoperi-

odic boundary conditions, the solution of the initial-boundary value

problem exhibits the phenomenon of revival at specific (“rational”)

times, meaning that it is a linear combination of a certain number

of copies of the initial datum. Equivalently, the fundamental solu-

tion at these times is a finite linear combination of delta functions.

At other (“irrational”) times, for suitably rough initial data, e.g.,

a step or more general piecewise constant function, the solution

exhibits a continuous but fractal-like profile. Further, we express

the solution for general homogenous linear boundary conditions in

terms of numerically computable eigenfunctions. Alternative so-

lution formulas are derived using the Uniform Transform Method

17

(UTM), that can prove useful in more general situations. We then

investigate the effects of general linear boundary conditions, in-

cluding Robin, and find novel “dissipative” revivals in the case of

energy decreasing conditions.

Well-posedness of initial-boundary value problemsfor dispersive equations via the Fokas method

A. Alexandrou Himonas


University of Notre Dame

[email protected]

Dionyssios Mantzavinos


University of Kansas

[email protected]

Fangchi Yan∗



[email protected]

We shall discuss the initial-boundary value problem for dispersive

equations. First, by applying the unified transform method (UTM),

which is also known as the Fokas method [F3], we shall solve the

initial-boundary value problem with forcing to obtain a formula

for the solution. Then, replacing the forcing with the nonlinearity

we will define the iteration map for the nonlinear equation. Fi-

nally, following the methodology developed for the cubic NLS in

[FHM2] or the KdV in [FHM1] (see also [HMY]), we shall prove

well-posedness in Sobolev spaces.

F3. A.S. Fokas, A unified approach to boundary value problems, SIAM,

2008.

FHM1. A.S. Fokas, A. Himonas and D. Mantzavinos, The Korteweg-de

Vries equation on the half-line. Nonlinearity 29 (2016), 489-527.

FHM2. A.S. Fokas, A. Himonas and D. Mantzavinos, The nonlinear

Schrodinger equation on the half-line. Trans. Amer. Math. Soc. 369

(2017), 681-709.

HMY. A. A. Himonas, D. Mantzavinos, F. Yan The nonlinear

Schrodinger equation on the half-line with Neumann boundary cond-

tions. Appl. Numer. Math (2018).

KPV1. C.E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial

value problem for the Korteweg-de Vries equation. J. AMS 4 (1991),

323-347.

Analysis of nonlinear evolution equations in domainswith a boundary

Athanassios S. Fokas

Department of Applied Mathematics and Theoretical Physics

University of Cambridge

[email protected]

A. Alexandrou Himonas



[email protected]

Dionyssios Mantzavinos∗



[email protected]

Fangchi Yan



[email protected]

The initial value problem for nonlinear evolution equations has

been studied extensively and from many points of view over the

last fifty years. On the other hand, the analysis of initial-boundary

value problems for these equations is rather limited, despite the

fact that such problems arise naturally in applications. This talk

will be devoted to a new approach for the well-posedness of non-

linear initial-boundary value problems, which combines the linear

solution formulae produced via the unified transform method of

Fokas with suitably adapted harmonic analysis techniques. Con-

crete examples to be discussed include the nonlinear Schrodinger

and Korteweg-de Vries equations, as well as a reaction-diffusion

equation with power nonlinearity.

Inverse scattering transform for the defocusingAblowitz-Ladik equation with arbitrary nonzerobackground

Barbara Prinari1,2,∗ and Alyssa K. Ortiz2

1 Department of Mathematics, University at Buffalo2 Department of Mathematics, University of Colorado Colorado Springs


In this talk we discuss the inverse scattering transform (IST) for the

defocusing Ablowitz-Ladik equation with arbitrarily large nonzero

boundary conditions at infinity. The IST was developed in the past

[1, 2] under the assumption that the amplitude of the background

intensity Qo satisfies a “small norm” condition 0 < Qo < 1. As

recently shown by Ohta and Yang [3], the defocusing AL system,

which is modulationally stable for 0 ≤ Qo < 1, becomes un-

stable if Qo > 1. And, in analogy with the focusing case, when

Qo > 1 the defocusing AL equation admits discrete rogue wave

solutions, some of which are regular for all times. Therefore, it

is clearly of importance to develop the IST for the defocusing AL

with Qo > 1, analyze the spectrum and characterize the soliton and

rational solutions from a spectral point of view. Both the direct and

the inverse problems are formulated in terms of a suitable uniform

variable; the inverse problem is posed as a Riemann-Hilbert prob-

lem in the complex plane, and solved by properly accounting for

the asymptotic dependence of eigenfunctions and scattering data

on the Ablowitz-Ladik potential.

1. V.E. Vekslerchik and V.V. Konotop, Discrete nonlinear Schrodinger

equation under non-vanishing boundary conditions, Inv. Probl., 8,

(1992) 889.

2. M.J. Ablowitz, G. Biondini and B. Prinari, Inverse scattering transform

for the integrable discrete nonlinear Schrodinger equation with non-

vanishing boundary conditions, Inv. Probl., 23, (2007) 1711.

3. Y. Ohta and J. Yang, General rogues waves in the focusing and defo-

cusing Ablowitz-Ladik equations, J. Phys. A, 47, (2014) 255201.

18

4. B. Prinari and F. Vitale, Inverse scattering transform for the focusing

Ablowitz-Ladik system with nonzero boundary conditions, Stud. App.

Math., 137, (2016) 28.

5. B. Prinari, Discrete solitons of the Ablowitz-Ladik equation with

nonzero boundary conditions via inverse scattering, J. Math. Phys., 57,

(2016) 083510.

Non-uniqueness and norm-inflation forCamassa-Holm-type equations

Curtis Holliman


The Catholic University of America

Washington, DC 20064

[email protected]

Alex Himonas


The University of Notre Dame


[email protected]

We consider a number of equations related to the Camassa-Holm

equation and will examine how well-posedness fails when the ini-

tial data are taken in Sobolev spaces with exponents less than 3/2.

Depending on the structure of the equation, the ill-posedness is ei-

ther norm-inflation or non-uniqueness and typically depends on the

Sobolev exponent.

1. A. Himonas and C. Holliman Non-Uniqueness for the Fokas-Olver-

Rosenau-Qiao equation. Journal of Mathematical Analysis and Appli-

cations 470 (1), 647-658.

2. A. Himonas, C. Holliman and C. Kenig Construction of 2-peakon solu-

tions and ill-posedness for the Novikov Equation. Siam J. Math. Anal.

Vol. 50, No. 3, pp. 2968–3006.

3. A. Himonas, C. Holliman and K. Grayshan, Norm inflation and ill-

posedness for the Degasperis-Procesi equation. Comm. Partial Differ-

ential Equations 39, 2198–2215, 2014.

4. A. Himonas, K. Grayshan and C. Holliman, Ill-posedness for the b-

family of equations. J. Nonlinear Sci. 26 (2016), 1175–1190.

The Cauchy problem for evolution equations withanalytic data

Alex A. Himonas

Department of Mathematics, University of Notre Dame


[email protected]

In this talk we will discuss analyticity properties in the spatial and

time variables for solutions to the Cauchy problem of evolution

equations with analytic initial data. In particular, lower bound esti-

mates for the uniform radius of spatial analyticity will be presented

for Camassa-Holm and Korteweg-de Vries type equations. The

talk is based on works with Professors G. Petronilho, R. Baros-

tichi, S. Selberg, H. Kalisch.

On the evolution of dark matter

Ryan C. Thompson


University of North Georgia

Dahlonega, GA

[email protected]

Dark matter is defined as nonluminous matter not yet directly de-

tected by astronomers that is hypothesized to exist to account for

various observed gravitational effects. In this talk, we will pro-

vide a brief history of the observations made by renowned scien-

tists since the late nineteenth century and the subsequent data col-

lected that led to the proposed concept of dark matter in the Uni-

verse. We now know that this invisible nondissipative dark matter

plays a decisive role in the formation of large scale structures in the

Universe such as galaxies, clusters of galaxies, and superclusters.

Since the corresponding nonlinear dynamics may be modeled by

hydrodynamic-like equations, this is where we shall focus the rest

of our attention and discuss results regarding these systems.

1. A. V. Gurevich, K.P. Zybin, Nondissipative gravitational turbulence,

Soviet Phys. JETP 67 No. 1 (1988), 1-12.

2. A. V. Gurevich, K.P. Zybin, Large-scale structure of the Universe: An-

alytic Theory, Soviet Phys. Usp. 38 No. 7 (1995), 687-722.

3. J. H. Jeans, Astronomy and Cosmology, Cambridge University Press,

London and New York, 1969.

4. Ya. B. Zeldovich, I.D. Novikov, Structure and Evolution of the Uni-

verse, Moscow, “Nauka” (1975), 736 pp.

Isometric immersions and self-similar buckling innon-Euclidean elastic sheets

John A. Gemmer and Maximilian Rezek

Wake Forest University, Department of Mathematics

127 Manchester Hall, Winston Salem, NC 27109


The edges of torn elastic sheets and growing leaves often display

hierarchical self-similar like buckling patterns. Within non-

Euclidean plate theory this complex morphology can be under-

stood as low bending energy isometric immersions of hyperbolic

Riemannian metrics. With this motivation we study the isometric

immersion problem in a strip with an asymptotically decaying met-

ric. By finding explicit piecewise smooth solutions of hyperbolic

Monge-Ampere equations on, we show there exist periodic isomet-

ric immersions of hyperbolic surfaces in the small slope regime.

We extend these solutions to exact isometric immersions through

resummation of a formal asymptotic expansion. Using this con-

struction, we identify the key role of branch-point (or monkey-

saddle) singularities, in complex wrinkling patterns within the class

of finite bending energy isometric immersions. Using these de-

fects we give an explicit construction of strain-free embeddings of

hyperbolic surfaces that are fractal like and have lower elastic en-

ergy than their smooth counterparts.For hyperbolic non-Euclidean

sheets, complex wrinkling patterns are thus possible within the

class of finite bending energy isometric immersions. Further, our

results identify the key role of the degree regularity of the isometric

immersion in determining the global structure of a non-Euclidean

elastic sheet in 3-space.

Solitary waves in dispersive evolution equations ofWhitham type with nonlinearities of mild regularity

Fredrik Hildrum


19

Norwegian University of Science and Technology,

7491 Trondheim, Norway

[email protected]

We show existence of small-amplitude solitary and periodic traveling-

wave solutions in fractional Sobolev spaces Hs to a class of non-

linear, dispersive integro-differential equations of the form

ut + (Lu + n(u))x = 0,

where L is a Fourier multiplier operator of any negative order whose

symbol is of KdV type at the origin and has integrable inverse

Fourier transform—so that L becomes convolution with integrable

kernel—and n is an inhomogeneous power-type nonlinearity of or-

der strictly greater than 1 at the origin. Notably, this class includes

Whitham’s model equation for surface gravity water-waves featur-

ing the exact linear dispersion relation, in which we obtain periodic

waves for s > 0 and solitary waves for s > 16 . Our tools involve

constrained variational methods, Lions’ concentration-compactness

principle, a fractional chain rule and a cut-off argument for n,

which enables us to go below the typical s > 12 regime. More-

over, we prove that most of the nonlocal estimates follow directly

from integrability of the kernel.

1. M. Ehrnstrm, M. D. Groves and E. Wahln, “On the existence and sta-

bility of solitary-wave solutions to a class of evolution equations of

Whitham type”, Nonlinearity, 25(10), 2903–2936, 2012.

2. P. L. Lions, “The concentration-compactness principle in the calculus

of variations. The locally compact case. I”, Ann. Inst. H. Poincare Anal.

Non Lineaire, 1(2), 109–145, 1984.

3. T. Runst and W. Sickel, “Sobolev Spaces of Fractional Order, Ne-

mytskij Operators, and Nonlinear Partial Differential Equations”,

De Gruyter Series in Nonlinear Analysis and Applications, Wal-

ter de Gruyter, 1996.

The Cauchy problem for the “good” Boussinesq equa-tion with analytic and Gevrey initial data

Rafael Barostichi∗


[email protected]

Alex Himonas


[email protected]

Renata Figueira

Federal University of Sao Carlos - Brazil

[email protected]

We shall consider the initial value problem for the “good” Boussi-

nesq equation with initial data belonging in a class of Gevrey func-

tions on both the line and the circle, which includes a class of ana-

lytic functions that can be extended holomorphically in a symmet-

ric strip of the complex plane around the real axis.

We shall talk about the history of this equation and present some

details of the proof of the local well-posedness in theses analytic-

Gevrey spaces. This is work in collaboration with Alex Himonas

and Renata Figueira.

1. J.L. Bona and R. Sachs, Global existence of smooth solutions and sta-

bility theory of solitary waves for a generalized Boussinesq equation,

Comm. Math. Phys., 118 (1988), 15-29.

2. L.G. Farah, Local solutions in Sobolev spaces with negative indices for

the “good” Boussinesq equation, CPDE, 34 (2009), 52-73.

3. F. Linares, Global existence of small solutions for a generalized

Boussinesq equation, J. Diff. Equations, 106 (1993), 257-293.

Gevrey regularity in time variable for solutions tothe “good” Boussinesq equation

Renata Figueira∗


Federal University of Sao Carlos

Sao Carlos, SP- Brazil

[email protected]

Alex Himonas




[email protected]

Rafael Barostichi




[email protected]

We shall consider the Cauchy problem for the “good” Boussinesq

equation and enunciate a result about its well-posedness in a class

of analytic Gevrey spaces, which guarantees the Gevrey regularity

of the solutions in space variable. The main discussion of this talk

concerns about regularity in time of these solutions. This work is

in collaboration with Rafael Barostichi and Alex Himonas.

1. L.G. Farah, Local solutions in Sobolev spaces with negative indices for

the “good” Boussinesq equation. CPDE 34 (2009), 52–73.

2. L.G. Farah and M. Scialom, On the periodic “good” Boussinesq equa-

tion. Proceedings of the American Math. Soc. 138(3) (2010), 953–964.

3. J. Gorsky, A. Himonas, C. Holliman and G. Petronilho, The Cauchy

problem of a periodic higher order KdV equation in analytic Gevrey

spaces. J. Math. Anal. Appl. 405 (2013), 349–361.

4. H. Hannah, A. Himonas and G. Petronilho, Gevrey regularity of the

periodic gKdV equation. J. Diff. Equations 250 (2011), 2581–2600.

Existence of solutions for conservation laws

John Holmes

231 West 18th Avenue

Columbus OH, 43210-1174

[email protected]

Systems of conservation laws in one spacial variable are locally

well-posed in the space of functions with bounded total variation

(BV). It is well known that classical solutions break down in finite

time; in particular, smoothness is lost and shocks form. However,

if the initial data is sufficiently small in BV, weak solutions ex-

ist (and when entropy conditions are imposed) are unique. There

20

have been several proofs of this result including the random choice

method and the vanishing viscosity method. These methods have

been extended to systems with forcing. Global in time results are

also found for systems with forcing under suitable constraints. We

will discuss some new results concerning the existence of solutions

to these systems, and the relationship between our results and these

previous methodologies.

Unified transform method with moving interfaces

Dave Smith∗

Yale-NUS College, Singapore

[email protected]

Tom Trogdon

University of California, Irvine CA

[email protected]

Vishal Vasan

International Centre for Theoretical Sciences, Bengaluru, India

[email protected]

The unified transform method was extended to interface problems

in the past 5 years, particularly by Sheils. Earlier work by Pelloni

and Fokas implemented the unified transform method on domains

with moving boundaries. We present a synthesis and extension

of these approaches, and an application to a new linearization of

the Korteweg-de Vries equation with step-like initial datum that

produces linear dispersive shocks.

Non-uniform continuous dependence for Euler equa-tions in Besov spaces

Jose Pastrana



[email protected]

The Cauchy problem governing the motion of an incompressible

and ideal fluid, in a domain Ω ⊆ Rd, is given by the system of

partial differential equations: ∂tu + (u · ∇)u +∇p = 0. Where

incompressibility translates to div u = 0 and u0(x) := u(x, 0) is

the initial configuration.

For local well-posedness theory see Bahouri, Chemin and Danchin

[1]. Ever since the papers of Kato and Ponce [3] there has been a lot

of interest in the regularity properties of the data to solution map,

u0 → u. We make use of the approximate solutions technique

which traces back to Kenig, Ponce and Vega [4] (when working

on KdV type equations) and a construction due to Himonas and

Misiolek [2]; to show that continuity of such map is the best you

can expect for the Besov spaces, Bsp,q. This is done for all relevant

scales in the periodic case and partially in Euclidean space; we

restrict to dimension d = 2. As a consequence we obtain the result

for the little Holder class, c1,σ(T2) ( C1,σ(T2), σ ∈ (0, 1) where

Misiolek and Yoneda [5] proved local well posedeness in the sense

of Hadamard.

1. Bahouri, H., Chemin, J., Danchin, R., Fourier Analysis and Nonlinear

Partial Differential Equations, Springer, New York 2011.

2. Himonas, A., Misiolek, G. Non-Uniform Dependence on Initial Data

of Solutions to the Euler Equations of Hydrodynamics. Commun. Math.

Phys. 296 (2010), 285-301.

3. Kato, T., Ponce, G., On non-stationary flows of Viscous and Ideal Flu-

ids. Duke Mathematical Journal, Vol. 55, No.3, 487-499 (1987)

4. Kenig, C., Ponce, G., Vega, L. On the (generalized) KdV Equation.

Duke Mathematical Journal, Vol. 59, No.3, 585-610 (1989)

5. Misiolek, G., Yoneda, T. Continuity of the solution map of the Euler

Equations in Holder spaces and weak norm inflation in Besov spaces.;

Trans. Amer. Math. Soc. 370 (2018), no. 7, 4709-4730.

Non-uniform dependence of the data-to-solution mapfor the Hunter–Saxton equation in Besov spaces

F. Tiglay, J. Holmes

Department of Mathematics, The Ohio State University, Columbus, OH

43210


The Cauchy problem for the Hunter-Saxton equation is known to

be locally well posed in Besov spaces Bs2,r on the circle. We prove

that the data-to-solution map is not uniformly continuous from any

bounded subset of Bs2,r to C([0, T]; Bs

2,r). We also show that the so-

lution map is Holder continuous with respect to a weaker topology.

Well-posedness and analyticity of solutions to a wa-ter wave problem with viscosity

Marieme Ngom and David P. Nicholls∗

Department of Mathematics, Statistics, and Computer Science

University of Illinois at Chicago

Chicago, IL 60607


The water wave problem models the free–surface evolution of an

ideal fluid under the influence of gravity and surface tension. The

governing equations are a central model in the study of open ocean

wave propagation, but they possess a surprisingly difficult and sub-

tle well–posedness theory. In this talk we discuss the existence

and uniqueness of spatially periodic solutions to the water wave

equations augmented with physically inspired viscosity suggested

in the recent work of Dias et al. (2008). As we show, this viscosity

(which can be arbitrarily weak) not only delivers an enormously

simplified well–posedness theory for the governing equations, but

also justifies a greatly stabilized numerical scheme for use in study-

ing solutions of the water wave problem.

Energy decay for the linear damped Klein-Gordonequation on unbounded domain

Satbir Malhi and Milena Stanislavova


[email protected]

In this talk, we consider energy decay for the damped Klein-Gordon

equation.

utt + γ(x)ut − uxx + u = 0. (x, t) ∈ R × R (3)

Where γ(x)ut represents a damping force proportional to the ve-

locity ut.

21

We give an explicit necessary and sufficient condition on the con-

tinuous damping functions λ ≥ 0 for which the energy E(t) =∫ ∞

−∞|ux|2 + |u|2 + |ut|2dx decays exponentially, whenever

(u(0), ut(0)) ∈ H2(R)× H1(R). The approach we use in this pa-

per is based on the asymptotic theory of C0 semigroups, in partic-

ular, the results by Gearhart-Pruss, and later Borichev and Tomilov

in which one can relate the decay rate of energy and the resolvent

growth of the semigroup generator. A key ingredient of our proof

is a projection method, in which we project the frequency domain

on appropriate regions and estimate the resolvent norms through

Fourier transformation. At the end of the talk, I will also show

some result on Fractional type Klein Gordon equation.

Riemann problems, solitons and dispersive shocks inmodulationally unstable media

Gino Biondini

State University of New York at Buffalo

[email protected]

The study of Riemann problems — i.e., the evolution of a jump

discontinuity between two uniform values of the initial datum —

is a well-established part of fluid dynamics, since understanding

the response of a system to such inputs is a step in characteriz-

ing its behavior. When nonlinearity and dissipation are the dom-

inant physical effects, these problems can give rise to classical

shocks. Conversely, when dissipation is negligible compared to

dispersion, Riemann problems can give rise to dispersive shock

waves (DSWs). This talk will discuss Riemann problems and DSW

formation in self-focusing media, using the cubic one-dimensional

nonlinear Schrodinger equation as a prototypical example. I will

show how a broad class of problems can bs effectively studied

using Whitham modulation theory. At the same time, however,

the full power of the inverse scattering method and the Deift-Zhou

nonlinear steepest descent method must be used in order to obtain

rigorous results.

1. “Universal nature of the nonlinear stage of modulational instability”,

G. Biondini and D. Mantzavinos, Phys. Rev. Lett. 116, 043902 (2016)

2. “Universal behavior of modulationally unstable media”, G. Biondini,

S. Li, D. Mantzavinos and S. Trillo, SIAM Review 60, 888–908 (2018)

3. G. Biondini, S. Li and D. Mantzavinos, “Soliton transmission, trapping

and wake in modulationally unstable media”, Phys. Rev. E 98, 042211

(2018)

4. G. Biondini, “Riemann problems and dispersive shocks in self-

focusing media”, Phys. Rev. E 98, 052220 (2018)

5. G. Biondini and J. Lottes, “Nonlinear interactions between solitons and

dispersive shocks in focusing media”, submitted (2019)

SESSION 6: “Random matrices, Painleve equations, and integrable

systems”

Entropy of beta random matrix ensembles

Alexander Bufetov

Laboratoire d’Analyse, Topologie, Probabilites, CNRS, Marseille

[email protected]

Sevak Mkrtchyan∗

Department of Mathematics, University of Rochester, Rochester, NY, USA

[email protected]

Maria Shcherbina

Institute for Low Temperature Physics Ukr. Ac. Sci., Kharkov, Ukraine

[email protected]

Alexander Soshnikov

Department of Mathematics, University of California at Davis, Davis,

USA

[email protected]

We will study the asymptotic properties of the density functions

of beta ensembles that arise in random matrix theory. We will

show that the ensembles have the asymptotic equipartition prop-

erty (AEP), and discuss the analogy with the Shannon-McMillan-

Breiman theorem and entropy. In addition to the AEP, the density

of the eigenvalues of these ensembles satisfy a Central Limit The-

orem. We will discuss the results in detail in the case of several

classical ensembles, and give a sketch for the case of beta ensem-

bles with generic real analytic potential.

Rational solutions of Painleve equations

Peter D. Miller∗

Dept. of Mathematics, University of Michigan

530 Church St., Ann Arbor, MI 48109

[email protected]

All of the six Painleve equations except the first have rational solu-

tions for certain parameter values. We survey some recent results

obtained in collaboration with T. Bothner, R. Buckingham, and Y.

Sheng on the asymptotic behavior of rational solutions of Painleve

II, III, and IV when the parameters are large. These results are ob-

tained by first computing the correct isomonodromy data for the

Jimbo-Miwa Lax pair associated with the family of rational solu-

tions with the help of classical special functions, their connection

formulæ, and Schlesinger transformations. Then it becomes possi-

ble to apply the Deift-Zhou steepest descent method to an appro-

priate Riemann-Hilbert problem characterizing the rational solu-

tions at hand. This allows the transitions between pole-free regions

and regions containing regular lattices of poles to be characterized

in terms of bifurcations of a suitable g-function, and provides ac-

curate asymptotic formulæ for the rational solutions valid in both

types of regions.

A representation of joint moments of CUE charac-teristic polynomials in terms of a Painleve-V solution

Robert Buckingham∗


University of Cincinnati

[email protected]

We establish a representation of the joint moments of the char-

acteristic polynomial of a CUE random matrix and its derivative

in terms of a solution of the σ-Painleve V equation. The deriva-

tion involves the analysis of a formula for the joint moments in

22

terms of a determinant of generalised Laguerre polynomials us-

ing the Riemann-Hilbert method. We use this connection with the

σ-Painleve V equation to derive explicit formulae for the joint mo-

ments and to show that in the large-matrix limit the joint moments

are related to a solution of the σ-Painleve III equation. This is joint

work with Estelle Basor, Pavel Bleher, Tamara Grava, Alexander

Its, Elizabeth Its, and Jonathan Keating.

Skew-orthogonal polynomials and continuum limitsof the Pfaff lattice

Virgil U. Pierce

University of Northern Colorado, School of Mathematical Sciences

[email protected]

The partition function of the Gaussian Orthogonal and Gaussian

Symplectic Ensembles (GOE and GSE) can be expressed in terms

of the skew-orthogonal polynomials with respect to a perturbed

Gaussian measure. As in the case of the Gaussian Unitary Ensem-

ble that has been studied extensively, this provides a connection

between the random matrix ensemble and a family of integrable lat-

tice hierarchies. In the case of GOE and GSE those hierarchies are

the so-called Pfaff lattices. In this presentation we will review re-

sults about the skew-orthogonal polynomials and their asymptotic

expansions. The goal is a description of the continuum limits of the

Pfaff lattice hierarchies as they pass from a differential-difference

system to a differential system by passing the discrete variable to a

continuous one. Ideally this computation is based upon a rigorous

foundation of the existence of such a limit, and will result in ex-

pressions for the generating functions enumerating Mobius maps

(ribbon graphs embedded on unoritented surfaces).

Classification of quad-equations on a cuboctahedron

Nalini Joshi and Nobutaka Nakazono∗

School of Mathematics and Statistics, The University of Sydney, New

South Wales, Australia.∗Department of Physics and Mathematics, Aoyama Gakuin University,

Kanagawa, Japan.

[email protected] and ∗[email protected]

In the theory of discrete integrable systems, the classification of

integrable partial difference equations (PDEs) by Adelr-Bobenko-

Suris (2003, 2009) and Boll (2011) are well known. They classi-

fied quad-equations1 on a cube using the CAC property. The CAC

property means a local property of Backlund transformations of

some integrable PDEs, including discrete Schwarzian KdV equa-

tion, lattice modified KdV equation, lattice potential KdV equation

and so on. Thus, repeated translation of a cube which has the CAC

property (CAC cube) leads to a space-filling cubic lattice (CAC cu-

bic lattice), on which integrable PDEs are iterated. Such PDEs are

collectively called ABS equations.

In our recent works, the mathematical connection between two

longstanding classifications of ABS equations and discrete Painleve

equations by Sakai (2001) have been investigated by using their

lattice structures. Our approach is as follows. First, we derive a

lattice, where quad-equations are observed, from the theory of dis-

crete Painleve equation. The derived lattice provides not only the

1An equation Q(x, y, z, w) = 0, where Q is an irreducible multi-affine polyno-

mial, is called a quad-equation.

type of quad-equation, but also the combinatorial structure of the

lattice before reduction. Then, we reconstruct the lattice from a

CAC cubic lattice via reduction.

For the lower types of discrete Painleve equations in the Sakai’s

classification, this approach works well. However, in a study of a

higher type of discrete Painleve equation, a different lattice struc-

ture from CAC cubic lattice appeared. The lattice can be obtained

from a reduction of the lattice. The new lattice locally has a cuboc-

tahedron structure (CACO property) instead of the CAC cubic struc-

ture, but such a structure has not been investigated until now.

In this talk, we give a more detailed description of the CACO prop-

erty and show a classification of quad-equations on a cuboctahe-

dron using the CACO property.

SLk character varieties and quantum cluster alge-bras

Leonid O. Chekhov∗ and Michael Z. Shapiro

Michigan State University, East Lansing, MI, 48824


We describe quantum algebras of monodromies of SLk Fuchsian

systems using the Fock-Goncharov construction [3] of higher Te-

ichmuller spaces. We prove that the monodromy matrices in the

disc with three marked points on the boundary, which corresponds

to configurations of three flags in Rn, satisfy the Lie-Poisson semi-

classical and quantum commutation relations, whereas a particular

combination of these matrices A = MT1 M2 enjoys the quantum

reflection equation. It is known that this equation naturally ap-

pears as a Poisson structure on the set of matrices of upper triangu-

lar groupoid studied by A. Bondal [1] that is compatible with the

braid-group action and with the dynamics governed by transforma-

tions of bilinear forms A 7→ BABT studied by Chekhov and Maz-

zocco [2]. In the mathematical physics literature particular Poisson

leaves of these algebras were identified by J. Nelson and T. Regge

[4] with algebras of geodesic functions on Riemann surfaces with

holes. Our approach enables us to find canonical (Darboux) coor-

dinate representation for general Poisson leaves of these algebras,

classify their central elements both in the upper-triangluar and in

the general case, and construct the cluster algebra representations

for the corresponding braid-group action.

1. A. Bondal, A symplectic groupoid of triangular bilinear forms and the

braid groups, preprint IHES/M/00/02 (Jan. 2000).

2. L.O. Chekhov, M. Mazzocco, Poisson algebras of block-upper-

triangular bilinear forms and braid group action, Commun. Math. Phys.

332 (2013) 49–71.

3. V. V. Fock and A. B. Goncharov, Moduli spaces of local systems and

higher Teichmuller theory, Publ. Math. Inst. Hautes Etudes Sci. 103

(2006), 1-211.

4. Nelson J.E., Regge T., Homotopy groups and (2+1)-dimensional

quantum gravity, Nucl. Phys. B 328 (1989), 190–199.

The space of initial conditions for some 4D Painlevesystems

Tomoyuki Takenawa

Faculty of Marine Technology, Tokyo University of Marine Science and

Technology,

23

2-1-6 Etchu-jima, Koto-ku, Tokyo, 135-8533, Japan

[email protected]

In recent years, research on 4D Painleve systems have progressed

mainly from the viewpoint of isomonodromy deformation of lin-

ear equations. In this talk we study the geometric aspects of 4D

Painleve systems by investigating the space of initial conditions in

Okamoto-Sakai’s sense, which was a powerful tool in the original

2D case. Specifically, starting from known discrete symmetries, we

construct the space of initial conditions for some 4D Painleve sys-

tems, and using the Neron-Severi bi-lattice, clarify the whole group

of their discrete symmetries. The examples include the directly

coupled 2D Painleve equations, Noumi-Yamada’s A(1)5 system and

the 4D Garnier system. The spaces of initial conditions for the first

two equations are obtained by 16 blow-ups from (P1)4, while for

the last equation, it is obtained by 21 blow-ups from (P2)2.

Asymptotic of solution of three-component Painleve-II equation.

Alexander Its and Andrei Prokhorov*

Indiana University-Purdue University Indianapolis

402 N Blackford St., Indianapolis, IN, 46202, USA

Saint Petersburg State University

Universitetskaya emb. 7/9, 199034, St. Petersburg, Russia


We consider the three-component Painleve equation. It was ob-

tained in [5] as degeneration of higher rank Inozemtsev rational

extension of Calogero system. It can be interpreted as the equation

of motion of 3 interacting particles in the external potential.

We are interested in its application in random matrix theory. More

precisely Tracy-Widom beta distribution with even β = 2r was de-

scribed in [4] using the particular solution of r-component Painleve-

II equation. Tracy-Widom beta law is the limiting distribution

of the largest eigenvalue of Hermite and Laguerre β-ensembles

of random matrices when the size of the matrix tends to infinity.

This distribution is well studied for β = 1, 2, 4 and is described

in these cases using Hastings-McLeod solution of one-component

Painleve-II equation. For arbitrary β > 0 the leading term in the

tail asymptotics was obtained rigorously in [3]. The full asymp-

totic expansion for left and right tail asymptotics was conjectured

in [2].

We study the solution of three-component Painleve-II equation men-

tioned above. We use the Riemann-Hilbert problem for multi-

component Painleve equations found recently in [1]. We perform

nonlinear steepest descent analysis to get asymptotic results.

1. M. Bertola, M. Cafasso, V. Roubtsov, Noncommutative Painleve equa-

tions and systems of Calogero type, Commun. Math. Phys. , 363:2,

(2018), 503–530.

2. G. Borot, C. Nadal, Right tail asymptotic expansion of Tracy-Widom

beta laws,Random Matrices: Theory and Applications, 01:03, 1250006

(2012).

3. J. Ramirez, B. Rider, and B. Virag, Beta ensembles, stochastic Airy

spectrum, and a diffusion, J. Amer. Math. Soc., 2011, (2011), 919–944.

4. I. Rumanov, Painleve Representation of Tracy-Widomβ distribution

for β = 6 , Commun. Math. Phys. , 342:3, (2016), 843–868.

5. K. Takasaki, Painleve-Calogero correspondence revisited, J. Math.

Phys., 42, (2001), 1443.

Algebro-geometric solutions to Schlesinger systems

Vladimir Dragovic

Department of Mathematical Sciences, University of Texas at Dallas, 800

West Campbell Road, Richardson TX 75080, USA.

Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia.

[email protected]

Renat Gontsov

M.S. Pinsker Laboratory no.1, Institute for Information Transmission Prob-

lems of the Russian Academy of Sciences, Bolshoy Karetny per. 19,

build.1, Moscow 127051 Russia.

[email protected]

Vasilisa Shramchenko∗

Department of mathematics, University of Sherbrooke, 2500, boul. de

l’Universite, J1K 2R1 Sherbrooke, Quebec, Canada.

[email protected]

We construct various algebro-geometric solutions to the Schlesinger

system. First, we discuss a rank two four point Schlesinger system

which we solve using a special meromorphic differential on an el-

liptic curve presented as a ramified double covering of the Riemann

sphere. This differential has a remarkable property: the common

projection of its two zeros on the base of the covering, regarded

as a function of the only moving branch point of the covering, is

a solution of a Painleve VI equation. This differential provides an

invariant formulation of one particular Okamoto transformation for

the Painleve VI equations.

Next, we study the Schlesinger system of partial differential equa-

tions in the case when the unknown matrices of arbitrary size (p ×p) are triangular and the eigenvalues of each matrix, called the

exponents of the system, form an arithmetic progression with a ra-

tional difference q, the same for all matrices. We show that such

a system possesses a family of solutions expressed via periods of

meromorphic differentials on the Riemann surfaces of superellip-

tic curves. We determine the values of the difference q, for which

our solutions lead to explicit polynomial or rational solutions of

the Schlesinger system. As an application of the (2 × 2)-case, we

obtain explicit sequences of rational solutions and one-parametric

families of rational solutions of Painleve VI equations.

Two discrete dynamical systems are discussed and analyzed whose

trajectories encode significant explicit information about a number

of problems in combinatorial probability. In this talk we will fo-

cus on applications to random walks in random environments. The

two models are integrable and our analysis uncovers the geometric

sources of this integrability and uses that to conceptually explain

the rigorous existence and structure of elegant closed form expres-

sions for the probability distributions for physically meaningful

random variables of these walks. The work here brings together

ideas from a variety of fields including dynamical systems theory,

probability theory, classical analogues of quantum spin systems,

addition laws on elliptic curves, and links between randomness and

symmetry.

24

Discrete Painleve equations in tiling problems

Anton Dzhamay∗

School of Mathematical Sciences, University of Northern Colorado, Gree-

ley, CO 80639, USA

[email protected]

Alisa Knizel

Department of Mathematics, Columbia University, New York, NY, USA

[email protected]

The notion of a gap probability is one of the main characteristics of

a probabilistic model. In [3] Borodin showed that for some discrete

probabilistic models of Random Matrix Type discrete gap prob-

abilities can be expressed through solutions of discrete Painleve

equations, which provides an effective way to compute them [1].

We discuss this correspondence for a particular class of models

of lozenge tilings of a hexagon. For uniform probability distribu-

tion, this is one of the most studied models of random surfaces.

Borodin, Gorin, and Rains [2] showed that it is possible to assign

a very general elliptic weight to the distribution and degenerations

of this weight correspond to the degeneration cascade of discrete

polynomial ensembles, such as Racah and Hahn ensembles and

their q-analogues. This also correspond to the degeneration scheme

of discrete Painleve equations, due to the work of Sakai. Con-

tinuing the approach of Knizel [4], we consider the q-Hahn and

q-Racah ensembles and corresponding discrete Painleve equations

of types q − P(A(1)2 ) and q − P(A

(1)1 ) [5]. We show how to use

the algebro-geometric techniques of Sakai’s theory to pass from the

isomonodromic coordinates of the model to the discrete Painleve

coordinates that is compatible with the degeneration.

1. Alexei Borodin and Dmitriy Boyarchenko, Distribution of the first par-

ticle in discrete orthogonal polynomial ensembles, Comm. Math. Phys.

234 (2003), no. 2, 287–338.

2. Alexei Borodin, Vadim Gorin, and Eric M. Rains, q-distributions on

boxed plane partitions, Selecta Math. (N.S.) 16 (2010), no. 4, 731–

789.

3. Alexei Borodin, Discrete gap probabilities and discrete Painleve equa-

tions, Duke Math. J. 117 (2003), no. 3, 489–542.

4. Alisa Knizel, Moduli spaces of q-connections and gap probabilities, In-

ternational Mathematics Research Notices (2016), no. 22, 1073–7928.

5. Kenji Kajiwara, Masatoshi Noumi, and Yasuhiko Yamada, Geometric

aspects of Painleve equations, J. Phys. A 50 (2017), no. 7, 073001,

164.

A survey of Bi-flat F-manifolds

Alessandro Arsie


The University of Toledo, 43606, Toledo, OH, USA

[email protected]

Paolo Lorenzoni

Dipartimento di Matematica e Applicazioni,

University of Milano-Bicocca, 20126 Milano, Italy

[email protected]

I will present a survey of the work done by Paolo Lorenzoni and

myself in the last few years developing the theory of bi-flat F-

manifolds and exploring their relationships with integrable hierar-

chies (dispersionless and dispersive), with Painleve transcendents,

and with complex reflection groups. If there is enough time, I

will address also very recent results about the existence of inte-

grable dispersive deformations in the non-Hamiltonian setting us-

ing tools from the so called cohomological field theory (these latter

results are being developed together also with Alexander Buryak

and Paolo Rossi).

SESSION 7: “Stability and traveling waves”

On the existence and instability of solitarywater waves with a finite dipole

Hung Le

Department of Mathematics, University of Missouri, Columbia, MO 65211

[email protected]

his paper considers the existence and stability properties of two-

dimensional solitary waves traversing an infinitely deep body of

water. We assume that above the water is vacuum, and that the

waves are acted upon by gravity with surface tension effects on the

air–water interface. In particular, we study the case where there is

a finite dipole in the bulk of the fluid, that is, the vorticity is a sum

of two weighted δ-functions. Using an implicit function theorem

argument, we construct a family of solitary waves solutions for this

system that is exhaustive in a neighborhood of 0. Our main result is

that this family is conditionally orbitally unstable. This is proved

using a modification of the Grillakis–Shatah–Strauss method re-

cently introduced by Varholm, Wahlen, and Walsh.

Double-periodic waves of the focusing NLS equationand rogue waves on the periodic background

Jinbing Chen

School of Mathematics, Southeast University, Nanjing, Jiangsu 210096,

P.R. China

[email protected]

Dmitry E. Pelinovsky∗

Department of Mathematics, McMaster University, Hamilton, Ontario,

Canada, L8S 4K1

[email protected]

We address Lax–Novikov equations derived from the cubic NLS

equation. Lax-Novikov equations of the lowest orders admit ex-

plicit periodic and double-periodic solutions expressed as rational

functions of Jacobian elliptic functions. By applying an algebraic

method which relates the periodic potentials and the squared peri-

odic eigenfunctions of the Lax operators, we characterize explicitly

the location of eigenvalues in the periodic spectral problem away

from the imaginary axis. We show that Darboux transformations

with the periodic eigenfunctions remain in the class of the same

periodic waves of the NLS equation. On the other hand, Darboux

transformations with the non-periodic solutions to the Lax equa-

tions produce rogue waves on the periodic background which are

25

formed in a finite region of the time-space plane. The results are

based on the recent papers [1, 2, 3].

1. J. Chen and D.E. Pelinovsky, “Rogue periodic waves in the modified

Korteweg-de Vries equation”, Nonlinearity 31 (2018), 1955–1980.

2. J. Chen and D.E. Pelinovsky, “Rogue periodic waves in the focusing

nonlinear Schrodinger equation”, Proceeding A of Roy. Soc. Lond. 474

(2018), 20170814 (18 pages).

3. J. Chen and D.E. Pelinovsky, “Periodic travelling waves of the modified

KdV equation and rogue waves on the periodic background”, (2018),

arXiv:1807.11361 (40 pages).

Formation of extreme events in NLS systems

Efstathios G. Charalampidis


Amherst

Amherst, MA 01003-4515, USA

[email protected]

his talk will focus on the formation and spatio-temporal evolution

of extreme events, called rogue waves in nonlinear Schrodinger

(NLS) equations and discrete variants thereof. Motivated by the

physics of ultracold atoms, i.e., atomic Bose-Einstein condensates

(BECs), we will attempt to address the question about what type

of experimental initial conditions should be utilized for producing

waveforms which are strongly reminiscent of the Peregrine soli-

ton. To do so, we will consider the initial boundary value problem

(IBVP) with Gaussian wavepacket initial data for the scalar (NLS)

and novel features will be presented. In particular, it will be shown

that as the width of the relevant Gaussian is varied, large ampli-

tude excitations strongly reminiscent of Peregrine, Kuznetsov-Ma

or regular solitons appear to form. This analysis will be comple-

mented by considering the Salerno model interpolating between

the discrete NLS (DNLS) and Ablowitz-Ladik (AL) models where

similar phenomenology is observed. Finally, and if time permits,

recent results on the stability of discrete Kuznetsov-Ma solitons

(via the use of Floquet theory) will be discussed as well. The

findings presented in this talk might be of particular importance

towards realizing experimentally extreme events in BECs.

Stability of planar fronts in a class of reaction-diffusion systems

Anna Ghazaryan

Department of Mathematics, Miami University, Oxford, OH 45056, USA

[email protected]

Yuri Latushkin

Mathematics Department, University of Missouri, Columbia, MO 65211,

USA

[email protected]

Xinyao Yang

Xi’an Jiaotong-Liverpool University, Suzhou, Jiangsu, P. R. China

[email protected]

For a class of reaction-diffusion equations we study the planar

fronts with the essential spectrum of the linearization in the direc-

tion of the front touching the imaginary axis. At the linear level,

the spectrum is stabilized by using an exponential weight. A-priori

estimates for the nonlinear terms of the equation governing the evo-

lution of the perturbations of the front are obtained when perturba-

tions belong to the intersection of the exponentially weighted space

with the original space without a weight. These estimates are then

used to show that in the original norm, initially small perturbations

to the front remain bounded, while in the exponentially weighted

norm, they algebraically decay in time.

Asymptotic stability for spectrally stableLugiato-Lefever solutions in periodic waveguides

Milena Stanislavova and Atanas Stefanov

Department of Mathematics, University of Kansas


We consider the Lugiato-Lefever model of optical fibers in the pe-

riodic context. Spectrally stable periodic steady states were con-

structed recently in [2] and [3], also by S. Hakkaev, M. Stanislavova

and A. Stefanov, [5]. The spectrum of the linearization around such

solitons consists of simple eigenvalues 0, −2α < 0, while the rest

of it is a subset of the vertical line µ : ℜµ = −α. Assuming

such property abstractly, we show that the linearized operator gen-

erates a C0 semigroup and more importantly, the semigroup obeys

(optimal) exponential decay estimates. Our approach is based on

the Gearhart-Pruss theorem, where the required resolvent estimates

may be of independent interest. These results are applied to the

proof of asymptotic stability with phase of the steady states.

1. Y.K. Chembo, C.R. Menyuk, Spatiotemporal Lugiato-Lefever formal-

ism for Kerr-comb generation in whispering-gallery-mode resonators,

Phys. Rev. A 87, (2010), 053852.

2. L. Delcey, M. Haragus, Periodic waves of the Lugiato-Lefever equation

at the onset of Turing instability, Phil. Trans. R. Soc. A 376, (2018),

20170188.

3. L. Delcey, M. Haragus, Instabilities of periodic waves for the Lugiato-

Lefever equation, to appear in Rev. Roumaine Maths. Pures Appl.

4. F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, M. Stanislavova,

A spectral mapping theorem and invariant manifolds for nonlinear

Schrodinger equations, Indiana Univ. Math. J. 49, (2000), no. 1, p.

221–243.

5. S. Hakkaev, M. Stanislavova, A. Stefanov, On the generation of stable

Kerr frequency combs in the Lugiato-Lefever model of periodic optical

waveguides, submitted, available at arXiv:1806.04821.

6. L. Lugiato, R. Lefever, Spatial dissipative structures in passive optical

systems. Phys. Rev. Lett. 58, (1987), p. 2209–2211.

7. R. Mandel, W. Reichel, A priori bounds and global bifurcation results

for frequency combs modeled by the Lugiato-Lefever equation. SIAM

J. Appl. Math. 77 (2017), no. 1, p. 315–345.

Fisher-KPP dynamics in diffusiveRosenzweig-MacArthur and Holling-Tanner models

Hong Cai

Department of Physics, Brown University,

182 Hope Street, Providence, RI 02912, USA,

Hong [email protected]

Anna Ghazaryan

Department of Mathematics, Miami University,

301 S. Patterson Ave, Oxford, OH 45056, USA

26

[email protected]

Vahagn Manukian

Department of Mathematical and Physical Sciences, Miami University,

1601 University Blvd, Hamilton, OH 45011, USA

[email protected]

We prove existence of traveling fronts in two known population dy-

namics models, Rosenzweig-MacArthur and Holling-Tanner, and

investigated the relation of these fronts with fronts in scalar Fisher-

KPP equation. More precisely, we prove existence of traveling

fronts in a modified diffusive Rosenzweig-MacArthur predator-

prey model in the two situations. One situation is when the prey

diffuses at the rate much smaller than that of the predator. In the

second situation both the predator and the prey diffuse very slowly.

Both situations can be captured as singular perturbations of the as-

sociated limiting systems. In the first situation we demonstrate a

clear relation of the fronts with the fronts in a scalar Fisher-KPP

equation. We show that a similar relation also holds for fronts in a

diffusive Holling-Tanner population model. The analysis suggests

that the scalar Fisher-KPP equation may serve as a normal form

for a variety of reaction-diffusion systems that rise in population

dynamics.

Bulk versus surface diffusion in highly amphiphilicpolymer networks

Yuan Chen and Keith Promislow∗


Michigan State University

[email protected]

Shibin Dai

Department of Mathematics, University of Alabama

[email protected]

Amphiphilic materials self assemble into complex networks, a fun-

damental example is the endoplasmic reticulum that serves as the

basis for intracellular transport and protein synthesis. A key prop-

erty of the ER network is it grows by transport of the network ma-

terial along the the network itself, by surface diffusion. This is

primarily due to the strongly hydrophobic nature of the lipids that

makes the energy of a single lipid in solvent prohibitively high. We

discuss several approaches to model this phenomena which include

wells with limited smoothness that induce compactly supported bi-

layers. Regularizations that include asymptotically strong convex-

ity support small densities of background lipids, but with enhanced

mobility that induces significant bulk flux. We show that balancing

strong convexity with degenerate mobility arrives at a model with

limited background density and weak bulk flux.

Recent results on application of the Maslov index inspectral theory of differential operators

Yuri Latushkin and Selim Sukhtaiev

Department of Mathematics, University of Missouri, Columbia, MO 65211,

USA

Department of Mathematics, Rice University, Houston, TX 77005, USA


We describe relations between the Maslov index and the count-

ing function for the spectrum of selfadjoint extensions of abstract

symmetric operators related to abstract boundary triples. We will

also discuss Hadamard’s type formulas expressing the derivative

of eigenvalues with respect to a parameter in terms of the respec-

tive Maslov crossing forms. Applications are given to multidimen-

sional Schrodinger operators on periodic and star-shaped domains.

The Maslov index is a geometric characteristic defined as the signed

number of intersections of a path in the space of Lagrangian planes

with the train of a given plane. The problem of relating this quan-

tity to the spectral count is rooted in Sturm’s Theory and has a long

history going back to the classical work by Arnold, Bott and Smale,

and has attracted recent attention of several groups of mathemati-

cians.

On some select Klein-Gordon problems: Internalmodes, fat tails, wave collisions and beyond

P.G. Kevrekidis

Department of Mathematics and Statistics

University of Massachusetts, Amherst, MA 01003, USA

[email protected]

In this work we will revisit the seemingly well-established story

of the φ4 kink collisions and discuss a (seemingly) fatal sign er-

ror. This will already expose some intriguing open questions for

what was previously thought to be well-known. This will serve

as a teaser for the development of further mathematical theory on

the subject. However, the emphasis of the work will be on a num-

ber of vignettes in cases that are even less well understood than

φ4, namely φ6, φ8, φ10 and φ12 models. The first of these models

can have kinks with either 0 or (controllably) many internal modes.

Some of the associated spectral and collisional phenomenology of

the relevant exponentially decaying kinks will be presented. Then,

we will venture into the remaining three models and unearth even

more complex features of the latter. For one thing, it is now pos-

sible to have power-law decaying kinks for which linearization

yields no information. In this case, many of the things we know

and trust go out the window: sum ansatze do not work to con-

struct proper initial conditions for interactions; if used, they yield

misleading results. Methods for evaluating interactions (including

variational ones etc.) do not properly work. Again, special care

needs to be used to unveil the power law interaction between the

kinks. We will thus attempt to provide a glimpse of the current state

of understanding and to offer a number of intriguing directions for

future study.

Steady concentrated vorticity and its stability of the2-dim Euler equation on bounded domains

Chongchun Zeng

Georgia Tech

[email protected]

On a smooth bounded domain Ω ⊂ R2, we consider steady solu-

tions of the incompressible Euler equation with concentrated vor-

ticity. More precisely, with prescribed integer n > 0, vortical do-

main sizes r1, . . . , rn > 0, and vorticity strengths µ1, . . . , µn 6= 0,

we seek steady vorticity distributions in the form of

ω = ∑nj=1 ωj(x) where

1.) the vortical domains satisfy Ωj = supp(ωj) ⊂ B(pj, 2rjǫ),

|Ωj| = πr2j ǫ2, with 0 < |ǫ| << 1 and distinct p1, . . . , pn ∈ Ω;

27

and 2.) µj =∫

ωjdx.

Since the dynamics of localized vorticity is approximated by the

point vortex dynamics, we take p1, . . . , pn close to a non-degene-

rate steady configuration of the point vortex system in Ω with pa-

rameters µ1, . . . , µn. Through a perturbation method applied to

Ωj parametrized by conformal mappings, we obtained two types

of steady solutions with smooth ∂Ωj being O(ǫ2) perturbations

to ∂B(pj, rjǫ): a.) infinitely many piecewise smooth solutions

ω ∈ C0,1(Ω); and b.) a unique steady vortex patch with piecewise

constant vorticity, i.e. ωj =µj

πr2j ǫ2 χ(Ωj). Moreover, the spectral

and evolutionary properties (stability, exponential trichotomy, etc.)

of the linearized vortex patch dynamics at the latter is determined

by those of the linearized point vortex dynamics at the steady con-

figuration p1, . . . , pn. This is a joint work with Yiming Long

and Yuchen Wang at Nankai University.

A bifurcation analysis of standing pulses and theMaslov index

Paul Cornwell

Johns Hopkins Applied Physics Laboratory

[email protected]

Claire Kiers∗

The University of North Carolina at Chapel Hill

[email protected]

The Maslov index is a powerful and insightful tool that can be used

to determine the stability of solutions for PDEs. We demonstrate

the robustness of a certain method of Maslov index calculation

by applying it to standing pulse solutions of a three-component

reaction-diffusion system. The Maslov index shows exactly why

the stability of a wave changes at a bifurcation due to the appear-

ance of a conjugate point. The calculation also indicates that the

Maslov index can see stable eigenvalues.

Dynamics of frequency combs modeled by theLugiato-Lefever equation

Mariana Haragus

Institut FEMTO-ST, Univ. Bourgogne-Franche Comte, France

[email protected]

The Lugiato-Lefever equation is a nonlinear Schrodinger-type equa-

tion with damping, detuning and driving, derived in nonlinear op-

tics by Lugiato and Lefever (1987). While extensively studied in

the physics literature, there are relatively few rigorous mathemat-

ical studies of this equation. Of particular interest for the physi-

cal problem is the formation and the dynamical behavior of Kerr

frequency combs (optical signals consisting of a super-position of

modes with equally spaced frequencies). The underlying mathe-

matical questions concern the existence and the stability of certain

particular steady solutions of the Lugiato-Lefever equation. In this

talk, I’ll focus on periodic steady waves for which I’ll show how

tools from bifurcation theory can be used to study their existence

and stability.

Renormalized oscillation theory for linear

Hamiltonian systems via the Maslov index

Peter Howard∗

Department of Mathematics, Texas A&M University, College Station, TX

77843, USA

[email protected]

Alim Sukhtayev

Department of Mathematics, Miami University, Oxford, OH 45056, USA

[email protected]

Working with a general class of linear Hamiltonian systems on

bounded intervals, we show that renormalized oscillation results

can be obtained in a natural way through consideration of the Maslov

index associated with appropriately chosen paths of Lagrangian

subspaces of C2n.

Stability of traveling waves in a model for a thinliquid film flow

Stephane Lafortune


College of Charleston

Charleston, SC 29424

[email protected]

Anna Ghazaryan


Miami University, 301 S. Patterson Ave

Oxford, OH 45056, USA, Ph. 1-513-529-0582

[email protected]

Vahagn Manukian


Miami University, 301 S. Patterson Ave

Oxford, OH 45056, USA

[email protected]

We consider a model for the flow of a thin liquid film down an

inclined plane in the presence of a surfactant. The model is known

to possess various families of traveling wave solutions. We use

a combination of analytical and numerical methods to study the

stability of the traveling waves. We show that for at least some

of these waves the spectra of the linearization of the system about

them are within the closed left-half complex plane.

Rigorous verification of wave stability

Blake Barker∗ and Taylor Paskett

Brigham Young University


Kevin Zumbrun

Indiana University

[email protected]

We discuss recent work regarding rigorous verification of stabil-

ity properties of traveling waves. In particular, we describe our

work developing computer assisted proof techniques to evaluate

the Evans function in order to prove spectral stability of waves in

the one-dimensional non-isentropic Navier-Stokes equations with

an ideal, polytropic gas equation of state. For this system, spectral

28

stability implies nonlinear stability. Proving spectral stability is the

last piece of a program begun over 20 years ago for establishing the

stability of traveling waves in this model.

Stability of travelling waves in a haptotaxis model

Kristen Harley, Peter van Heijster and Graeme Pettet

Queensland University of Technology


[email protected]

Robby Marangell∗, Tim Roberts and Martin Wechselberger

University of Sydney

[email protected], [email protected] and mar-

[email protected]

I will examine the spectral stability of travelling waves in a hap-

totaxis model for tumor invasion [1]. In the process, I will show

how to apply Lienard coordinates to the linearised stability prob-

lem and show some further developments in a geometrically in-

spired method for numerically computing the point spectrum of a

linearised operator.

1. K.E. Harley, P. v Heijster, R. Marangell, G. J. Pettet, and M. Wech-

selberger, Existence of traveling wave solutions for a model of tumor

invasion. SIAM Journal on Applied Dynamical Systems. 13 1. (2014),

366-396.

Solitary waves for weakly dispersive equations withinhomogeneous nonlinearities

Ola Maehlen


Norwegian University of Science and Technology,


[email protected]

We show existence of solitary-wave solutions to the equation

ut + (Lu − n(u))x = 0 ,

for weak assumptions on the dispersion L and the nonlinearity n.

The symbol m of the Fourier multiplier L is allowed to be of low

positive order (s > 0), while n need only be locally Lipschitz and

asymptotically homogeneous at zero. We shall discover such solu-

tions in Sobolev spaces contained in H1+s.

1. M. N. Arnesen, Existence of solitary-wave solutions to nonlocal equa-

tions, Discrete Contin. Dyn. Syst., 36 (2016), pp. 3483-3510.

2. M. Ehrnstrom, M. D. Groves, and E. Wahln, On the existence and sta-

bility of solitary-wave solutions to a class of evolution equations of

Whitham type, Nonlinearity, 25 (2012), pp. 2903-2936.

3. M.I.Weinstein, Existence and dynamic stability of solitary wave solu-

tions of equations arising in long wave propogation, Comm. Partial

Differential Equations, 12 (1987), pp. 1133-1173.

Spectral stability of hydraulic shock profiles

Alim Sukhtayev∗ and Zhao Yang

Miami University and Indiana University


Kevin Zumbrun

Indiana University

[email protected]

By reduction to a generalized Sturm Liouville problem, we es-

tablish spectral stability of hydraulic shock profiles of the Saint-

Venant equations for inclined shallow-water flow, over the full pa-

rameter range of their existence, for both smooth-type profiles and

discontinuous-type profiles containing subshocks. Together with

work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear

and nonlinear H2 ∩ L1 → H2 stability with sharp rates of decay in

Lp, p ≥ 2, the first complete stability results for large-amplitude

shock profiles of a hyperbolic relaxation system.

Turning point principle for the stability of stellarmodels

Zhiwu Lin

School of Mathematics

Georgia Institute of Technology

Atlanta, GA, 30332

[email protected]

I will discuss some recent results on the linear stability criterion of

spherically symmetric equilibria of several stellar models, includ-

ing Euler-Poisson, Einstein-Euler and Einstein-Vlasov models. For

Euler-Poisson and Einstein-Euler models, a turning point principle

for the sharp stability criterion will be given. For Vlasov-Einstein

model, the stability part of the turning point principle is obtained

and the linear instability in the strong relativisitic limit will also be

discussed. For these models, a combination of first order and 2nd

order Hamiltonian formulations is used to derive the stability crite-

rion and study the linearized equation for initial data in the energy

space. This is joint work with Chongchun Zeng (on Euler-Poisson)

and with Hadzic and Rein (on Einstein-Euler and Einstein-Vlasov).

Solitary wave solutions of a Whitham-Bousinessqsystem

Dag Nilsson∗

Norwegian University of Science and Technology

[email protected]

Evgueni Dinvay


[email protected]

We consider a Whitham-Boussinesq type system that was recently

introduced in [2] as a fully dispersive model for bidirectional sur-

face waves. Moreover, the system was shown to be locally well

posed in [1].

In this paper we prove existence of solitary wave solutions of this

system, and in addition show that these solutions are approximated

by scalings of KdV-type solitary waves. This is proved using a

variational approach, where solitary waves are identified as critical

points of a certain functional, and proceed to show that there exist

minimizers of this functional, using the concentration-compactness

theorem.

29

1. Dinvay, E., On well-posedness of a dispersive system of the Whitham–

Boussinesq type, Applied Mathematics Letters, Volume 88, February

2019, Pages 13-20.

2. Dinvay, E., Dutykh, D., Kalisch, H. A comparative study of bi-

directional Whitham systems. Applied Numerical Mathematics.

Viewing spectral problems through the lens of theKrein matrix

Todd Kapitula∗


Calvin College

[email protected]

Ross Parker

Division of Applied Mathematics

Brown University

ross [email protected]

Bjorn Sandstede


Brown University

bjorn [email protected]

When considering the problem of finding point spectrum for the

linearization about a wave for a Hamiltonian system, it is of inter-

est to not only find those eigenvalues with positive real part, but

also those purely imaginary eigenvalues with negative Krein sig-

nature. The Krein matrix is a meromorphic-valued function of the

spectral parameter which has the property that it is singular. More-

over, it can be constructed so that the Krein signature of purely

imaginary eigenvalues can be graphically determined via the sign

of a derivative. Here we construct the Krein matrix for linear and

quadratic eigenvalue problems, and show how it can be used:

• to locate possible Hamiltonian-Hopf bifurcations (collision

of purely imaginary eigenvalues with opposite Krein signa-

ture)

• locate small eigenvalues which arise through some type of

bifurcation.

More details associated with the applications will be presented by

Ross Parker in a subsequent talk.

Spectral stability of multi-pulses via the Kreinmatrix

Ross Parker∗


Brown University

ross [email protected]

Todd Kapitula

Deparment of Mathematics and Statistics

Calvin College

[email protected]

Bjorn Sandstede


Brown University

bjorn [email protected]

The Chen-Mckenna suspension bridge equation is a nonlinear PDE

which is 2nd order in time and is used to model traveling waves on

a suspended beam. For certain parameter regimes, it admits multi-

pulse traveling wave solutions, which are small perturbations of the

stable, primary pulse solution. Linear stability of these multi-pulse

solutions is determined by eigenvalues near the origin representing

the interaction between the individual pulses. Linearization about

these multi-pulse solutions yields a quadratic eigenvalue problem.

To study this problem, we use a reformulated version of the Krein

matrix, which was presented by Todd Kapitula in a previous talk.

Using an appropriate leading order expansion of the Krein matrix,

we are able to give analytical criteria for the stability of these multi-

pulse solutions. We also present numerical results to support our

analysis.

Coriolis forces and particle trajectories for waveswith stratification and vorticity

Miles H. Wheeler∗

University of Vienna

Faculty of Mathematics

Oskar-Morgenstern-Platz 1

1080 Wien, Austria

[email protected]

In recent years there has been much mathematical interest in gen-

eralizations of the classical water wave problem which take into

account the Coriolis force due to the rotation of the Earth, and in

particular in a two-dimensional model for waves traveling along

the equator. In the first part of this talk we will observe that, for

waves which travel at a constant speed, the Coriolis terms in this

two-dimensional model can in fact be completely removed by a

change of variables. This fact does not seem to appear in the exist-

ing literature, and it allows for many proofs and calculations to be

dramatically simplified.

If time permits we will also discuss ongoing work with Biswajit

Basu (University of Vienna) on the particle trajectories and related

properties of solitary waves with vorticity and/or stratification.

On the stability of solitary water waves with a pointvortex

Kristoffer Varholm∗

Department of Mathematical Sciences, Norwegian University of Science

and Technology,


[email protected]

Erik Wahlen

Centre for Mathematical Sciences, Lund University, PO Box 118, 22100

Lund, Sweden

[email protected]

Samuel Walsh

Department of Mathematics, University of Missouri, Columbia, MO 65211,

USA

[email protected]

This paper investigates the stability of solutions of the steady water

wave problem with a submerged point vortex. We prove that waves

30

with sufficiently small amplitude and vortex strength are condition-

ally orbitally stable. In the process of obtaining this result, we de-

velop a quite general stability theory for bound state solutions of

a large class of Hamiltonian systems in the presence of symmetry.

This is in the spirit of the seminal work of Grillakis, Shatah, and

Strauss [2], but with hypotheses that are relaxed in a number of

ways necessary for the point vortex system, and for other hydro-

dynamical applications more broadly. In particular, we are able to

allow the Poisson map to be state-dependent, and to have merely

dense range.

As a second application of the general theory, we consider a fam-

ily of nonlinear dispersive equations that includes the generalized

KdV and Benjamin–Ono equations. The stability (or instability)

of solitary waves for these systems has been studied extensively,

notably by Bona, Souganidis, and Strauss [1], who used a modifi-

cation of the GSS method. We provide a new, more direct proof of

these results that follows as a straightforward consequence of our

abstract theory.

1. J. L. BONA, P. E. SOUGANIDIS, AND W. A. STRAUSS, Stability and

instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc.

London Ser. A, 411 (1987), pp. 395–412.

2. M. GRILLAKIS, J. SHATAH, AND W. STRAUSS, Stability theory of

solitary waves in the presence of symmetry. I, J. Funct. Anal., 74

(1987), pp. 160–197.

Modulational instability of viscous fluid conduitperiodic waves

Mathew A. Johnson and Wesley R. Perkins∗

Department of Mathematics, University of Kansas, 1460 Jayhawk Boule-

vard, Lawrence, KS 66045


The Whitham modulation equations are widely used to describe

the behavior of modulated periodic waves on large space and time

scales; hence, they are expected to give insight into the stability

of spatially periodic structures. However, the derivation of these

equations are based on formal asymptotic (WKB) methods, thus

removing a layer of rigor that would otherwise support their pre-

dictions. In this study, we aim at rigorously verifying the predic-

tions of the Whitham modulation equations in the context of the so-

called conduit equation, a nonlinear dispersive PDE governing the

evolution of the circular interface separating a light, viscous fluid

rising buoyantly through a heavy, more viscous, miscible fluid at

small Reynolds numbers. In particular, using rigorous spectral per-

turbation theory, we connect the predictions of the Whitham mod-

ulation equations to the rigorous spectral (in particular, modula-

tional) stability of the underlying wave trains. This makes rigorous

recent formal results on the conduit equation obtained by Maiden

and Hoefer.

Localization for Anderson models on tree graphs

David Damanik, Selim Sukhtaiev*

Department of Mathematics, Rice University, Houston, TX 77005, USA

[email protected]

Jake Fillman

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA

In this talk, I will discuss Anderson localization for Bernoulli–type

random models on metric and discrete radial graphs. Dynamical

localization is proved on compact intervals contained in the com-

plement of a discrete set of exceptional energies. This is based on

joint work with D. Damanik and J. Fillman.

A Maslov index for non-Hamiltonian systems

Graham Cox∗


Memorial University of Newfoundland

St. John’s, NL Canada

[email protected]

The Maslov index is a powerful and well known tool in the study of

Hamiltonian systems, providing a generalization of Sturm-Liouville

theory to systems of equations. For non-Hamiltonian systems, one

no longer has the symplectic structure needed to define the Maslov

index. In this talk I will describe a recent construction of a “gener-

alized Maslov index” for a very broad class of differential equa-

tions. The key observation is that the manifold of Lagrangian

planes can be enlarged considerably without altering its topolog-

ical structure, and in particular its fundamental group. This is

joint work with Tom Baird, Paul Cornwell, Chris Jones and Robert

Marangell.

Nonlinear stability of layers in precipitation modelsAlin Pogan

Miami University


301 S. Patterson Ave.

Oxford, OH 45056, USA

[email protected]

Standing layers are known to exist in models arising in chemical

conversion equations in closed reactors. We explore various con-

cepts of stability such as spectral, linear and nonlinear stability.

Periodic traveling hydroelastic waves

David M. Ambrose

Department of Mathematics, Drexel University

Philadelphia, PA 19104 USA

[email protected]

Recent work of the presenter, Benjamin Akers, and J. Douglas

Wright developed a formulation for traveling waves in interfacial

fluid dynamics which allows the free fluid surface to have multi-

valued height. This formulation was shown to be amenable to

efficient computation of bifurcation branches as well as develop-

ment of local and global bifurcation theory for interfacial capillary-

gravity waves. All of this work has then been adapted to the hy-

droelastic case, allowing elastic effects at the fluid interface, such

as those present in ice sheets, cellular membranes, or thin struc-

tures such as flags. With Akers and David Sulon, we have proved

existence of families of traveling waves and computed the same.

The analysis in the hydroelastic case also proves existence in the

Wilton ripple case, in which the kernel of the relevant linearization

is two-dimensional.

31

Invariant Manifolds of Traveling Waves of the 3DGross-Pitaevskii Equation in the Energy Space

Jiayin Jin and Zhiwu Lin

School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332


Chongchun Zeng

School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332

[email protected]

We study the local dynamics near general unstable traveling waves

of the 3D Gross-Pitaevskii equation in the energy space by con-

structing smooth local invariant center-stable, center-unstable and

center manifolds. We also prove that (i) the center-unstable mani-

fold attracts nearby orbits exponentially before they go away from

the traveling waves along the center or unstable directions and (ii)

if an initial data is not on the center-stable manifolds, then the for-

ward orbit leaves traveling waves exponentially fast. Furthermore,

under an additional non-degeneracy assumption, we show the or-

bital stability of the traveling waves on the center manifolds, which

also implies the uniqueness of the local invariant manifolds. Our

method based on a geometric bundle coordinate

Generalized solitary wave solutions of the capillary-gravity Whitham equation

J. Douglas Wright

Drexel University

[email protected]

“Whitham” equations have enjoyed a recent resurgence of popu-

larity as models for free surface fluid flows. They are, roughly

speaking, obtained by using the full linear part of the appropriate

Euler equation together with a simpler “KdV”- type nonlinearity.

Generalized solitary waves are traveling wave solutions which are

the superposition of a classical solitary wave with a “small beyond

all orders” periodic wave. Such waves are known to exist for the

full capillary-gravity wave problem and in this talk we discuss re-

cent work on establishing their existence for the “Whithamized”

version. (This work is joint with A. Stefanov and M. Johnson.)

Modulational dynamics of spectrally stableLugiato-Lefever periodic waves

Mathew A. Johnson∗ and Wesley R. Perkins



Mariana Haragus

Univ. Bourgogne Franche?Comt?e

[email protected]

e consider the dynamics of periodic steady waves of the Lugiato-

Lefever equation, which is an equation derived in nonlinear optics

of NLS type with damping, detuning, and driving. Using Floquet-

Bloch theory, we are able to describe at the linear level the modula-

tional dynamics of periodic steady waves that are spectrally stable

to general bounded perturbations on the line. We will also discuss

important challenges towards describing the associated nonlinear

dynamics.

Asymptotic stability of the Novikov peakons

Ming Chen∗

Department of Mathematics, University of Pittsburgh, PA 15260, USA

[email protected]

Wei Lian

College of Science, Harbin Engineering University, Harbin 150001, P. R.

China

lianwei [email protected]

Dehua Wang

Department of Mathematics, University of Pittsburgh, PA 15260, USA

[email protected]

Runzhang Xu


China

[email protected]

We prove that the peakons of the Novikov equation are asymp-

totically H1-stable in the class of functions with the momentum

density m := u − uxx belonging to the set of non-negative finite

Radon measure M+. The key ingredient in the argument is a Liou-

ville property for the uniformly (up to translation) almost localized

global solutions satisfying m ∈ M+, that is, we prove that such a

solution must be a peakon.

SESSION 8: “Dispersive wave equations and their soliton interac-

tions: Theory and applications”

Backward behavior of a dissipative KdV equation

Yanqiu Guo and Edriss S. Titi

Florida International University and Texas A&M University


In this talk, I will discuss the backward-in-time behavior of a KdV

equation influenced by dissipation and source terms. In particular,

we prove that every solution of a KdV-Burgers-Sivashinsky type

equation blows up in the energy space, backward in time, if the so-

lution does not belong to the global attractor. In addition, we pro-

vide some physical interpretation of various backward behaviors

of several perturbations of the KdV equation by studying explicit

soliton-type solutions. This is a joint work with E. S. Titi.

On the energy cascade of acoustic wave turbulence:Beyond Kolmogorov-Zakharov solutions

Avy Soffer and Minh-Binh Tran

Mathematics Department, Rutgers University, New Brunswick, NJ 08903

USA

Department of Mathematics, Southern Methodist University, Dallas, TX

75275, USA


In weak turbulence theory, the Kolmogorov-Zakharov spectra is a

class of time-independent solutions to the kinetic wave equations.

32

In this paper, we construct a new class of time-dependent solu-

tions to those kinetic equations. These solutions exhibit the inter-

esting property that the energy is cascaded from small wavenum-

bers to large wavenumbers. We can prove that starting with a reg-

ular initial condition whose energy at the infinity wave number

p = ∞ is 0, as time evolves, the energy is gradually accumulated

at p = ∞. Finally, all the energy of the system is concentrated

at p = ∞ and the energy function becomes a Dirac function

at infinity Eδp=∞, where E is the total energy. The existence of

this class of solutions is, in some sense, a rigorous mathematical

proof based on the kinetic description for the energy cascade phe-

nomenon. We restrict our attention in this paper to the statistical

description of acoustic waves. However, the technique is quite ro-

bust and can be applied to other types of wave turbulence kinetic

equations.

Keyword: weak turbulence theory, acoustic wave, Kolmogorov-

Zakharov spectra, energy cascade

Dynamics of a heavy quantum tracer particle in aBose gas

Thomas Chen∗


University of Texas at Austin

[email protected]

Avy Soffer


Rutgers University

[email protected]

We consider the dynamics of a heavy quantum tracer particle cou-

pled to a non-relativistic boson field in R3. The pair interactions of

the bosons are of mean-field type, with coupling strength propor-

tional to 1/N where N is the expected particle number. Assuming

that the mass of the tracer particle is proportional to N, we derive

generalized Hartree equations in the limit where N tends to infinity.

Moreover, we prove the global well-posedness of the associated

Cauchy problem for sufficiently weak interaction potentials. This

is joint work with Avy Soffer (Rutgers University).

1. T. Chen and A. Soffer, Mean field dynamics of a quantum tracer par-

ticle interacting with a boson gas, J. Funct. Anal., 276 (3), 971-1006,

2019.

Soliton Potential interaction of NLS in R3

Qingquan Deng

The School of Mathematics and Statistics, Central China Normal Univer-

sity

152 Luoyu Street, Wuhan, 430079, P. R. China

[email protected]

We consider the following equation

i∂tψ = −1/2∆ψ + Vψ − Fǫ(|ψ|2)ψ.

In this work we mainly focus on the dynamics and scattering of a

narrow soliton of the above NLS equation with a potential in R3,

where the asymptotic state of the system can be far from the initial

state in parameter space. Specifically, if we let a narrow soliton

state with initial velocity υ0 of order 1 to interact with an exter-

nal potential V(x), then the velocity υ+ of outgoing solitary wave

in infinite time will in general be very different from υ0. In con-

trast to our present work, previous results proved that the soliton is

asymptotically stable so that υ+ stays close to υ0 for all times.

Stable blow-up dynamics in the generalizedL2-critical Hartree equation

Svetlana Roudenko∗, Anudeep Kumar Arora and Kai Yang

Department of Mathematics and Statistics, DM430

Florida International University, Miami, FL 33199

[email protected] and [email protected] and [email protected]

We study stable blow-up dynamics in the nonlinear Schrodinger

(NLS) equation and generalized Hartree equation in the L2-critical

regime. The NLS equation is with pure power nonlinearity iut +∆u + |u|2σu = 0, and the generalized Hartree equation is a

Schrodinger-type equation with a nonlocal, convolution-type non-

linearity in dimension d: iut +∆u+(|x|−(d−2) ∗ |u|p

)|u|p−2u =

0, p ≥ 2.

First, we consider the L2-critical case of the NLS equation in di-

mensions 4 ≤ d ≤ 12 and give a numerically-assisted proof of the

spectral property, which completes the log-log blow-up theory of

Merle-Raphael for the mass-critical NLS up to the dimension 12.

We next consider the generalized Hartree equation in the L2-critical

regime and investigate spectral properties needed to understand the

blow-up dynamics of the solutions. We then show that similar to

NLS, solutions with mass slightly above the corresponding ground

state and negative energy, will blow-up with the “log-log” dynam-

ics in the 3d generalized Hartree equation.

Knocking out teeth in one-dimensional periodic NLS:Local and global wellposedness results

L. Chaichenets, D. Hundertmark

Karlsruhe Institute of Technology


P. Kunstmann


[email protected]

N. Pattakos


[email protected]

In this talk local and global wellposedness results of the 1-dimension-

al nonlinear Schrodinger equation

iut − uxx ± |u|α−1u = 0

will be discussed with initial data u0 ∈ Hs(R) + Hs(T), where

s ≥ 0, α ∈ (1, 5) and T is the one dimensional torus.

In the case of the cubic nonlinearity, α = 3, local existence of weak

solutions in the extended sense is shown through a differentiation

by parts argument and in the case of the quadratic nonlinearity,

α = 2, global existence is established with the use of Strichartz

type estimates and a conserved quantity argument.

33

1. L. CHAICHENETS, D. HUNDERTMARK, P. KUNSTMANN AND N.

PATTAKOS, Knocking out teeth in one-dimensional periodic NLS.

arXiv:1808.03055 (2018), submitted to Analysis and PDE.

2. L. CHAICHENETS, D. HUNDERTMARK, P. KUNSTMANN AND N.

PATTAKOS, Global wellposedness of the quadratic NLS in one dimen-

sion with initial data in L2(R) + H1(T). preprint (2019).

Conservation laws and asymptotics for the waveequation

Stefanos Aretakis

University of Toronto, Toronto, Canada

[email protected]

We will present results regarding the precise late-time asymptotics

for solutions to the wave equation on black hole backgrounds. Our

method relies on purely physical space techniques and makes use

of conservation laws for the wave equation along null hypersur-

faces. We will present results for both extremal and sub-extremal

black hole backgrounds ([1, 2]). In the case of extremal black

holes, we will show that deriving precise asymptotics leads to some

interesting conclusions such as the existence of observational sig-

natures of extremal event horizons ([3]).

1. Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic, Late-

time asymptotics for the wave equation on extremal Reissner-

Nordstrom backgrounds, arXiv:1807.03802 (2018)

2. Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic, Late-

time asymptotics for the wave equation on spherically symmetric, sta-

tionary spacetimes, to appear in Advances in Mathematics, 323 (2018),

529-621

3. Yannis Angelopoulos and Stefanos Aretakis and Dejan Gajic, Horizon

hair of extremal black holes and measurements at null infinity , Phys.

Rev. Lett., 121 (2018), 131102.

Derivation of the Schrodinger-Klein-Gordonequations

Nikolai Leopold

IST Austria (Institute of Science and Technology Austria), Am Campus 1,

3400 Klosterneuburg, Austria.

[email protected]

Soren Petrat∗

Jacobs University, Department of Mathematics, Campus Ring 1, 28759

Bremen, Germany.

[email protected]

This talk is about an example of how to derive non-linear Schro-

dinger equations in a mathematically rigorous way, starting from

the linear interacting many-body Schrodinger equation.

Here, I will present the recent result [1] where we start with the

Nelson model with ultraviolet cutoff. This model is linear and de-

scribes a quantum system of non-relativistic particles coupled to

a positive or zero mass quantized scalar field. We take the non-

relativistic particles to obey Fermi statistics and discuss the time

evolution in a mean-field limit of many fermions which is coupled

to a semiclassical limit. At time zero, we assume that the bosons

of the radiation field are close to a coherent state and that the state

of the fermions is close to a Slater determinant with a certain semi-

classical structure. We prove that the many-body state approxi-

mately retains its Slater determinant and semiclassical structure at

later times and that its time evolution can be approximated by the

fermionic Schrodinger-Klein-Gordon equations. These are a non-

linear system of two equations: a nonlinear Schrodinger equation

and a wave equation with source term. We prove the convergence

for reduced densities with explicit rates and for all semiclassical

times.

1. N. Leopold and S. Petrat, Mean-field Dynamics for the Nelson Model

with Fermions, Preprint, [arXiv:1807.06781] (2018).

The effect of threshold energy obstructions on the

L1 → L∞ dispersive estimates for some Schrodingertype equations

M. Burak Erdogan

University of Illinois at Urbana Champaign

[email protected]

Willam. R. Green

Rose-Hulman Institute of Technology

[email protected]

Ebru Toprak ∗

Rutgers University

[email protected]

In this talk, I will discuss the differential equation iut = Hu,

H := H0 +V , where V is a decaying potential and H0 is a Lapla-

cian related operator. In particular, I will focus on when H0 is

Laplacian, Bilaplacian and Dirac operators. I will discuss how the

threshold energy obstructions, eigenvalues and resonances, effect

the L1 → L∞ behavior of eitH Pac(H). The threshold obstructions

are known as the distributional solutions of Hψ = 0 in certain

dimension dependent spaces. Due to its unwanted effects on the

dispersive estimates, its absence has been assumed in many works.

I will mention our previous results on Dirac operator, [1, 2] and

recent results on Bilaplacian operator, [3] under different assump-

tions on threshold energy obstructions.

1. Erdogan, M. B., and Green, W. R., Toprak, E. Dispersive estimates

for Dirac operators in dimension three with obstructions at threshold

energies , to appear American Journal of Mathematics,

2. Erdogan, M. B., and Green, W. R., Toprak, E. Dispersive estimates

for massive Dirac operators in dimension two ,J. Differntial Equations

(2018), Volume 264, 5802–5837.

3. Green, W. R., Toprak, E. On the Fourth order Schrodinger equation

in four dimensions: dispersive estimates and zero energy resonances,

arxiv.org/abs/1810.03678.

Quantization of energy of blow up for wave maps

Hao Jia

University of Minnesota

[email protected]

The two dimensional wave map equation is an important geometric

wave equation. Soliton and dispersion are two fundamental fea-

tures for wave maps. We will report a recent idea to show that

34

when a wave map blows up, all the concentrated energy is in the

form of traveling waves.

Local smoothing estimates for Schrodinger equationson hyperbolic space and applications

Jonas Luhrmann

Johns Hopkins University

[email protected]

We establish frequency-localized local smoothing estimates for

Schrodinger equations on hyperbolic space. The proof is based on

the positive commutator method and a heat flow based Littlewood-

Paley theory. Our results and techniques are motivated by appli-

cations to the problem of stability of solitary waves to nonlinear

Schrodinger-type equations on hyperbolic space.

This is joint work with Andrew Lawrie, Sung-Jin Oh, and Sohrab

Shahshahani.

Nonlinear waves on vortex filaments in quantumliquids: A geometric perspective

Scott A. Strong and Lincoln D. Carr

Department of Applied Mathematics and Statistics and Department of

Physics

Colorado School of Mines


A vortex filament is modeled as a one-dimensional region of a

quantum liquid about which the otherwise irrotational fluid circu-

lates. The vortex filament equation, i.e., the local induction ap-

proximation, asserts that points on a vortex filament are transported

by the velocity field in the direction of the local binormal vector

and at a speed proportional to local curvature. Its simplest non-

trivial prediction is that vortex rings with smaller curvature (larger

radius) travel slower than those with larger curvature. That said,

the result should be thought of as an arclength conserving flow

which evolves the curvature and torsion variables according to an

integrable Schrodinger equation. While this allows the vortex line

to support a wide variety of nonlinear waves, the integrability is

thought to restrict energy transfer between helical Kelvin modes.

In this talk, we go beyond the local induction approximation by

defining corrections which lead to a non-Hamiltonian evolution of

the curvature and torsion variables. These corrections are asso-

ciated with an emergent curvature gain/loss mechanism and en-

hanced dispersion on the vortex medium. Altogether we find that

regions of localized curvature seek to transport their bending into

the vortex in the form of helical Kelvin waves, which provides a

necessary ingredient for modeling vortex dynamics in turbulent ul-

tracold quantum fluids.

Global well-posedness and scattering for the Davey-Stewartson system at critical regularity

Matthew Rosenzweig



2515 Speedway, Stop C1200 Austin, TX 78712

[email protected]

In this talk, I will discuss a two-dimensional nonlinear dispersive

PDE arising in the study of water waves called the Davey-Stewartson

system (DS), which is formally similar to the L2-critical cubic non-

linear Schrodinger equation (NLS) but differs by an additional non-

local term. Specifically, I will discuss recent work on the global

well-posedness and scattering for a particular case of DS with ini-

tial data in the critical L2 space, which is inspired by Benjamin

Dodson’s breakthrough work on the cubic NLS. Finally, I will dis-

cuss the question of the rigorous justification of DS as a multiple

scales approximation for wave packet solutions to the water waves

equation.

Semi-linear Schrodinger’s equation with randomtime-dependent potentials

Marius Beceanu∗

University at Albany SUNY Mathematics and Statistics Department

1400 Washington Ave., Albany, NY 12222, USA

[email protected]

Avy Soffer

Rutgers University Department of Mathematics

110 Frelinghuysen Rd., Piscataway, NJ 08854, USA

[email protected]

Jurg Frohlich

Institute for Advanced Study School of Mathematics

1 Einstein Drive, Prnceton, NJ 08540, USA

[email protected]

This talk will be a presentation of results, obtained together with

Jurg Frohlich and Avy Soffer, pertaining to the semi-linear Schrodinger

equation with random time-dependent potential

i∂tψ − ∆ψ + Vωψ(x, t)ψ = N(ψ).

In general, the interaction between a nonlinear term N(ψ) and the

bound states of a time-dependent linear potential can be compli-

cated to describe. There exist small standing-wave solutions and

growth in norm of the solutions is possible. However, for the case

of a random time-dependent short-range potential on Euclidean

space, driven by a Markov process, we show that, with proba-

bility one, all solutions scatter (i.e. nonlinear wave operators are

bounded) and disperse at the same rate as for the free equation.

Long time dynamics for nonlinear dispersive equa-tions

Baoping

Peking University

[email protected]

Dispersive equations usually admit solutions with quite different

asymptotic behaviors, such as scattering solutions and solitons. So

it is rather difficult to describe the long time dynamics for general

solutions. In this talk, we will discuss few cases for which we are

able to get a definite answer.

Global well-posedness for mass-subcritical NLS incritical Sobolev space

Yifei Wu

Center for Applied Mathematics

35

Tianjin University

[email protected]

In this talk, we consider the mass-subcritical nonlinear Schrodinger

equation. It was known that the solution is global if the initial

data is small in critical Sobolev space, or the solution is uniformly

bounded in whole lifespan in critical Sobolev space. In this talk, we

show that if any initial data in critical Sobolev space with compact

suppoerted, then the corresponding solution is global.

Higher order corrections to mean field dynamics ofBose cold gases

Peter Pickl

Mathematical Institute LMU Munich// Theresienstr. 39//80333 Muenchen

[email protected]

It is well known that the the dynamics of ultra-cold Bose gases

in the weak coupling regime is given by its respective mean-field

limit, i.e. the Hartree equation. Recent developments in mathemat-

ical research made it possible to prove the validity of the next order

correction, the Boguliubov dynamics, in many situations.

While convergence to the Hartree equation is typically proven to

be valid in trace norm, it has been shown by several authors that

the N-body solution is close to the solution of the Boguliubov

time evolution in L2. In contrast to the mean-field description, the

Boguliubov time evolutions takes pair correlations into account.

In this talk I will prove the validity of higher order corrections in

the high density limit. We shall show that the rate of convergence

convergence gets better when higher order correlations are taken

into account.

The estimates hold even in situations where the volume and the

density of the gas go to infinity. It is a joint work with Lea Boß-

mann, Natasa Pavlovic, and Avy Soffer based on [1] Our result is

similar to but more explicit than recent findings by Paul and Pul-

virenti [2].

1. S. Petrat, P.Pickl, A. Soffer, Derivation of the Bogoliubov Time Evolu-

tion for Gases with Finite Speed of Sound, arXiv:1711.01591 , (2017).

2. H. Paul, M. Pulvirenti, Asymptotic expansion of the mean-field ap-

proximation, Disc. & Cont. Dyn. Sys. - A, 39 (4) (2019), 1891-1921.

SESSION 9: “Nonlinear evolutionary equations: Theory, numerics

and experiments”

Snakes and lattices: Understanding the bifurcationstructure of localized solutions to lattice dynamicalsystems

Jason J. Bramburger∗ and Bjorn Sanstede

170 Hope Street

Providence, Rhode Island, 02906, USA

jason [email protected] and bjorn [email protected]

A wide variety of spatially localized steady-state solutions to par-

tial differential equations (PDEs) are known to exhibit a bifurca-

tion phenomenon termed snaking. That is, these solutions bounce

between two different values of the bifurcation parameter while

expanding the region of localization and hence ascending in norm.

The mechanism that drives snaking in PDEs has been understood

by analyzing the evolution of the ordinary differential equation in

the spatial variable governing steady-state solutions to the PDE.

In this talk we extend this theory to lattice dynamical systems by

showing that the associated steady-state equations in this context

can be written as a discrete dynamical system. We can then inter-

pret localized solutions to the lattice system as homoclinic orbits

of the associated discrete dynamical system, and show that the bi-

furcation structure is determined by bifurcations of nearby hetero-

clinic orbits. We supplement these results with examples from a

well-studied bistable lattice differential equation which has been

the focus of many works to date.

Growing stripes, with and without wrinkles

Ryan Goh∗


Boston University

[email protected]

The interplay between growth processes and spatial patterns has

arisen as a topic of recent interest in many fields, such as directional

quenching in alloy melts, growing interfaces in biological systems,

moving masks in ion milling, eutectic lamellar crystal growth, and

traveling reaction fronts, where such processes have been shown to

select spatially periodic patterns, and mediate the formation of de-

fects. Mathematically, they can be encoded in a step-like parameter

dependence that allows patterns in a subset of the spatial domain,

and suppresses them in the complement, while the boundary of the

pattern-forming region propagates with fixed normal velocity.

In this talk, I will show how techniques from dynamical systems,

functional analysis, and numerical continuation, can be used to

study the effect of these traveling heterogeneities on patterns left

in the wake; finding for example how the speed of the parame-

ter interface affects orientation and deformation of stripes. I will

also show how periodic wrinkles can form on top of pure stripes,

with frequency behavior similar to that of a saddle-node on a limit

cycle. I will explain this approach in the context of the Swift-

Hohenberg PDE, a prototypical model for many pattern forming

systems, posed in one and two spatial dimensions. I will also dis-

cuss recent work which uses techniques from geometric desingu-

larization and modulational theory to study the stability and dy-

namics of these structures.

1. Avery, M and Goh, R and Goodloe, O and Milewski, A and

Scheel, A, Growing stripes, with and without wrinkles, arXiv preprint

arXiv:1810.08688, (2018).

2. R. Goh, A. Scheel. Pattern-forming fronts in a Swift-Hohenberg equa-

tion with directional quenching - parallel and oblique stripes, J. London

Math. Soc., 98 (2018), 104-128.

Nonlinear eigenvalue problems in biologically moti-vated PDEs

Zoi Rapti∗ and Jared C. Bronski


University of Illinois, Urbana-Champaign

36


Andrea K. Barreiro


Southern Methodist University

[email protected]

This paper is focused on the spectral properties of certain classes of

coupled nonlinear PDEs arising in biology. We will present results

that show the existence of only real spectrum in the corresponding

non-selfadjoint eigenvalue problem. Our proof relies on the theory

of operator pencils and Herglozt functions. Concrete applications

will be demonstrated in models of rabies epidemics in fox popula-

tions, plant-herbivore interactions and morphogen diffusion.

Grain boundaries of the Swift-Hohenberg equation:simulations and analysis

Joceline Lega

Department of Mathematics, University of Arizona, 617 N. Santa Rita

Avenue, Tucson, AZ 85721

[email protected]

I will summarize the results of [1], which describes an analytical

and numerical investigation of the phase structure of some stable

grain boundary solutions of the Swift-Hohenberg equation.

I will then introduce new analytical and numerical tools to explore

properties of the phase of the pattern in the vicinity of the disloca-

tions that form at the core of such grain boundaries in the strong

bending limit.

This work is joint with Nick Ercolani.

1. Nicholas M. Ercolani, Nikola Kamburov, Joceline Lega, The phase

structure of grain boundaries, Phil. Trans. R. Soc. A 376, 20170193

(2018).

Bifurcations on an NLS dumbbell graph

Roy H. Goodman

Department of Mathematical Sciences, New Jersey Institute of Technol-

ogy, University Heights, Newark, NJ 07102

[email protected]

We consider the bifurcations of standing wave solutions to the

nonlinear Schrodinger equation (NLS) posed on a quantum graph

consisting of two loops connected by a single edge, the so-called

dumbbell, recently studied by Marzuola and Pelinovsky. The au-

thors of that study found the ground state undergoes two bifurca-

tions, first a symmetry-breaking, and the second which they call a

symmetry-preserving bifurcation. We clarify the type of the

symmetry-preserving bifurcation, showing it to be transcritical. We

then reduce the question, and show that the phenomena described

in that paper can be reproduced in a simple discrete self-trapping

equation on a combinatorial graph of bowtie shape. This allows

for complete analysis both by geometric methods and by parame-

terizing the full solution space. We then expand the question, and

describe the bifurcations of all the standing waves of this system,

which can be classified into three families, and of which there ex-

ists a countably infinite set.

Traveling waves in the fifth order Korteweg-de Vriesequation and discontinuous shock solutions of theWhitham modulation equations

Patrick Sprenger, Mark Hoefer

Department of Applied Mathematics, University of Colorado Boulder

[email protected]

Whitham modulation theory is a powerful mathematical tool to de-

scribe the slow evolution of a nonlinear, periodic wave. It yields

a system of hyperbolic partial differential equations for the evo-

lution of the wave’s parameters. The typical solution of interest

in applications to dispersive shock waves is a weak, self-similar

expansion wave solution to the hyperbolic Whitham system. This

talk will focus on the fifth order Korteweg-de Vries (KdV5) equa-

tion and its rich family of traveling wave solutions. It is shown that

discontinuous shock solutions of the Whitham modulation system

which represent the zero dispersion limit of traveling wave solu-

tions of the KdV5 equation. These shock correspond to a rapid

transition joining two disparate periodic waves copropagating at a

fixed velocity. These traveling waves necessarily satisfy classical

jump conditions for the far-field wave parameters and shock ve-

locity. These solutions have recently been observed numerically in

applications to water waves and nonlinear optics.

Nonlinear instability of spectrally stable shifted stateson star graphs

Adilbek Kairzhan and Dmitry E. Pelinovsky

Department of Mathematics, McMaster University

Hamilton, Ontario L8S4K1, Canada


Roy Goodman


New Jersey Institute of Technology

Newark NJ, USA

[email protected]

When coefficients of the cubic terms match coefficients in the

boundary conditions at a vertex of a star graph and satisfy a cer-

tain constraint, the nonlinear Schrodinger (NLS) equation on the

star graph can be transformed to the NLS equation on a real line.

Such balanced star graphs appeared in the context of reflectionless

transmission of solitary waves. The steady states can be translated

along the edges of a balanced star graph with a translational pa-

rameter and are referred to as the shifted states. When the star

graph has exactly one incoming edge and several outgoing edges,

the steady states are spectrally stable if their monotonic tails are lo-

cated at the outgoing edges. Nonlinear stability of these spectrally

stable states has been an open problem up to now. In this talk,

we show that these spectrally stable states are nonlinearly unstable

because of the irreversible drift along the incoming edge towards

the vertex of the star graph. These spectrally stable states are de-

generate minimizers of the action functional with the degeneracy

due to the symmetry of the NLS equation on a balanced star graph.

When the shifted states reach the vertex as a result of the drift,

they become saddle points of the action functional, in which case

the nonlinear instability leads to destruction of the shifted states.

In addition to the rigorous mathematical results, we use numerical

37

simulations to illustrate the drift instability and destruction of the

shifted states on the balanced star graph.

Curve lengthening and shortening in strong FCH

Yuan Chen and Keith Promislow

Michigan State University


We show that nearly circular, codimension one interfaces evolving

under the L2-gradient flow of the strong scaling of the functional-

ized Cahn Hilliard gradient flow enjoy a sharp-interface limit cor-

responding to a curve shortening or regularized curve-lengthening

flow. Depending upon the distribution of mass, the interface by

absorbing or releasing mass from the far-field may expand against

interface that induces interfacial meandering or shrink. More pre-

cisely, we show that the leading order interfacial evolution can be

described by an asymptotically large but finite dimension, Galerkin

reduction of motion against curvature regularized by higher order

Willmore terms.

Observation of phase domain walls in deep watersurface gravity waves

F. Tsitoura


Amherst, Amherst, MA 01003-4515, USA

[email protected]

Experiments of nonlinear phase domain walls in weakly nonlin-

ear deep water surface gravity waves are presented. The domain

walls presented are connecting homogeneous zones of weakly non-

linear plane Stokes waves of identical amplitude and wave vector

but differences in phase. By exploiting symmetry transformations

within the framework of the nonlinear Schrodinger equation we

demonstrate the existence of exact analytical solutions represent-

ing such domain walls in the weakly nonlinear limit. The walls are

in general oblique to the direction of the wave vector and stationary

in moving reference frames. Experimental and numerical studies

confirm and visualize the findings.

1. F. Tsitoura, U. Gietz, A. Chabchoub and N. Hoffmann, Phase Domain

Walls in Weakly Nonlinear Deep Water Surface Gravity Waves, Phys.

Rev. Lett., 120 (2018), 224102.

Models for 3D Euler’s equations

Hang Yang

Euler’s equation is one of the most important mathematical prob-

lems in fluids. The global regularity of 2D Euler has been solved

by Yudovich [1] in late 60’s. Yet in 3D, due to the competition of

quadratic non-linear terms of different natures, the dynamics of Eu-

ler’s equations remains still unclear nowadays. In 2013, Hou-Luo

[2] investigated 3D Euler’s equations in the axisymmetric settings

and observed numerical blow up. Their numerical simulation has

shed significant light on the study of a few important fluids prob-

lems centered around Euler’s equations. In this talk, we will in

particular discuss theoretical developments on Boussinesq equa-

tions and SQG equations that followed thereafter.

1. V. I. Yudovich, ?Non-stationary flows of an ideal incompressible fluid?,

Zh. Vychisl. Mat. Mat. Fiz., 3:6 (1963), 1032-1066; U.S.S.R. Comput.

Math. Math. Phys., 3:6, 1407-1456 (1963).

2. T. Hou and G. Luo: Toward the finite-time blowup of the 3d axisym-

metric Euler equations: A numerical investigation, Multiscale Model.

Simul., 12(4):1722–1776 (2014).

New PT-symmetric systems with solitons: nonlinearDirac and Landau-Lifshitz equations

Igor Barashenkov

Department of Mathematics, University of Cape Town, South Africa

[email protected]

Although the spinor field in (1+1) dimensions has the right struc-

ture to model a dispersive bimodal system with gain and loss, the

plain addition of gain to one component of the field and loss to

the other one results in an unstable dispersion relation. In this

talk, we advocate a different recipe for the PT-symmetric exten-

sion of spinor models — the recipe that does not produce insta-

bility of the Dirac equation. We consider the PT-symmetric ex-

tensions of nonlinear spinor models and demonstrate a remarkable

sturdiness of spinor solitons in two dimensions. Another new class

of PT-symmetric systems comprises the Heisenberg ferromagnet

with spin torque transfer. In the vicinity of the exceptional point,

the corresponding Landau-Lifshitz equation reduces to a nonlinear

Schroedinger equation with a quadratic nonlinearity. In the sim-

plest, isotropic, case the equation has the form iψt + ψxx − ψ +ψ2 = 0. We show that this PT-symmetric Schrodinger equation

has stable soliton solutions.

1. N V Alexeeva, I V Barashenkov and A Saxena, Spinor soli-

tons and their PT-symmetric offspring, Ann Phys (2018),

https://doi.org/10.1016/j.aop.2018.11.010.

2. I V Barashenkov and A Chernyavsky, A PT-symmetric Heisenberg fer-

romagnet and a quadratic nonlinear Schrodinger equation. Submitted

for publication.

Parity-time and other symmetries in optics and pho-tonics

Demetrios Christodoulides

CREOL-The College of Optics & Photonics

University of Central Florida

Orlando, FL 32816, USA

[email protected]

The prospect of judiciously utilizing both optical gain and loss has

been recently suggested as a means to control the flow of light.

This proposition makes use of some newly developed concepts

based on non-Hermiticity and parity-time (PT) symmetry-ideas first

conceived within quantum field theories. By harnessing such no-

tions, recent works indicate that novel synthetic structures and de-

vices with counter-intuitive properties can be realized, potentially

enabling new possibilities in the field of optics and integrated pho-

tonics. Non-Hermitian degeneracies, also known as exceptional

points (EPs), have also emerged as a new paradigm for engineer-

ing the response of optical systems. In this talk, we provide an

overview of recent developments in this newly emerging field. The

use of other type symmetries in photonics will be also discussed.

38

Generation, propagation and interaction of solitarywaves in integrable versus non-integrable nonlinearlattices

Guo Deng∗, Gino Biondini and Surajit Sen

Department of Physics, University at Buffalo

Department of Mathematics, University at Buffalo

[email protected], [email protected], [email protected]

The study of lattice dynamics, i.e., the motion of a spatially discrete

system governed by a system of differential-difference equations,

is a classical subject. Of particular interest are lattices that support

the propagation of solitary waves [1]. In this talk, we will com-

pare the properties of two kinds of lattices, one integrable and one

non-integrable: the Toda lattice and the Hertzian chain. As is well

known, the Toda lattice is an integrable system and has exact soli-

ton solutions [2]. In contrast, the Hertzian chain, which has many

physical and engineering applications, is a non-integrable system

and no exact solitary-wave solutions are known [3]. Here we will

analyze the similarities and differences between the solitary waves

in these two systems, we will discuss how each of these systems

respond to a velocity perturbation, and we will compare the inter-

action dynamics of solitary wave.

1. G. Friesecke and J.A.D. Wattis, Existence theorem for solitary waves

on lattices, Commun. Math. Phys., 161, pp. 391–418 (1994)

2. M. Toda, Theory of nonlinear lattices, Springer-Verlag, (1981)

3. V.F. Nesterenko, Dynamics of heterogeneous materials, Springer-

Verlag, (2001)

SESSION 10: “Recent advances in PDEs from fluid dynamics and

other dynamical models”

Godbillon-Vey helicity in magnetohydrodynamicsand fluid dynamics

G. M. Webb∗, Q. Hu and A. Prasad

Center for Space Plasma and Aeronomic Research,

The University of Alabama in Huntsville, Huntsville AL 35805

[email protected]

S. C. Anco

Department of Mathematics, Brock University, St. Catharines

ON L2S3A1, Canada

[email protected]

The Godbillon-Vey invariant occurs in the theory of foliations. The

magnetic Godbillon-Vey invariant in magnetohydrodynamics

(MHD) for the magnetic field B occurs if the magnetic helicity

density hm = A · B = A·∇ × A = 0. This implies that the Pfaf-

fian A·dx = 0 admits an integrating factor µ, where µA·dx = dΦ

and the family of surfaces Φ(x, y, z) = const. is a foliation. The

Godbillon-Vey field η = A × B/|A|2 lies in the surface and the

Godbillon-Vey helicity density is defined as hgv = η·∇ × η. We

obtain evolution equations for the Godbillon-Vey helicity density

hgv and the Godbillon-Vey invariant Hgv =∫

V hgvd3x for a vol-

ume V moving with the fluid for the case where hm = 0. We also

write down the evolution equation for hgv for flows with hm 6= 0,

and show how hgv and hm are coupled via the shear tensor of the

background fluid flow. An application of the Godbillon-Vey helic-

ity to the nonlinear force free fields is described.

Dispersion and attenuation in a poroelastic modelfor gravity waves on an ice-covered ocean

Hua Chen∗, Robert P. Gilbert

Department of Mathematical Sciences, University of Delaware

Institute of Mechanics and Materials, Ruhr-Universitat Bochum


Philippe Guyenne

Department of Mathematical Sciences, University of Delaware

[email protected]

The recurrent interactions between ocean waves and sea ice are a

widespread feature of the polar regions, and their impact on sea-

ice dynamics and morphology has been increasingly recognized

as evidenced by the surge of research activity during the last two

decades. The rapid decline of summer ice extent that has occurred

in the Arctic Ocean over recent years has contributed to the re-

newed interest in this subject. Continuum models have recently

gained popularity to describe wave propagation in various types of

ice cover and across a wide range of length scales. In this talk, we

propose a continuum wave-ice model where the floating ice cap is

described as a homogeneous poroelastic material and the underly-

ing ocean is viewed as a slightly compressible fluid. The linear dis-

persion relation for time-harmonic wave solutions of this coupled

system is established and compared to predictions from existing

theories.

Wave model for Poiseuille flow of nematic liquid crys-tals

Geng Chen∗

Department of Mathematics, University of Kansas, Lawrence, KS 66045,

U.S.A.

[email protected]

Tao Huang

Department of Mathematics, Wayne State University, Detroit, MI, 48201,

U.S.A.

[email protected]

Weishi Liu

Department of Mathematics, University of Kansas, Lawrence, KS 66045,

U.S.A.

[email protected]

In this talk, we will discuss the global existence of Holder contin-

uous solution for the Poiseuille flow of full Ericksen-Leslie system

modeling nematic liquid crystals. Different from many previous

results which omit the kinetic energy, the full system we consider

includes a quasilinear wave equation, which may form cusp singu-

larity in general. The strong coupling on the second order parabolic

equation on the velocity of flow and the quasilinear wave equation

on the direction field of mean orientation of the liquid crystal gives

the main challenge for the global existence, which will be solved

39

by a new method. This is a joint work with Weishi Liu and Tao

Huang.

Finite time blow up of compressible Navier-Stokesequations on half space or outside a fixed ball

Dongfen Bian∗

School of Mathematics and Statistics, Beijing Institute of Technology, Bei-

jing 100081, China;

Division of Applied Mathematics, Brown University, Providence, Rhode

Island 02912.

dongfen [email protected] and [email protected]

Jinkai Li

South China Research Center for Applied Mathematics and Interdisci-

plinary Studies, South China Normal University, Zhong Shan Avenue West

55, Tianhe District, Guangzhou 510631, China

[email protected]

In this paper, we consider the initial-boundary value problem to the

compressible Navier-Stokes equations for ideal gases without heat

conduction in the half space or outside a fixed ball in RN , with

N ≥ 1. We prove that any classical solutions (ρ, u, θ), in the class

C1([0, T]; Hm(Ω)), m > [N2 ] + 2, with bounded from below ini-

tial entropy and compactly supported initial density, which allows

to touch the physical boundary, must blow-up in finite time, as long

as the initial mass is positive.

Global well-posedness of coupled parabolic systems

Wei Lian∗


China

lianwei [email protected]

Runzhang Xu


China

[email protected]

Yi Niu

School of Information Science and Engineering,

Shandong Normal University, Jinan 250001, P. R. China

yanyee [email protected]

The initial boundary value problem of a class of reaction-diffusion

systems (coupled parabolic systems) with nonlinear coupled source

terms is considered in order to classify the initial data for the global

existence, finite time blowup and long time decay of the solution.

The whole study is conducted by considering three cases accord-

ing to initial energy: low initial energy case, critical initial energy

case and high initial energy case. For the low initial energy case

and critical initial energy case the sufficient initial conditions of

global existence, long time decay and finite time blowup are given

to show a sharp-like condition. And for the high initial energy

case the possibility of both global existence and finite time blowup

is proved first, and then some sufficient initial conditions of finite

time blowup and global existence are obtained respectively.

SESSION 11: Moved to Session 26

SESSION 12: “Dispersive shocks, semiclassical limits and appli-

cations”

Universal behavior of modulationally unstable me-dia with non-zero boundary conditions

Gino Biondini


[email protected]

Sitai Li∗

University of Michigan

[email protected]

Dionyssios Mantzavinos


[email protected]

Stefano Trillo

University of Ferrara

[email protected]

This talk is divided into three parts. First, I will briefly describe the

inverse scattering transform for the focusing nonlinear Schrodinger

(NLS) equation with nonzero boundary conditions at infinity, and

then I will present the long-time asymptotics of pure soliton solu-

tions on the nonzero background. Second, I will describe in detail

the properties of the asymptotic state of the modulationally unsta-

ble solutions of the NLS equation, including the number of oscilla-

tions and the local structure of the solution near each peak, showing

in particular that in the long-time limit the solution tends to an en-

semble of classical (i.e., sech-shaped) solutions of the NLS equa-

tion. Third, I will show that a similar asymptotic state is shared

among a broad class of systems of NLS-type possessing modula-

tional instability.

Modulational instability of a plane wave in the pres-ence of localized perturbations: some experimentalresults in nonlinear fiber optics

Stphane Randoux∗, Adrien E. Kraych, Pierre Suret

Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes

et Molecules, F-59000 Lille, France

[email protected] and [email protected] and

[email protected]

Gennady El

Department of Mathematics, Physics and Electrical Engineering, Northum-

bria University, Newcastle upon Tyne, NE1 8ST, United Kingdom

[email protected]

We report an optical fiber experiment in which we study nonlinear

stage of modulational instability of a plane wave in the presence of

a localized perturbation [1]. Using a recirculating fiber loop as ex-

perimental platform, we show that the initial perturbation evolves

into expanding nonlinear oscillatory structure exhibiting some uni-

versal characteristics that agree with theoretical predictions based

40

on integrability properties of the focusing nonlinear Schrodinger

equation [2]. Our experimental results demonstrate persistence of

the universal evolution scenario, even in the presence of small dis-

sipation and noise in an experimental system that is not rigorously

of an integrable nature.

1. A. E. Kraych, P. Suret, G. El, S. Randoux Nonlinear evolution of the

locally induced modulational instability in fiber optics Accepted for

publication in Phys. Rev. Lett. (2019) [arXiv:1805.05074]

2. G. Biondini and D. Mantzavinos, Universal nature of the nonlinear

stage of modulational instability Phys. rev. Lett. 116, 043902 (2016)

Towards kinetic equation for soliton and breathergases for the focusing nonlinear Schroedinger equa-tion

Alexander Tovbis∗

Department of Mathematics, University of Central Florida, Orlando, FL,

USA

[email protected]

Gennady El

Department of Mathematics, Physics and Electrical Engineering

Northumbria University Newcastle, UK

[email protected]

inetic equation for a soliton gas for the Korteweg - de Vries equa-

tion was first proposed by V. Zakharov and later derived by G.

El using the thermodynamic limit of the KdV-Whitham equations.

Later, G. El and A. Kamchatnov proposed kinetic equation for the

soliton gas for the focusing Nonlinear Schroedinger (fNLS) equa-

tion using physical reasoning.

In this talk, we consider the large N limit of nonlinear N-phase

wave solutions to the fNLS equation subject to a certain scaling of

the corresponding bands and gaps. In this limit, we obtain integral

equations for the scaled wavenumbers and frequences and, as a

consequence, derive the kinetic equation for soliton/breather gases,

which takes into account soliton-soliton and soliton-background

interactions. Our approach can be used to derive kinetic equation

for the soliton gas on the background of any finite gap solution.

This work is still in progress.

Wave-mean flow interactions in dispersive hydrody-namics

Gennady El and Thibault Congy




Mark Hoefer

Department of Applied Mathematics, University of Colorado Boulder, USA

[email protected]

The interaction of waves with a mean flow is a fundamental and

longstanding problem in fluid mechanics. The key to the study of

such an interaction is the scale separation, whereby the length and

time scales of the waves are much shorter than those of the mean

flow. The wave-mean flow interaction has been extensively studied

for the cases when the mean flow is prescribed externally—as a

stationary or time-dependent current (a “potential barrier”).

In this talk, I will describe a new type of the wave-mean flow in-

teraction whereby a short-scale wave projectile—a soliton or a lin-

ear wave packet—is incident on the evolving large-scale nonlin-

ear dispersive hydrodynamic state: a rarefaction wave or a disper-

sive shock wave (DSW). Modulation equations are derived for the

coupling between the soliton (wavepacket) and the mean flow in

the nonlinear dispersive hydrodynamic state. These equations ad-

mit particular classes of solutions that describe the transmission or

trapping of the wave projectile by an unsteady hydrodynamic state.

Two adiabatic invariants of motion are identified in both cases that

determine the transmission, trapping conditions and show that soli-

tons (wavepackets) incident upon smooth expansion waves or com-

pressive, rapidly oscillating DSWs exhibit so-called hydrodynamic

reciprocity. The latter is confirmed in a laboratory fluids experi-

ment for soliton-hydrodynamic state interactions.

The developed theory is general and can be applied to integrable

and non-integrable nonlinear dispersive wave equations in various

physical contexts including nonlinear optics and cold atom physics.

As concrete examples we consider the Korteweg-de Vries and the

viscous fluid conduit equations. The talk is based on recent papers

[1, 2]

1. M. D. Maiden, D. V. Anderson, N. A. Franco, G. A. El, & M. A. Hoe-

fer, Solitonic dispersive hydrodynamics: theory and observation. Phys.

Rev. Lett., 120 (2018) 144101.

2. T. Congy, G.A. El and M.A. Hoefer, Interaction of linear

modulated waves with unsteady dispersive hydrodynamic states,

arXiv:1812.06593.

The universality of the semi-classical sine-Gordonequation at the gradient catastrophe

Bingying Lu∗

Institute of Mathematics, Academia Sinica, Taipei

[email protected]

Peter Miller

Department of Mathematics, University of Michigan in Ann Arbor

[email protected]

We study the semi-classical sine-Gordon equation with pure im-

pulse initial data below the threshold of rotation: ǫ2utt − ǫ2uxx +sin(u) = 0, u(x, 0) ≡ 0, ǫut(x, 0) = G(x) ≤ 0, and |G(0)| <2. A dispersively-regularized shock forms in finite time. Using

Riemann–Hilbert analysis, we rigorously studied the asymptotics

near a certain gradient catastrophe. In accordance with a con-

jecture made by Dubrovin et. al., the asymptotics in the this re-

gion is universally (insensitive to initial condition) described by

the tritronquee solution to the Painleve-I equation. Furthermore,

we are able to universally characterize the shapes of the spike-like

local structures (rogue wave on periodic background) on top of the

poles of the tritronquee solution.

Semiclassical Lax spectrum of Zakharov-Shabat sys-tems with periodic potentials

Jeffrey Oregero and Gino Biondini

University at Buffalo, SUNY


41

The semiclassical limit of the focusing nonlinear Schrodinger equa-

tion with periodic initial conditions is studied analytically and nu-

merically. First, through a comprehensive set of careful numerical

simulations, it is demonstrated that solutions arising from many

different initial conditions share the same qualitative features, which

also coincide with those of solutions arising from localized initial

conditions. Rigorous bounds on the location of eigenvalues of the

associated scattering problem are derived and it is shown that the

spectrum is a subset of the real and imaginary axes of the spectral

variable in the semiclassical limit. Finally, by employing a suitable

Wentzel-Kramers-Brillouin expansion for the scattering eigenfunc-

tions, asymptotic formulae are derived for the number and location

of the bands and gaps in the spectrum, as well as for the relative

band and gap widths.

Nonlinear Schrodinger equations and the universaldescription of dispersive shock wave structure

Thibault Congy and Gennady El




Mark Hoefer


University of Colorado, Boulder, USA

[email protected]

Michael Shearer


North Carolina State University, Raleigh, USA

[email protected]

A dispersive shock wave (DSW) is an expanding, modulated non-

linear wavetrain that connects two disparate hydrodynamic states,

and can be viewed as a dispersive counterpart to the dissipative,

classical shock. DSWs have raised a lot of interest in the recent

years, due to the growing recognition of their fundamental na-

ture and ubiquity in physical applications, examples being found in

oceanography, meteorology, geophysics, nonlinear optics, plasma

physics and condensed matter physics. Although well-established

methods, such as the Whitham modulation theory, have proved

particularly effective for the determination of DSW solutions of

certain nonlinear wave equations, a universal description of these

objects is still lacking.

The nonlinear Schrodinger (NLS) equation and the Whitham mod-

ulation equations both describe slowly varying, locally periodic

nonlinear wavetrains, albeit in differing amplitude-frequency do-

mains. Taking advantage of the overlapping asymptotic regime

that applies to both the NLS and Whitham modulation descrip-

tions, we developed a universal analytical description of DSWs

generated in Riemann problems for a broad class of integrable and

non-integrable nonlinear dispersive equations [1]. The proposed

method extends DSW fitting theory that prescribes the motion of

a DSW’s edges into the DSW’s interior, that is, this work reveals

the DSW structure. I will present this new method and illustrate its

efficacy by considering various physically relevant examples.

1. T. Congy, G. A. El, M. A. Hoefer and M. Shearer, Nonlinear

Schrodinger equations and the universal description of dispersive shock

wave structure. Stud. Appl. Math. 2018;1–28.

Nonlinear interactions between solitons and disper-sive shocks in focusing media

Gino Biondini and Jonathan Lottes

University at Buffalo


Nonlinear interactions in focusing media between traveling soli-

tons and the dispersive shocks produced by an initial discontinuity

are studied numerically and analytically using the one-dimensional

nonlinear Schrodinger equation.

Evolution of broad initial profiles—solitary wave fis-sion and solitary wave phase shift

Michelle Maiden and Mark A. Hoefer∗

Department of Applied Mathematics, University of Colorado, Boulder,

USA


Gennady El

Department of Mathematics, Physics and Electrical Engineering, Northum-

bria University, Newcastle, UK

[email protected]

The temporal evolution of a large, localized initial disturbance is

considered in the context of scalar, dispersive nonlinear equations

in the dispersive hydrodynamic regime. Modulation theory for

solitary wave fission in long time evolution of the broad distur-

bance is developed and certain universal properties of the dynam-

ics are identified. The theory, asymptotically valid for the gener-

ation of a large number of solitary waves, yields predictions for

the number of solitary waves and their amplitude distribution. The

number of solitary waves universally depends linearly on the initial

profile’s width. The normalized cumulative amplitude distribution

function is independent of the initial profile’s width. These prop-

erties are verified quantitatively in experiments involving the inter-

facial dynamics of two miscible Stokes fluids with high viscosity

contrast. The number of observed solitary waves is consistently

within 1-2 waves of the prediction, and the amplitude distribution

shows remarkable agreement. Additionally, using Whitham mod-

ulation theory, a universal phase shift formula for the interaction

of a solitary wave that is initially well-separated from a broad dis-

turbance is presented and shown to agree with numerical compu-

tations. All of the modulation theory predictions are agnostic to

integrable structure of the underlying PDE model.

Dispersive shocks dynamics of phase diagrams

Costanza Benassi and Antonio Moro



[email protected]

The theory of Nonlinear Conservation Laws arises as a universal

paradigm for the description of phase transitions, cooperative and

catastrophic behaviours in many body systems at the crossroad of

integrable systems, statistical mechanics and random matrix the-

ory.

In classical magnetic and fluid models the free energy can be ob-

tained as a solution of a viscous integrable hierarchy of PDEs and

42

phase transitions are associated to the classical shock dynamics of

order parameters in the space of thermodynamics variables.

We show that for Hermitian Matrix Models, where the partition

function is given by a tau function of the Toda hierarchy, phase

transitions are associated to the dispersive shock dynamics of the

continuum limit of the Toda hierarchy.

SESSION 13: “Recent advances in numerical methods of pdes and

applications in life science, material science”

A second-order fully-discrete linear energy stablescheme for a binary compressible viscous fluid model

Xueping Zhao

Department of Mathematics, University of South Carolina, Columbia, SC

29208, USA

[email protected]

Qi Wang

Department of Mathematics, University of South Carolina, Columbia, SC

29208, USA

and Beijing Computational Science Research Center, Beijing 100193, China

[email protected]

We present a linear, second order fully discrete numerical scheme

on a staggered grid for a thermodynamically consistent hydrody-

namic phase field model of binary compressible fluid flow mix-

tures derived from the generalized Onsager Principle. The hydro-

dynamic model not only possesses the variational structure, but

also warrants the mass, linear momentum conservation as well as

energy dissipation. We first reformulate the model in an equivalent

form using the energy quadratization method and then discretize

the reformulated model to obtain a semi-discrete partial differen-

tial equation system using the Crank-Nicolson method in time. The

numerical scheme so derived preserves the mass conservation and

energy dissipation law at the semi-discrete level. Then, we dis-

cretize the semi-discrete PDE system on a staggered grid in space

to arrive at a fully discrete scheme using the 2nd order finite differ-

ence method, which respects a discrete energy dissipation law. We

prove the unique solvability of the linear system resulting from the

fully discrete scheme. Mesh refinements and two numerical exam-

ples on phase separation due to the spinodal decomposition in two

polymeric fluids and interface evolution in the gas-liquid mixture

are presented to show the convergence property and the usefulness

of the new scheme in applications.

Efficient schemes with unconditionally energy sta-bilities for anisotropic phase field models: S-IEQ andS-SAV

Xiaofeng Yang

1523 Greene Street, Columbia, SC, 29208

[email protected]

We consider numerical approximations for anisotropic phase field

models, by taking the anisotropic Cahn-Hilliard/Allen-Cahn equa-

tions with their applications to the faceted pyramids on nanoscale

crystal surfaces and the dendritic crystal growth problems, as spe-

cial examples. The main challenge of constructing numerical

schemes with unconditional energy stabilities for these type of mod-

els is how to design proper temporal discretizations for the nonlin-

ear terms with the strong anisotropy. We combine the recently de-

veloped IEQ/SAV approach with the stabilization technique, where

some linear stabilization terms are added, which are shown to be

crucial to remove the oscillations caused by the anisotropic coeffi-

cients, numerically. The novelty of the proposed schemes is that all

nonlinear terms can be treated semi-explicitly, and one only needs

to solve some coupled/decoupled, but linear equations at each time

step. We further prove the unconditional energy stabilities rigor-

ously, and present various 2D and 3D numerical simulations to

demonstrate the stability and accuracy.

Efficient and stable numerical methods for a class ofstiff reaction-diffusion systems with free boundaries

Shuang Liu and Xinfeng Liu∗

Department of Mathematics, University of South Carolina, Columbia, USA


The systems of reaction-diffusion equations coupled with moving

boundaries defined by Stefan condition have been widely used to

describe the dynamics of spreading population. There are several

numerical difficulties to efficiently handle such systems. Firstly

extremely small time steps are usually needed due to the stiff-

ness of the system. Secondly it is always difficult to efficiently

and accurately handle the moving boundaries. To overcome these

difficulties, we first transform the one-dimensional problem with

moving boundaries into a system with fixed computational domain,

and then introduce four different temporal schemes: Runge-Kutta,

Crank-Nicolsn, implicit integration factor (IIF) and Krylov IIF, for

handling such stiff systems. Numerical examples are examined to

illustrate the efficiency, accuracy and consistency for different ap-

proaches, and it can be shown that Krylov IIF is superior to other

approaches in terms of stability and efficiency by direct compari-

son.

1. Du, Y., and Lin, Z. (2010). Spreading-vanishing dichotomy in the dif-

fusive logistic model with a free boundary. SIAM Journal on Mathe-

matical Analysis, 42(1), 377-405.

2. Nie, Q., Zhang, Y.-T. and Zhao, R. (2006). Efficient semi-implicit

schemes for stiff systems. Journal of Computational Physics, 214, 521-

537.

3. Chen, S. Q. and Zhang, Y. T. (2011). Krylov implicit integration fac-

tor methods for spatial discretization on high dimensional unstructured

meshes: application to discontinuous Galerkin methods. Journal of

Computational Physics, 230(11), 4336-4352.

4. Piqueras, M.-A., Company, R., Jodar, L. (2017). A front-fixing numer-

ical method for a free boundary nonlinear diffusion logistic population

model. Journal of Computational and Applied Mathematics, 309, 473-

481.

Approximating nonlinear reaction-diffusionproblems with multiple solutions

Tom Lewis∗The University of North Carolina at Greensboro

43

[email protected]

In this paper we introduce the class of positone boundary value

problems and the analytic issues that must be addressed when us-

ing an approximation method. In particular, we will consider the

problem of approximating a function u that solves the semilinear

elliptic boundary value problem

−∆u = λ f (u) in Ω

with u > 0 in Ω and u = 0 on ∂Ω, where λ > 0 is a constant; Ω

is an open, bounded, convex domain; and f is a postitone operator

with sublinear growth, i.e., f satisfies the three conditions f (w) >

0 for all w ≥ 0, f is nondecreasing, and limw→∞f (w)

w = 0.

Such problems arise in mathematical biology and the theory of

nonlinear heat generation. Under certain conditions, the problems

may have multiple positive solutions or even nonexistence of a pos-

itive solution. We will discuss new analytic techniques for proving

admissibility, stability, and convergence of finite difference meth-

ods for approximating sublinear positone problems. The admis-

sibility and stability results will be based on adapting the method

of sub- and supersolutions typically used to analyze the underly-

ing PDEs. Since most known approximation methods for posi-

tone boundary value problems rely upon shooting techniques, they

are restricted to one-dimensional problems and/or radial solutions.

The new tools will serve as a foundation for approximating posi-

tone boundary value problems in higher dimensions and on more

general domains.

A hybrid model for simulating sprouting angiogene-sis in biofabrication

Yi Sun and Qi Wang

Department of Mathematics, University of South Carolina


We present a 2D hybrid model to study sprouting angiogenesis

of multicellular aggregates during vascularization in biofabrica-

tion. This model is developed to describe and predict the time evo-

lution of angiogenic sprouting from endothelial spheroids during

tissue or organ maturation in a novel biofabrication technology–

bioprinting. Here we employ typically coarse-grained continuum

models (reaction-diffusion systems) to describe the dynamics of

vascular-endothelial-growth-factors, a mechanical model for the

extra-cellular matrix based on the finite element method and cou-

ple a cellular Potts model to describe the cellular dynamics. The

model can reproduce sprouting from endothelial spheroids and net-

work formation from individual cells.

A parallel approach to kinetic viscoelastic modelling

Paula Vasquez and Erik Palmer

411 LeConte College

Columbia, SC 29208, United States


Viscoelastic materials are characterized by the coupling of micro-

structural changes to macroscale deformations. We present an elas-

tic dumbbell model that leverages the parallel processing power of

High Performance Computing (HPC) Graphics Processing Units

(GPUs) to create a unique micro-macro scale driven design which

incorporates the nonlinear nature of viscoelastic responses as well

as the stochastic processes which describe the breaking and reform-

ing of entanglements in the underlying microscopic network. The

model allows a full reconstruction of the microstructure-flow cou-

pling thereby creating a platform with the ability to investigate how

microscopic changes affect macroscopic responses. In this talk we

focus on oscillatory flow and show both evolution of stress and

species distribution as functions of frequency and strain.

SESSION 14: “Nonlinear kinetic self-organized plasma dynamics

driven by coherent, intense electromagnetic fields”

Spectrally accurate methods for kineticelectron plasma wave dynamics

Jon Wilkening and Rockford Sison


University of California

Berkeley, CA 94720-3840


Bedros Afeyan

Polymath Research Inc.

827 Bonde Court

Pleasanton, CA 94566

[email protected]

We present two numerical methods for computing solutions of the

Vlasov-Fokker-Planck-Poisson equations that are spectrally accu-

rate in all three variables (time, space and velocity). The first is a

Chebyshev collocation method for solving the Volterra equation for

the space-time evolution of the plasma density for the linearized,

collisionless case. This is then used to efficiently represent the

velocity distribution function in Case-van Kampen normal modes,

building on the work of Li and Spies. The second is an arbitrary-

order exponential time differencing scheme that makes use of the

Duhamel principle to fold in the effects of collisions and nonlinear-

ity. We investigate the emergence of a continuous spectrum in the

collisionless limit and the embedding of Landau’s poles in this gen-

eral setting. We resolve the effects of filamentation, phase mixing,

and Landau and collisional damping to arbitrary order of accuracy,

focusing on echoes and trapping phenomena.

Improving the performance of plasma kinetic simu-lations by iteratively learned phase space tiling:variational constrained optimization meetmachine learning

B. Afeyan∗ and R.D. Sydora

Polymath Research Inc., Pleasanton, California


We describe a general method of constrained optimization to or-

ganically change the equations being solved, given prior knowl-

edge on nearby problems (differing via parameter choices, res-

olution, modeling simplifications, etc.). The general method is

called NSCAR: Nearby Skeleton Constrained Accelerated Recom-

puting. We then specialize to plasma kinetic equations and focus

44

on two new methods which improve the performance of PIC codes

and Vlasov Codes, called BARS and APOSTLE. BARS stands for

Bidirectional Adaptive Refinement Scheme and APOSTLE stands

for Adaptive Particle Orbit Sampling Technique for Lagrangian

Evolution. We demonstrate the advantages of these techniques by

applying them to the learned, sparse (non-uniformly sampled phase

space) representation of accurate solutions of nonlinear plasma

waves in the kinetic/trapping regime. as well as for KEEN waves.

Extensions of this method to the multidimensional setting where

magnetic fields, the Weibel instability and nonlinear plasma waves

interact inexorably will also be described.

Work supported by a grant from AFOSR and the DOE FES-NNSA

Joint Program in HEDLP.

Nonlinear instabilities due to drifting species andmagnetic fields in high energy density plasmas

B. A. Shadwick∗ and Alexander Stamm

Department of Physics and Astronomy

University of Nebraska–Lincoln

sh [email protected] [email protected]

Relative drifts between particles species are fundamental driving

forces behind many plasma instabilities. For example, the Bune-

man instability arises due to an election-ions drift. We study the

nonlinear evolution of this processes in the presence of externally

imposed transverse magnetic fields. Our results are primarily drawn

from simulations using both Vlasov–Maxwell and macro-particle

methods. We compare electrostatically driven modes to full elec-

tromagnetic treatments. Ion to electron mass ratios of 1, 10 and

100 will be included.

Work supported by the DOE NNSA-FES Joint program in HEDLP.

Geostrophic turbulence and the formation of largescale structure

Edgar Knobloch

Department of Physics, University of California, Berkeley CA 94720

[email protected]

Low Rossby number convection is studied using an asymptotically

reduced system of equations valid in the limit of strong rotation

[1]. The equations describe four regimes as the Rayleigh number

Ra increases: a disordered cellular regime near threshold, a regime

of weakly interacting convective Taylor columns at larger Ra, fol-

lowed for yet larger Ra by a breakdown of the convective Taylor

columns into a disordered plume regime characterized by reduced

heat transport efficiency, and finally by a new type of turbulence

called geostrophic turbulence. Properties of this state will be de-

scribed and illustrated using direct numerical simulations of the

reduced equations. These simulations reveal that geostrophic tur-

bulence is unstable to the formation of large scale barotropic vor-

tices [1] or jets [2], via a process known as spectral condensation

[3]. The details of this process are quantified and its implications

explored. The results are corroborated by direct numerical simula-

tions of the Navier-Stokes equations [4]–[6].

1. K. Julien, A. M. Rubio, I. Grooms and E. Knobloch, Statistical and

physical balances in low Rossby number Rayleigh-Bnard convection,

Geophys. Astrophys. Fluid Dyn., 106 (2012), 392–428.

2. K. Julien, E. Knobloch and M. Plumley, Impact of domain anisotropy

on the inverse cascade in geostrophic turbulent convection, J. Fluid

Mech., 837, R4 (2018).

3. M. Chertkov, C. Connaughton, I. Kolokolov, V. Lebedev, Dynamics of

energy condensation in two-dimensional turbulence, Phys. Rev. Lett.,

99 (2007), 084501.

4. C. Guervilly, D. W. Hughes and C. A. Jones, Large-scale vortices

in rapidly rotating Rayleigh-Benard convection, J. Fluid Mech., 758

(2014), 407–435.

5. B. Favier, L. J. Silvers and M. R. E. Proctor, Inverse cascade and sym-

metry breaking in rapidly rotating Boussinesq convection, Phys. Fluids,

26 (2014), 096605.

6. B. Favier, C. Guervilly and E. Knobloch, Subcritical turbulent conden-

sate in rapidly rotating Rayleigh-Benard convection, J. Fluid Mech., in

press.

7. C. Guervilly and D. W. Hughes, Jets and large-scale vortices in rotating

RayleighBenard convection, Phys. Rev. Fluids, 2 (2017), 113503.

Impact of cyclotron harmonic wave instabilities onstability of self-organized nonlinear kinetic plasmastructures

R.D. Sydora∗ and B. Afeyan

Polymath Research Inc., Pleasanton, California


B.A. Shadwick

University of Nebraska, Lincoln

[email protected]

Crossing, intense laser beamsin high energy density plasmas lead

to the generation of nonlinear kinetic electron plasma waves (NL-

EPW). The usefulness of such plasma structures depends on their

long-time stability. Externally imposed magnetic fields is one

method to confine multidimensional NL-EPW both in the trapping

regime and when vortex merger is prevalent. However, magnetic

fields introduce cyclotron harmonic waves that may be driven un-

stable (Harris instability) by velocity space anisotropies formed

through different plasma heating processes along and across the

magnetic field. The Harris instability causes transverse electro-

static perturbations that leads to the escape of trapped particles in

NL-EPW, contributing to their rapid dissipation. In this work we

assess the importance of the Harris instability on NL-EPW and find

regimes where its impact is minimized. These studies employ self-

consistent particle simulations and the use of reconstructed parti-

cle orbit dynamics from the self-consistent electric and magnetic

fields.

Work supported by a grant from AFOSR and by the DOE NNSA-

FES joint program in HEDLP.

Internal wave energy flux from density perturbations

Frank M. Lee2,a and Michael R. Allshouse1,b

aDepartment of Physics and Astronomy, University of Nebraska-Lincoln,

Lincoln, NE 68508, USAbDepartment of Mechanical and Industrial Engineering, Northeastern Uni-

versity, Boston, MA 02115, USA

45


Harry L. Swinney1

1Center for Nonlinear Dynamics and Department of Physics, University

of Texas at Austin, Austin, TX 78712, USA

[email protected]

Philip J. Morrison2

2Institute for Fusion Studies and Department of Physics, University of

Texas at Austin, Austin, TX 78712, USA

[email protected]

Internal gravity waves arise from buoyancy restoration forces within

a fluid whose density varies with height. The energy of such waves

is of interest due to its significant presence in the energy budget

of the ocean, and affects mixing and the thermohaline circulation.

The energy flux of linear internal waves requires the pressure per-

turbation field, which is at present not an easily measurable quan-

tity in either laboratory or field observations.

We present a method using Green’s functions that gives the in-

stantaneous energy flux solely from the density perturbation field,

which is measurable in the laboratory using synthetic schlieren

[1, 2]. We use simulations of the Navier-Stokes equations to verify

the method, which show good agreement, and check the usability

of the method with laboratory data. We give arguments for the er-

ror scaling due to nonlinearity. Using the solution to the linear sys-

tem as a baseline, it may be possible to use a perturbative method

to find corrections to the Green’s function and the energy flux for

weakly nonlinear waves in future studies.

1. B. R. Sutherland, S. B. Dalziel, G. O. Hughes and P. F. Linden, Visu-

alization and measurement of internal waves by ‘synthetic schlieren.’

Part 1. Vertically oscillating cylinder., J. Fluid Mech., 390 (1999), 93-

126.

2. S. B. Dalziel, G. O. Hughes and B. R. Sutherland, Whole-field density

measurements by ‘synthetic schlieren’, Exp. Fluids, 28 (2000), 322-

335.

SESSION 15: “Waves in Topological Materials”

Wave-packet dynamics in slowly modulated photonicgraphene

Peng Xie and Yi Zhu

Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Bei-

jing 100084, China


Mathematical analysis on electromagnetic waves in photonic

graphene, a photonic topological material which has a honeycomb

structure, is one of the most important current research topics [1].

By modulating the honeycomb structure, numerous topological phe-

nomena have been observed recently [2]. The electromagnetic

waves in such a media are generally described by the 2-dimensional

wave equation. It has been shown that the corresponding ellip-

tic operator with a honeycomb material weight has Dirac points in

its dispersion surfaces [3]. In this article, we study the time evo-

lution of the wave-packets spectrally concentrated at such Dirac

points in a modulated honeycomb material weight. We prove that

such wave-packet dynamics is governed by the Dirac equation with

a varying mass in a large but finite time. Our analysis provides

mathematical insights to those topological phenomena in photonic

graphene.

1. Mark J. Ablowitz, Christopher W. Curtis and Yi Zhu, On tight-binding

approximations in optical lattices, Studies in Applied Mathematics,

129 (4) 2012, 362-388.

2. Charles L. Fefferman and Michael I. Weinstein, Honeycomb lattice po-

tentials and Dirac points, Journal of the American Mathematical Soci-

ety, 25(4) 2012, 1169-1220.

3. James P. Lee-Thorp, Michael I. Weinstein and Yi Zhu, Elliptic oper-

ators with honeycomb symmetry: Dirac points, Edge States and Ap-

plications to Photonic Graphene, Archive for Rational Mechanics and

Analysis, 2018, 1-63.

Unfitted Nitsche’s method for computing edge modesin photonic graphene

Hailong Guo

School of Mathematics and Statistics, The University of Melbourne

[email protected]

Xu Yang

Department of Mathematics, University of California, Santa Barbara

[email protected]

Yi Zhu

Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University

[email protected]

Photonic graphene, a photonic crystal with honeycomb structures,

has been intensively studied in both theoretical and applied fields.

Similar to graphene which admits Dirac Fermions and topological

edge states, photonic graphene supports novel and subtle propagat-

ing modes (edge modes) of electromagnetic waves. These modes

have wide applications in many optical systems. In this paper, we

propose a new unfitted Nitsche’s method to computing edge modes

in photonic graphene with some defect. The unique feather of the

methods is that it can arbitrary handle high contrast with geometric

unfitted meshes. We establish the optimal convergence of meth-

ods. Numerical examples are presented to validate the theoretical

results and to numerically verify the existence of the edge modes.

1. H. Guo, X. Yang, Y. Zhu, Bloch theory-based gradient recovery method

for computing topological edge modes in photonic graphene, Journal

of Computational Physics, 95(2019), 403–420.

2. H. Guo, X. Yang, Y. Zhu, Unfitted Nitsche’e method for computing

edge modes in photonic graphene, 2019, preprint.

Topologically protected edge modes in longitudinallydriven waveguides

Mark Ablowitz and Justin Cole∗


University of Colorado, Boulder

[email protected]

A tight-binding approximation is developed for deep longitudinally

driven photonic lattices. The physical system considered is that of

46

a laser-etched waveguide array which is helically-varying in the di-

rection of propagation. The lattice is decomposed into sublattices

each of which are allowed move independently of one another. The

linear Floquet bands are constructed for various rotation patterns

such as: different radii, different frequency, phase offset and quasi

one-dimensional motion. Bulk spectral bands with nonzero Chern

number are calculated and found to support topologically protected

edge wave envelopes which can propagate scatter-free around de-

fects. Finally, nonlinear soliton modes are found to propagate uni-

directionally and scatter-free at lattice defects.

Frozen Gaussian approximation for the Dirac equa-tion in semi-classical regime

Lihui Chai

Sun Yat-sen University, Guangzhou, China

[email protected]

Emmanuel Lorin∗

Carleton University, Ottawa, Canada

[email protected]

Xu Yang

University of California, Santa Barbabra, US

[email protected]

This work is devoted to the derivation and analysis of the Frozen

Gaussian Approximation (FGA) for the Dirac equation in the semi-

classical regime. Unlike the strictly hyperbolic system studied in

[1], the Dirac equation possesses eigenfunction spaces of multi-

plicity two, which demands more delicate expansions for deriv-

ing the amplitude equations in FGA. Moreover, we prove that the

nonrelativistic limit of the FGA for the Dirac equation is the FGA

of the Schrodinger equation, which shows that the nonrelativis-

tic limit is asymptotically preserved after one applies FGA as the

semiclassical approximation.

1. J. Lu and X. Yang, Convergence of frozen Gaussian approximation for

high frequency wave propagation, Comm. Pure Appl. Math., 65 (2012),

759-789.

Edge states in near-honeycomb structures

Alexis Drouot

Mathematics Department, Columbia University

[email protected]

I will study aspects of wave propagation in a continuous honey-

comb structure with a line defect. In a perturbative regime, I will

give a full description of edge states (time harmonic waves prop-

agating along the line defect). This shows that all possible edge

states are adiabatic combinations of Dirac point Bloch modes. This

improves work of Fefferman, Lee-Thorp, Weinstein and Zhu who

constructed edge states of this form.

I will then extend the result outside the perturbative regime. This

amounts to prove topological protection of edge states, a result

known as the bulk-edge correspondence.

1. A. Drouot, Characterization of edge states in perturbed honeycomb

structures. Preprint, arXiv:1811.08218.

2. A. Drouot, The bulk-edge correspondence for continuous honeycomb

lattices. Preprint available on demand.

3. C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein, Edge states in

honeycomb structures. Ann. PDE 2(2016), no. 2, Art. 12, 80 pp.

4. J. P. Lee-Thorp, M. I. Weinstein and Y. Zhu, Elliptic operators with

honeycomb symmetry: Dirac points, Edge States and Applications to

Photonic Graphene. To appear in Archives for Rational Mechanics and

Analysis; preprint arXiv:1710.03389.

Embedded eigenvalues and Fano resonance for metal-lic structures with small holes

Junshan Lin ∗

Department of Mathematics and Statistics, Auburn University, Auburn,

AL 36849

[email protected]

Stephen Shipman

Department of Mathematics, Louisiana State University, Baton Rouge, LA

70803

[email protected]

Hai Zhang

Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong

Kong

[email protected]

Fano resonance, which was initially discovered in quantum me-

chanics by Ugo Fano, has been extensively explored in photonics

since the past decade due to its unique resonant feature of a sharp

transition from total transmission to total reflection. Mathemati-

cally, Fano resonance is related to certain eigenvalues embedded in

the continuum spectrum of the underlying differential operator. For

photonic structures, the quantitative studies of embedded eigenval-

ues mostly rely on numerical approaches. In this talk, based on

layer potential technique and asymptotic analysis, I will present

quantitative analysis of embedded eigenvalues and their perturba-

tion as resonances for a periodic array of subwavelength metallic

structure. From a quantitative analysis of the wave field for the

scattering problem, a rigorous proof of Fano resonance will be

given. In addition, the field enhancement at Fano resonance fre-

quencies will be discussed.

Linear and nonlinear waves in honeycomb photonicmaterials

Yi Zhu

Tsinghua University

[email protected]

The past few years have witnessed an explosion of researches on

topological phenomena in different fields. One striking featur is

the existence of wave motions that are immune to defects and dis-

orders. In this talk, I will introduce our recent progresses on the

analysis of such novel and subtle wave dynamics in topological

photonic materials. Specifically, we prove the existence of Dirac

points in the honeycomb lattices and the existence of topological

edge modes by introducing a line defect. We then derive the corre-

sponding envelope equations to understand the subtle topological

wave dynamics. Both linear and nonlinear wave dynamics are in-

vestigated.

47

Computing edge spectrum in the presence of disor-der without spectral pollution

Kyle Thicke, Alexander Watson∗, and Jianfeng Lu

Mathematics Department, Duke University, NC


Edge states, electronic states localized at the edge of a two-dimen-

sional material, are defined mathematically as bound states of a

semi-infinite edge Hamiltonian. Accurate numerical computation

of such states is complicated by the fact that computing using ar-

bitrarily large finite truncations of the Hamiltonian yields spurious

edge states localized at the truncation. We present a method which

avoids this problem by properly accounting for the effect of the in-

finite bulk structure. Using this method we are able to probe com-

putationally the robustness of edge states of a graphene-like struc-

ture, modeled both by a continuum PDE and in the tight-binding

limit, to a broad class of perturbations. Robustness of such states

is of interest for applications because of their potential utility for

wave-guiding [1].

1. K. Thicke, A. B. Watson, and J. Lu, Computation of bound states of

semi-infinite matrix Hamiltonians with applications to edge states of

two-dimensional materials,

https://arxiv.org/abs/1810.07082 (2018)

SESSION 16: “Existence and stability of peaked waves in nonlin-

ear evolution equations”

Stability of standing waves for a nonlinear Klein-Gordon equation with delta potentials

Elek Csobo

Delft University of Technology

[email protected]

Francois Genoud

Ecole Polytechnique Federale de Lausanne

[email protected]

Masahito Ohta

Tokyo University of Science

[email protected]

Julien Royer

Universite Paul Sabatier

[email protected]

We study the orbital stability of standing wave solutions of a one-

dimensional nonlinear Klein-Gordon equation with Dirac poten-

tials. The general theory to study orbital stability of Hamiltonian

systems was initiated by the seminal papers of Grillakis, Shatah,

and Strauss [1, 2], newly revisited by De Bievre et. al. in [3]. I

present the Hamiltonian structure of the above system and the or-

bital stability of the standing wave solutions of the equation. A

major difficulty is to determine the number of negative eigenvalues

of the linearized operator around the stationary solution, which we

overcome by a perturbation argument.

1. M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves

in the presence of symmetry, I. J. of Funct. Anal. 74 (1987), no. 1,

160–197.

2. M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves

in the presence of symmetry, II. J. of Funct. Anal. 94 (1990), no. 2,

308–348.

3. S. De Bievre, F. Genoud, S. Rota-Nodari, Orbital stability: analysis

meets geometry, in Nonlinear optical and atomic systems, 2146 (2015),

147–273.

Convergence of Petviashvili’s method near periodicwaves in the fractional Korteweg-deVries equation

Uyen Le and Dmitry E. Pelinovsky

McMaster University


The fractional Korteweg-De Vries equation is a nonlinear partial

differential equation which has several applications in fluid dy-

namics. There are many iterative methods to approximate the soli-

tary wave solution of this equation. One robust iterative scheme

is the classical Petviashvili’s method. However, it has been nu-

merically found that the method may not converge in the case of

periodic waves. In this presentation we will explain the failure

of the classical Petviashvili’s method in approximating periodic

waves in the fractional KdV equation from the spectrum of the

generalized eigenvalue problem. We will also show that by modi-

fying the method with a mean value shift, we achieve unconditional

convergence for the Petviashvili’s method.

Convexity of Whitham’s highest cusped wave

Alberto Enciso and Bruno Vergara∗

Institute of Mathematical Sciences-ICMAT


Javier Gomez-Serrano

Princeton University

[email protected]

Whitham’s model [3] of shallow water waves is a non-local dis-

persive equation that features travelling wave solutions as well as

singularities. In this talk we will discuss a conjecture of Ehrnstrom

and Wahlen [1] on the profile of solutions of extreme form to this

equation and see that there exists a highest, cusped and periodic

solution, which is convex between consecutive peaks [2].

1. M. Ehrnstrom and E. Wahlen, On Whitham’s conjecture of a highest

cusped wave for a nonlocal shallow water wave equation, Ann. Inst. H.

Poincare Anal. Non. Lineaire, (in press), arXiv:1602.05384.

2. A. Enciso, J. Gomez-Serrano and B. Vergara, Convexity of Whitham’s

highest cusped wave, Submitted, (2018), arXiv:1810.10935.

3. G.B. Whitham, Variational methods and applications to water waves,

Proc. R. Soc. Lond. Ser. A , 299 (1967), 6-25.

Evolution equations with distinct sectors of peakon-type solutions

Stephen Anco


Brock University, Canada

48

[email protected]

Peakon-type solutions are studied for a family of nonlinear dis-

persive wave equations mt + f (u, ux)m + (g(u, ux)m)x = 0,

m = u − uxx. When the nonlinearities f m or (gm)x are higher

than quadratic, the equation is shown to possess infinitely many

distinct sectors of peakon-type solutions. The sectors arise from

freedom in how to regularize product of distributions, specifically

a Dirac delta function multiplied by a power of a Heaviside step

function. Only one choice of regularization coincides with the stan-

dard notion of a weak solution, but a different choice of regular-

ization appears to be necessary to preserve Hamiltonian structures

and integrability structure when they exist for smooth solutions. A

generalized Camassa-Holm equation, with p-power nonlinearities,

is used as an example to illustrate the results.

1. S.C. Anco and E. Recio, A general family of multi-peakon equations

and their properties, Accepted in J. Phys. A: Math. Theor. (2018).

arXiv:math-ph/1609.04354 math-ph

2. S.C. Anco and D. Kraus, Hamiltonian structure of peakons as weak

solutions for the modified Camassa-Holm equation, Discrete and Con-

tinuous Dynamical Systems (Series A) 38(9), (2018) 4449–4465.

Regular patterns and defects for the Rayleigh-Benardconvection

Mariana Haragus

Institut FEMTO-ST, Univ. Bourgogne-Franche Comte, France

[email protected]

We investigate pattern formation in the classical Rayleigh-Benard

convection problem. We focus on regular patterns such as rolls

and squares, and domain walls which are defects arising between

rolls with different orientations. The mathematical problem con-

sists in solving the Navier-Stokes equations for the fluid velocity

coupled with an additional equation for the deviation of the tem-

perature from the conduction profile in a cylindrical domain. Our

analysis relies upon a spatial dynamics formulation of the existence

problem and a centre-manifold reduction. In this setting, regular

patterns and domain walls are found as equilibria and heteroclinic

orbits, respectively, of a reduced system of ODEs. A normal form

transformation allows us to identify a leading-order approximation,

solutions of which are then shown to persist using transversality ar-

guments.

This is a joint work with Gerard Iooss (Nice).

Large-amplitude solitary water waves for theWhitham equation

Tien Truong

Slvegatan 18, SE-22100 Lund, room: 515

[email protected]

Erik Wahln

Slvegatan 18, SE-22100 Lund, room: 508

[email protected]

Miles H. Wheeler

3.120, Oskar-Morgenstern-Platz 1

[email protected]

In this talk we outline the main ideas behind proving the existence

of large-amplitude solitary wave solutions to the steady Whitham

equation −cφ + φ2 + K ∗ φ = 0 in the absence of surface tension.

The strategy is to use global bifurcation theory. To construct a

local curve, we modify and use a center manifold theorem for a

class of nonlocal equations. Then, we apply a version of the global

bifurcation theorem, which gives us an extra alternative related to

the loss of compactness to exclude. This issue is dealt with using a

Hamiltonian identity.

Krein signature without eigenfunctions and withouteigenvalues. What is Krein signature and what doesit measure?

Richard Kollar

Comenius University, Bratislava, Slovakia

[email protected]

Krein signature is a frequently used tool to study spectral stability

in Hamiltonian problems. Typically it is perceived as a sign of the

linearized (relative) energy of the corresponding eigenstate. We

present four different ways the Krein signature can be calculated

and interpreted without eigenfunctions or even without any corre-

sponding eigenvalue. The different perspectives explain how Krein

signature relates to robustness of the spectral stability results. One

of the examples presented is periodic travelling waves for general-

ized KdV-type equations.

A non-local approach to waves of maximal height forthe Degasperis–Procesi equation

Mathias Nikolai Arnesen

Department of Mathematical Sciences, Norwegian University of Science

and Technology


[email protected]

We consider the non-local formulation of the Degasperis-Procesi

equation ut +uux + L( 32 u2)x = 0, where L is the non-local Fourier

multiplier operator with symbol m(ξ) = (1 + ξ2)−1. We show

that all L∞, pointwise travelling-wave solutions are bounded above

by the wave-speed and that if the maximal height is achieved they

are peaked at those points, otherwise they are smooth. For suffi-

ciently small periods we find the highest, peaked, travelling-wave

solution as the limiting case at the end of the main bifurcation curve

of P-periodic solutions. The results imply that the Degasperis-

Procesi equation does not admit cuspon solutions in L∞.

Periodic traveling-wave solutions for regularized dis-persive equations: Sufficient conditions for orbitalstability with applications

Fabio Natali

Departament of Mathematics - State University of Maringa

Avenida Colombo, 5790, Maringa, PR, Brazil, CEP 87020-900

[email protected]

In this talk, we establish a new criterion for the orbital stability

of periodic waves related to a general class of regularized disper-

sive equations. More specifically, we present sufficient conditions

for the stability without knowing the positiveness of the associated

49

hessian matrix. As application of our method, we show the orbital

stability for a dispersive fifth-order model. The orbital stability

of periodic waves resulting from a minimization of a convenient

functional is also presented.

Waves of maximal height for a class nonlocal equa-tions with homogeneous symbol

Gabriele Bruell and Raj Narayan Dhara

Institute for Analysis, Karlsruher Institute of Technology

Department of Mathematics, University of West Bohemia


We discuss the existence and regularity of periodic traveling wave

solutions of a class of nonlocal equations with homogeneous sym-

bol of order −r, where r > 1. Based on the properties of the non-

local convolution operator, we apply analytic bifurcation theory

and show that a highest, peaked periodic traveling wave solution

is reached as the limiting case at the end of the main bifurcation

curve. The regularity of the highest wave is proved to be exactly

Lipschitz. As an application of our analysis, we reformulate the

steady reduced Ostrovsky equation in a nonlocal form in terms of

a Fourier multiplier operator with symbol m(k) = k−2. Thereby

we recover its unique highest 2π-periodic, peaked traveling wave

solution, having the property of being exactly Lipschitz at the crest.

1. L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Oceanol-

ogy, 18 (1978), 119–125.

2. M. Ehrnstrom, M. Johnson, and K. Claasen, Existence of a highest

wave in a fully dispersive two-way shallow water model, Arch Rational

Mech Anal, (2018).

3. B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation,

Princeton Series in Applied Mathematics, Princeton University Press,

Princeton, NJ, 2003.

Quansheng Liu and Zhijun Qiao

School of Mathematical and Statistical Sciences, University of Texas - Rio

Grande Valley

[email protected]

In this talk, we will talk about some recent developments in inte-

grable peakon systems, including the well-known CH, DP,

FORQ/MCH, NE, and other models. Some high order peakon

models will be reported first time. This is the joint work with Quan-

sheng Liu.

SESSION 17: “Nonlinear Dynamics of Mathematical Models in

Neuroscience”

Network reconstruction: Architectural andfunctional connectivity in the cerebral cortex

Paulina Volosov and Gregor Kovacic

Rensselaer Polytechnic Institute

110 Eighth Street, Troy, NY 12180


The extent of the relation between architectural and functional con-

nectivity in the cerebral cortex is a question which has attracted

much attention in recent years. Neuroscientists frequently use the

functional connectivity of neurons, i.e. the measures of causality

or correlations between the neuronal activities of certain parts of

a network, to infer the architectural connectivity of the network,

which indicates the locations of underlying synaptic connections

between neurons. Architectural connectivity can be used in the

modeling of neuronal processing and in the forming of conjectures

about the nature of the neural code. These two types of connectiv-

ity are by no means identical, and no one-to-one correspondence or

mapping exists from one to the other. In particular, certain trivial

measures of functional connectivity, such as correlations, give rise

to an undirected network, while synaptic architectural connectivity

is always directed. Nevertheless, architectural connectivity can be

inferred from functional connectivity, and this work is one attempt

to determine how to do so.

We begin by reconstructing the entire network using time-delayed

spike-train correlation, and we determine the time required before

an adequate reconstruction becomes possible and compare this to

time spans employed by experimentalists. We then sample the ma-

trix randomly and use the tool of matrix completion to fill-in the

rest of the network. To be more experimentally valid, we next ex-

amine a small slice or submatrix of the network and determine how

much information we can deduce about the whole network from

this small piece. An examination of the spectral properties of con-

nectivity matrices forms a major part of this analysis.

Network microstructure dominates global networkconnectivity in synchronous event initiation

Duane Nykamp∗ and Brittany Baker

School of Mathematics, University of Minnesota, Minneapolis, MN 55455

USA


Using a network model where one can modulate both network mi-

crostructure and global features of network connectivity, we ex-

amine the effects of both on the initiation of synchronous events.

The local microstructure is based on the SONET model [1], where

one can specify the frequencies of different two-edge motifs in the

network. By combining these local features with global structure

based on an underlying geometry, we investigated the interplay be-

tween the microstructure and the macrostructure as synchronous

events emerge in the network. We discovered that the microstruc-

ture played the dominate role in determining synchronous event

initiation.

1. L. Zhao, B. Beverlin II, T. Netoff and D. Q. Nykamp, Synchroniza-

tion from second order network connectivity statistics Frontiers Comp.

Neurosci., 5 (2011), 28.

Idealized models of insect olfaction

Pamela B. Pyzza∗

Ohio Wesleyan University, 61 S. Sandusky Street, Delaware, OH 43015

[email protected]

Katie Newhall

University of North Carolina at Chapel Hill, Chapel Hill, NC

50

[email protected]

Douglas Zhou

Shanghai Jiao Tong University, Shanghai, China

[email protected]

Gregor Kovacic

Rensselaer Polytechnic Institute, Troy, NY

[email protected]

David Cai

Deceased October 21, 2017

When a locust detects an odor, the stimulus triggers a specific se-

quence of network dynamics of the neurons in its antennal lobe.

The odor response begins with a series of synchronous oscillations,

followed by a short quiescent period, with a transition to slow pat-

terning of the neuronal firing rates, before the system finally returns

to a background level of activity. We begin modeling this behavior

using an integrate-and-fire neuronal network, composed of exci-

tatory and inhibitory neurons, each of which has fast-excitatory,

and fast- and slow-inhibitory conductance responses. We further

derive a firing-rate model for each (excitatory and inhibitory) neu-

ronal population, which allows for more detailed analysis of and in-

sight into the plausible olfaction mechanisms seen in experiments,

prior models, and our numerical model. We conclude that the tran-

sition of the network dynamics through fast oscillations, a pause

in network activity, and the slow modulation of firing rates can

be described by a system which has a limit cycle of the fast vari-

ables, slowly passes through a saddle-node-on-a-circle bifurcation

eliminating the oscillations, and, eventually, slowly passes again

through the bifurcation point, producing a new limit cycle with a

slower period – a process modulated by the slow variable.

On the dynamics of coupled Morris-Lecar neurons

Shelby Wilson∗

Morehouse College


830 Westview Dr.

Atlanta, GA 30314

[email protected]

In this work, we study the synchronization dynamics that arise

from an architecture where Morris-Lecar neurons are globally cou-

pled. We highlight a diverse set of asymptotic behavior for the cou-

pled system, and we analyze these outcomes as a function of the

system parameters. We will briefly present the nonlinear dynam-

ics and bifurcation behavior of Morris-Lecar neurons, thereby evi-

dencing Class I and Class II oscillatory behaviors. We also present

the formalism that we use to investigate the globally coupled net-

work.

We continue by analyzing how the interplay between the coupling

strength and the size of the neuronal ensemble determines the asymp-

totic dynamics of the coupled system. It is found that this collective

dynamics strongly depends on the topological nature of the limit-

cycle where the neurons are individually oscillating. Our analysis

shows that near the subcritical bifurcations to or from these limit

cycles, the ensemble dynamics can converge to one of three case :

total synchronization, to quenching, or to a non-trivial cluster syn-

chronization state where two distinct oscillating behaviors coexist

in the network.

Nonlinear wave equations of shear radial wave prop-agation in fiber-reinforced cylindrically symmetricmedia

Alexei Cheviakov∗ and Caylin Lee

Department of Mathematics and Statistics, University of Saskatchewan

Saskatoon, SK, Canada S7N 5E6


The framework of nonlinear elasticity can be systematically ap-

plied to model complex materials, including biomembranes [1, 2].

While the governing equations describe finite material displace-

ments without the assumption of their smallness, the mechanical

properties of specific materials are defined in terms of constitu-

tive functions. In this talk, we consider a model of an axially-

symmetric elastic solid undergoing radially-propagating shear dis-

placements. We focus on anisotropic fiber-reinforced materials

with two embedded families of interacting, helically-oriented elas-

tic fibers, commonly found in arterial walls [3]. As a first result, we

observe that for a wide class of constitutive functions, radial wave

models considered in a fully non-linear setting lead to linear wave

equations, which, moreover, do not contain any fiber-related terms.

The corresponding boundary value problems can be solved exactly.

Second, we consider a modified-fiber model, where the fibers have

a nonzero radial projection. In this case, the shear displacements

G(R, t) are shown to satisfy nonlinear wave equations of the form

Gtt =1

R

∂

∂R

[R(

N1GR + N2G2R + N3G3

R + N4

)],

where Ni are constant material parameters. A further extension of

the model, incorporating viscoelastic effects, leads to generalized

third-order nonlinear wave equations containing mixed space-time

derivatives GtR, GtRR. Further analysis of these nonlinear wave

models is of interest for a better understanding of shear wave prop-

agation in complex ideal and dissipative media, in particular, in

biological tissues.

1. G. A. Holzapfel and R. W. Ogden, Mechanics of biological tissue,

Springer Science & Business Media (2006).

2. A. F. Cheviakov and J-F. Ganghoffer, One-dimensional nonlinear elas-

todynamic models and their local conservation laws with applications

to biological membranes, J. Mech. Behav. Biomed. Mater., 58 (2016),

105-121.

3. G. A. Holzapfel, T. C. Gasser, and R. W. Ogden, A new constitutive

framework for arterial wall mechanics and a comparative study of ma-

terial models, J. Elast. Phys. Sci. Sol., 61, 1-3 (2000), 1-48.

SESSION 18: “Negative flows, peakons, integrable systems, and

their applications”

51

Global well-posedness for a nonlocal semilinearpseudo-parabolic equation with conical degeneration

Huafei Di and Yadong Shang

School of Mathematics and Information Science, Guangzhou University,

Guangdong, Guangzhou 510006, P R China

E-mail:[email protected]; [email protected]

This paper deals with a class of nonlocal semilinear pseudo-

parabolic equation with conical degeneration

ut −But −Bu = |u|p−1u − 1

|B|∫

B

|u|p−1udx1

x1dx′,

on a manifold with conical singularity, where B is Fuchsian type

Laplace operator with totally characteristic degeneracy on the

boundary x1 = 0. By using the modified methods of potential

well with Galerkin approximation and concavity, global existence,

uniqueness, finite time blow up and asymptotic behavior of solu-

tions will be discussed at the low initial energy J(u0) < d and crit-

ical initial energy J(u0) = d, respectively. Furthermore, we also

derive the threshold results of global existence and nonexistence

for the sign-changing solutions under some certain conditions. Fi-

nally, we investigate the global existence and finite time blow up of

solutions with the high initial energy J(u0) > d by the variational

method.

1. H. Chen, X. Liu, Y. Wei; Cone Sobolev inequality and Dirichlet prob-

lem for nonlinear elliptic equations on a manifold with conical singu-

larities, Calculus of Variations & Partial Differential Equations, 43(3-4)

(2012), 463-484.

2. M. Alimohammady, M.k. Koozehgar, G. Karamali; Global results for

semilinear hyperbolic equations with damping term on manifolds with

conical singularity, Mathematical Methods in the Applied Sciences,

40(11) (2017), 4160-4178.

Accelerating dynamical peakons and their behaviour

Stephen C. Anco

Department of Mathematics and Statistics, Brock University

St. Catharines, ON L2S3A1, Canada

[email protected]

Elena Recio∗

Department of Mathematics, Universidad de Cadiz

Puerto Real, Cadiz, Spain, 11510

[email protected]

Peakons are peaked travelling waves of the form

u(x, t) = a exp(−|x − ct|)

which were first found as weak solutions for the Camassa-Holm

equation. Several other similar peakon equations are well known:

Degasperis-Procesi equation, Novikov equation, modified Camassa-

Holm equation (also known as FORQ equation). Much of the in-

terest in these equations is that, firstly, they are integrable systems

having a Lax pair, bi-Hamiltonian structure, hierarchies of sym-

metries and conservation laws; secondly, they possess N-peakon

weak solutions given by a linear superposition of single peakons

with time-dependent amplitudes and speeds; and thirdly, they ex-

hibit wave breaking in which certain smooth initial data yields so-

lutions whose gradient ux blows up in a finite time while u stays

bounded.

All of these equations, and their various modified versions and non-

linear generalizations, belong to the general family of nonlinear

dispersive wave equations mt + f (u, ux)m + (g(u, ux)m)x = 0,

m = u − uxx, where f and g are arbitrary non-singular functions

of u and ux. Remarkably, every equation in this family possesses

N-peakon weak solutions [1].

In this work, a wide class of nonlinear dispersive wave equations

are shown to possess a novel type of peakon solution in which

the amplitude and speed of the peakon are time-dependent. These

novel dynamical peakons exhibit a wide variety of different be-

haviours for their amplitude, speed, and acceleration, including an

oscillatory amplitude and constant speed which describes a peakon

breather. Examples are presented of families of nonlinear disper-

sive wave equations that illustrate various interesting behaviours,

such as asymptotic travelling-wave peakons, dissipating/anti-

dissipating peakons, direction-reversing peakons, runaway and

blow up peakons, among others.

1. S.C. Anco, E. Recio, A general family of multi-peakon equations and

their properties. arXiv: 1609.04354 math-ph

Instability and uniqueness of the peaked periodictraveling wave in the reduced Ostrovsky equation

Dmitry Pelinovsky

Department of Mathematics, McMaster University,

Hamilton, ON L8S 4K1, Canada

[email protected]

Anna Geyer∗

Delft Institute of Applied Mathematics, TU Delft,

Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands

[email protected]

The existence of peaked periodic waves in the reduced Ostrovksy

equation has been known since the late 1970’s, see [1]. In our

recent paper we answer the long standing open question whether

these solutions are stable and prove linear instability of the peaked

periodic waves using semi-group theory and energy estimates. More-

over, we prove that the peaked wave is unique in the space of peri-

odic L2 functions with zero mean and a single minimum per period,

and that the equation does not admit Hlder continuous solutions,

i.e. there are no cusps. Our analysis relies on Fourier theory and

the existence of a first integral, together with sharp estimates of the

solution at the singularity at the peak.

1. L.A. Ostrovsky, Nonlinear internal waves a in rotating ocean,

Okeanologiya. 18 (1978) 181191.

Qilao Zha, Qiaoyi Hu, and Zhijun Qiao


Grande Valley

[email protected]

52

In this paper, we study a two-component short pulse system, which

was produced through a negative integrable flow associated with

the WKI hierarchy. The multi-soliton solutions for the two short

pulse system investigated, in particular, one-, two-, three-loop soli-

ton, and breather soliton solutions are discussed in details with in-

teresting dynamical interactions and shown through figures.

Some analysis results for the U(1)-invariant equation

Stephen Anco and Huijun He∗

Address (Department of Mathematics and Statistics, Brock University,

St. Catharines, Ontario, L2S 3A1, Canada)


Zhijun Qiao

Address (School of Mathematical and Statistical Sciences,

University of Texas C Rio Grande Valley (UTRGV), Edinburg, TX, 78539,

USA)

[email protected]

We study the peakon-like equation (U(1)-invariant equation) [1]:

mt +(

Re(eiθ(u + ux)(u − ux))m)

x

− i Im(eiθ(u + ux)(u − ux))m = 0.

(1) By applying the Littlewood-Paley theory and the transport the-

ory to this complex equation, we can obtain the Local well-posedness

of U(1)-invariant equation in some certain Besov spaces.

(2) We study the blow-up phenomenon of the equation according

to its L1 conservation laws.

(3) We study the analyticity and Gervey regularity of the equation.

(4) We study the persistence (the asymptotic behavior when the

spatial variable |x| large) of the equation.

(5) We study the orbital stability of the U(1)-invariant equation.

1. S. C. Anco and F. Mobasheramini, Integrable U(1)-invariant peakon

equations from the NLS hierarchy, Physica D, 355 (2017), 1–23.

Some properties of Wronskian solutions of nonlin-ear differential equations

Vesselin Vatchev

University of Texas Rio Grande Valley

[email protected]

Wronskian solutions are known for many nonlinear partial dif-

ferential equations including the well studied KdV and Boussi-

nessq Equations. In the talk we present some properties of solu-

tions obtained from Wronskian determinants W(φ1, φ2, . . . , φN)with generating functions φj(x, t) = cosh γj(x, t) or φj(x, t) =sinh γj(x, t) for γj(x, t) = pjx + σj(t), for real x and t and arbi-

trary functions σj.

By following the Hirota bi-linear method we study the properties

of the multi-soliton functions u = (log W)xx, including charac-

terization of the non-singular choices of sinh and cosh. We also

present an explicit decomposition u = ∑Nj=1 k jψ

2j for k j > 0 and

ψj the eigenfunctions of the eigenvalue operator in the Lax Pair for

KdV and the Boussinessq equations.

We also discuss particular non-linear choices of the functions σj.

Liouville correspondences between multi-componentintegrable hierarchies

Jing Kang and Xiaochuan Liu

School of Mathematics, Northwest University, Xi’an 710069, P.R. China

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an

710049, P.R. China


Peter J. Olver∗

School of Mathematics, University of Minnesota, Minneapolis, MN 55455,

USA

[email protected]

Changzheng Qu

Department of Mathematics, Ningbo University, Ningbo 315211, P.R. China

[email protected]

In this talk, we establish Liouville correspondences for the inte-

grable two-component Camassa-Holm hierarchy, the two-

component Novikov (Geng-Xue) hierarchy, and the two-compo-

nent dual dispersive water wave hierarchy by means of the related

Liouville transformations. This extends previous results on the

scalar Camassa-Holm and KdV hierarchies, and the Novikov and

Sawada-Kotera hierarchies to the multi-component case.

Lax algebraic representation for an integrable hier-archy

Shuxia Li and Zhijun Qiao


Grande Valley

[email protected]

Using the functional gradient approach of eigenvalues, this talk

presents a pair of Lenards operators for the Levis vector fields and

establishes commutator representations for hierarchies of Levisy

equations. The relationship between potential and stationary Levi’s

system is discussed in the end.

A new perspective in anomalous viscoelasticity fromthe derivative with respect to another function viewpoint

Xiao-Jun Yang

State Key Laboratory for Geomechanics and Deep Underground Engineer-

ing, China University of Mining and Technology, Xuzhou 221116, China

[email protected]

Feng Gao



[email protected]

Hong-Wen Jing



[email protected]

In this article, we address the new perspective in anomalous vis-

coelasticity containing the derivative with respect to another func-

tion for the first time. The Newton-like, Maxwell-like, Kelvin-

53

Voigt-like, Burgers-like and Zener-like models via the new deriva-

tives with respect to another functions are discussed in detail. The

results are accurate and efficient in the descriptions of the scale

behaviors of the complex materials involving the power law.

Some new exact solutions for the extended (3+1)-di-mensional Jimbo-Miwa equation

Wenhao Liua, Binlu Fengb, Yufeng Zhanga,∗

a School of Mathematics, China University of Mining and Technology,

Xuzhou, Jiangsu, 221116, Peoples Republic of China

b School of Mathematics and Information Sciences, Weifang University,

Weifang, Shandong, 261061, Peoples Republic of China

n this paper, firstly, the bilinear form of the extended (3+1)-dimen-

sional Jimbo-Miwa equation is provided, and its transformation of

dependent variable also is given. Secondly, we derived different so-

lutions of the equation by using the homoclinic test approach, the

three-wave method and the multiple exp-function method, respec-

tively. Finally, all these solutions are presented via 3-dimensional

plots with choices some special parameters to show the dynamic

characteristics.

New integrable peakon equations from a modifiedAKNS scheme

Evans Boadi, Sicheng Zhao, and Stephen Anco


Brock University, Canada

The standard AKNS scheme for generating integrable evolution

systems is modified to obtain integrable peakon systems. In the

simplest situation given by sl(2,R) matrices, the modified scheme

in the 1-component case yields the well-known Camassa-Holm

equation, the modified Camassa-Holm (FORQ) equation, and a

quadratic peakon equation on Novikov’s list. Large families of

integrable peakon equations which contain arbitrary functions of

the dynamical variables are obtained in the 2-component case. A

reduction of a family yields the U(1)-invariant integrable peakon

equations found recently [1, 2] by the tri-Hamiltonian splitting

method.

Recent results on work in progress for su(2), sl(2,C), sl(3,R), su(3),

su(2,1) will be presented as well.

1. S.C. Anco and F. Mobasheramini, Integrable U(1)-invariant peakon

equations from the NLS hierarchy, Physica D 355 (2017), 1–23.

2. S.C. Anco, X. Chang, J. Szmigielski, The dynamics of conservative

peakons in the NLS hierarchy, Studies in Applied Math. (2018), 1–34.

Hamiltonian structure of peakons as weak solutionsfor the modified Camassa-Holm equation

Stephen Anco


Brock University

[email protected]

Daniel Kraus*


SUNY Oswego

[email protected]

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian

system possessing N-peakon weak solutions for all N ≥ 1 in

the setting of an integral formulation which is used in analysis for

studying local well-posedness, global existence, and wave break-

ing for non-peakon solutions. Unlike the original Camassa-Holm

equation [1], the two Hamiltonians of the mCH equation do not

reduce to conserved integrals (constants of motion) for 2-peakon

weak solutions.

In this talk, we address this perplexing situation by finding an

explicit conserved integral for N-peakon weak solutions for all

N ≥ 2. When N is even, the conserved integral is shown to

provide a Hamiltonian structure with the use of a natural Poisson

bracket that arises from reduction of one of the Hamiltonian struc-

tures of the mCH equation. But when N is odd, the Hamiltonian

equations of motion arising from the conserved integral using this

Poisson bracket are found to differ from the dynamical equations

for the mCH N-peakon weak solutions.

Moreover, we show that the lack of conservation of the two Hamil-

tonians of the mCH equation when they are reduced to 2-peakon

weak solutions extends to N-peakon weak solutions for all N ≥ 2,

and we discuss the connection between this loss of integrability

structure and related work by Chang and Szmigielski on the Lax

pair for the mCH equation [2].

1. R. Camassa and D. D. Holm, An integrable shallow water equation

with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664

2. X. Chang and J. Szmigielski, Lax integrability and the peakon problem

for the modified Camassa-Holm equation, Commun. Math. Phys., 358

(2018), 295-341.

SESSION 19: “Network dynamics”

Configurational stability for the Kuramoto-Sakagu-chi model

Jared Bronski and Lee DeVille


University of Illinois at Urbana-Champaign, IL, 61801


Thomas Carty∗


Bradley University, Peoria, IL, 61625

[email protected]

The Kuramoto–Sakaguchi model is a generalization of the well-

known Kuramoto model that adds a phase-lag parameter, or “frus-

tration” to a network of phase-coupled oscillators. The Kuramoto

model is a flow of gradient type, but adding a phase-lag breaks

the gradient structure, significantly complicating the analysis of

the model. We present several results determining the stability of

phase-locked configurations: the first of these gives a sufficient

condition for stability, and the second a sufficient condition for in-

stability. In fact, the instability criterion gives a count, modulo 2,

54

of the dimension of the unstable manifold to a fixed point and hav-

ing an odd count is a sufficient condition for instability of the fixed

point.

1. J.C. Bronski, T. Carty, and L. DeVille, Configurational stability for the

Kuramoto-Sakaguchi model, Chaos 28, 103109 (2018), 16 99.

Adaptive zero determinant strategies in the iteratedprisoners dilemma tournament

Emmanuel Estrada and Dashiell Fryer∗


San Jose State University, CA, 95192


We have created an adaptive zero determinant strategy that changes

its parameters using the outcome of the last round as input. We then

ran this adaptive zero determinant strategy against a tournament of

other zero determinant strategies. We observed that the adaptive

strategy had a higher average score than the other zero determi-

nant strategies when we ran the tournament for a large amount of

rounds.

A matrix valued Kuramoto model

Jared Bronski


University of Illinois at Urbana-Champaign, IL, 61801

[email protected]

Thomas Carty and Sarah Simpson∗


Bradley University, Peoria, IL, 61625


A need to better approximate quantum mechanical oscillatory phe-

nomena motivated Lohe [1], [2] and others [3], [4] to derive non-

Abelian generalizations of the Kuramoto model for phase-locking.

Here we propose and analyze a purely real-valued model of this

type in which we consider a collection of symmetric matrix-valued

variables. This is a gradient flow where the matrices evolve to min-

imize energy in such a way as to try to align their eigenframes. The

phase-locked state is one where the eigenframes all align, and thus

the matrices all commute. We analyze the stability of the phase-

locked state for n × n matrices, and show that it is stable. We

also show that in the case of 2 × 2 matrices the model reduces to

a form of the Kuramoto model with dynamic coupling. Addition-

ally, we show that in the case of 2 × 2 matrices, the model has

a dynamically unstable set of fixed points analogous to the twist

states arising in the standard Kuramoto model.

1. M.A. Lohe, Non-Abelian Kuramoto models and synchronization,

Journal of Physics A 42 (39), 395101 (2009).

http://iopscience.iop.org/article/10.1088/1751-

8113/42/39/395101/meta

2. M.A. Lohe, Quantum synchronization over quantum networks, Journal

of Physics A 43 (46), 465301 (2010).


8113/43/46/465301/meta

3. Sun-Ho Choi and Seung-Yeal Ha, Quantum synchronization of the

schrodinger lohe model, Journal of Physics A 47 (35), 355104 (2014).


8113/47/35/355104/meta

4. Lee DeVille, Synchronization and stability for quantum kuramoto,

Journal of Statistical Physics (2018).

https://doi.org/10.1007/s10955-018-2168-9

The universal covariant representation andamenability

Mamoon Ahmed

Amman, 11941, Jordan

[email protected]

Let (G, P) be a quasi-lattice ordered group. In this paper we give a

modified proof of Laca and Raeburn’s theorem about the covariant

isometric representations of amenable quasi-lattice ordered groups

[1, Theorem 3.7]. First, we construct a universal covariant repre-

sentation for a given quasi-lattice ordered group (G, P) and show

that it is unique. Then we show if (G, P) is amenable, true rep-

resentations of (G, P) generate C∗-algebras that are isomorphic to

the universal object.

1. M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz

algebras of nonabelian groups, J. Funct. Anal. 139 (1996), 415–440.

2. M. Ahmed and A. Pryde, The structure Theorem and the Commutator

Ideal of Toeplitz Algebras, To appear Glasg. Math. J.

Bistability in the Kuramoto model

Timothy Ferguson∗


Arizona State University, AZ, 85281

[email protected]

The Kuramoto model is a general model for the behavior of net-

work coupled oscillators. For such a system stable phase-locked

solutions are of critical importance to the global long-time behavior

of the system. In particular, we consider the question of bistability,

namely, when two such stable solutions exist simultaneously. In

this regard, we give a generic condition for a bistability forming

bifurcation to occur in ring networks with positive coupling, and

apply it to produce examples of bistability. Furthermore, we derive

a necessary condition for this bifurcation in terms of the phase-

angles and numerically demonstrate that this condition is closely

related to the order parameter for N = 3 oscillators.

SESSION 20: “Dynamical Systems and integrability”

Hidden solutions of discrete systems

Nalini Joshi

School of Mathematics and Statistics F07,

The University of Sydney, NSW 2006, Australia

[email protected]

Christopher J. Lustri

Department of Mathematics, Macquarie University, NSW 2109, Australia

55

[email protected]

Steven Luu

School of Mathematics and Statistics F07,

The University of Sydney, NSW 2006, Australia

[email protected]

Hidden solutions are well known in irregular singular limits of dif-

ferential equations. Such solutions are not able to be identified

uniquely through conventional analysis, because free parameters

identifying the solution lie hidden beyond all orders of a diver-

gent asymptotic expansion. We identify such solutions of discrete

Painleve equations, specifically q-PI, d-PI, and d-PII, in the limits

where their independent variable goes to infinity and extend the in-

vestigation to further solutions and to partial difference equations.

Through such analysis, we determine regions of the complex plane

in which the asymptotic behaviour is described by a power series

expression, and find that the behaviour of these asymptotic solu-

tions shares a number of features with the tronquee and tri-tronquee

solutions of corresponding differential Painleve equation.

JL15. N. Joshi and C.J. Lustri. Stokes phenomena in discrete Painleve I.

Proceedings of the Royal Society of London A: Mathematical, Physi-

cal and Engineering Sciences, 471(2177):20140874, 2015.

JLL17. N. Josh and C.J. Lustri and S. Luu. Stokes phenomena in discrete

Painleve II. Proc. R. Soc. A, 473(2198):20160539, 2017.

JLL18. N. Joshi, C.J. Lustri, and S. Luu. Nonlinear q-Stokes phenomena

for q-Painleve I. arXiv:1807.00450 [math-ph], 2018.

JL18. N. Joshi, C.J. Lustri. Generalized Solitary Waves in a Finite-

Difference Korteweg-de Vries Equation. arXiv:1808.09654 [math-

ph], 2018.

Two dimensional stationary vorticity distribution andintegrable system

Yasuhiro Ohta

Department of Mathematics, Kobe University

Rokko, Kobe 657-8501, Japan

[email protected]

In two dimensional inviscid incompressible fluid, stationary flows

are described by solutions of the nonlinear Klein-Gordon equation

for stream function. It is well-known that in the two dimensional

Toda lattice hierarchy there are some integrable systems of the

form of nonlinear Klein-Gordon equation, namely Liouville equa-

tion, sine-Gordon equation, sinh-Gordon equation, Tzitzeica equa-

tion. Many solutions for these equations and stationary vorticity

distributions have been widely and deeply investigated in the con-

text of fluid dynamics. See for example [1], [2].

We study a class of solutions of integrable system which are related

with two dimensional vorticity distributions similar to the Stuart

vortex street. The solutions correspond to deformed vortex streets

and some of them have singularities which are regarded as an array

of point vortices. Relevance of such solutions as steady fluid flow

is also discussed.

1. M. C. Haslam, C. J. Smith, G. Alobaidi and R. Mallier, Some nonlinear

vortex solutions, Int. J. Diff. Eq., 2012 (2012), 929626.

2. K. W. Chow, S. C. Tsang and C. C. Mak, Another exact solution for

two-dimensional, inviscid sinh Poisson vortex arrays, Phys. Fluids, 15

(2003), 2437-2440.

Ellipsoidal billiards and Chebyshev-type polynomi-als

Vladimir Dragovic∗

The University of Texas at Dallas, Richardson, TX

[email protected]

Milena Radnovic

University of Sydney, Sydney, Australia

[email protected]

A comprehensive study of periodic trajectories of the billiards within

ellipsoids in the d-dimensional Euclidean space is presented. The

novelty of the approach is based on a relationship established be-

tween the periodic billiard trajectories and the extremal polynomi-

als of the Chebyshev type on the systems of d intervals on the real

line. As a byproduct, for d = 2 a new proof of the monotonicity of

the rotation number is obtained and the result is generalized for any

d. The case study of trajectories of small periods T, d ≤ T ≤ 2d is

given. In particular, it is proven that all d-periodic trajectories are

contained in a coordinate-hyperplane and that for a given ellipsoid,

there is a unique set of caustics which generates d + 1-periodic

trajectories. A complete catalog of billiard trajectories with small

periods is provided for d = 2 [2] and d = 3 [1].

Surprisingly enough, the Cayley type conditions for d = 2 appear

to be connected to the so-called discriminantly separable polyno-

mials, a class of polynomials introduced by the first author in his

study [3] of the classical Kowalevski integration of the Kowalevski

top.

1. V. Dragovic, M. Radnovic, Periodic ellipsoidal billiard trajectories and

extremal polynomials, arXiv 1804.02515

2. V. Dragovic, M. Radnovic, Caustics of Poncelet polygons and classical

extremal polynomials, arXiv 1812.02907 Regular and Chaotic Dynam-

ics, Vol. 24, 2019.

3. V. Dragovic, Geometrization and Generalization of the Kowalevski top,

Communications in Mathematical Physics, Vol. 298, no. 1, p. 37-64,

2010.

A discrete analogue of the Toda hierarchy and itssome properties

Masato Shinjo and Koichi Kondo

Faculty of Science and Engineering, Doshisha University,

1-3 Tatara miyakodani, Kyotanabe, Kyoto 610-0394, Japan


The Toda equation describing motions governed by nonlinear

springs is well-known as famous soliton equation in the study of

integrable systems. Flaschka’s variables [1] lead to Lax dynamics

of the Toda equation with tridiagonal matrix. In [2], a skillful dis-

cretization of the Toda equation is presented. The discrete Toda

equation contributes to computing eigenvalues of tridiagonal ma-

trices.

One of extensions of the Toda equation with associated tridiagonal

matrix is called the Toda hierarchy [3]. In this talk, we propose

56

a discrete analogue of the Toda hierarchy, which corresponds to a

generalization of the discrete Toda equation in [2]. We comprehen-

sively clarify Lax dynamics and solutions to both of the continuous

and discrete equations.

1. H. Flaschka, The Toda lattice. II. Existence of integrals, Phys. Rev. B,

9 (1974), 1924–1925.

2. R. Hirota, Conserved quantities of “random-time Toda equation”, J.

Phys. Soc. Jpn., 66 (1997), 283–284.

3. J. Moser, Finitely many mass points on the line under the influence

of an exponential potential-An integrable system, Dynamic Systems,

Theory and Applications, Lecture Notes in Phys., 38 (1975), 467–497.

In this talk I shall introduce the idea of a quasi-pfaffian, this is the

pfaffian equavalent to a quasi-determinant, these quasi-pfaffians

have identities analogous to quasi-determinant identities and in the

commutative case, they reduce to ratios of pfaffians. We will look

at some quasi-pfaffian identities and look at the connection be-

tween these quasi-pfaffians and noncommutative integrable sys-

tems.

On the inverse problem of the discrete calculus ofvariations

G. Gubbiotti

School of Mathematics and Statistics,The University of Sydney, Carslaw

Building, F07, 2006, Sydney (NSW), Australia

[email protected]

One of the most powerful tools in Mathematical Physics since Eu-

ler and Lagrange is the calculus of variations. The variational for-

mulation of mechanics where the equations of motion arise as the

minimum of an action functional (the so-called Hamilton’s princi-

ple), is fundamental in the development of theoretical mechanics

and its foundations are present in each textbook on this subject

[1, 2, 4]. Beside this, the application of calculus of variations goes

beyond mechanics as many important mathematical problems, e.g.

the isoperimetrical problem and the catenary, can be formulated in

terms of calculus of variations.

An important problem regarding the calculus of variations is to de-

termine which system of differential equations are Euler–Lagrange

equations for some variational problem. This problem has a long

and interesting history, see e.g. [3]. The general case of this prob-

lem remains unsolved, whereas several important results for par-

ticular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a La-

grangian in the discrete scalar setting. We will introduce a set of

differential operators called annihilation operators. We will use

these operators to reduce the functional equation governing of ex-

istence of a Lagrangian for a scalar difference equation of arbitrary

even order 2k, with k > 1 to the solution of a system of linear par-

tial differential equations. Solving such differential equations one

can either find the Lagrangian or conclude that it does not exist.

1. H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson

Education, 2002.

2. L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical

Physics. Elsevier Science, 1982.

3. P. J. Olver. Applications of Lie Groups to Differential Equations.

Springer-Verlag, Berlin, 1986.

4. E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles

and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

SESSION 21: “Stochastic Dynamics in Nonlinear Systems”

A network of transition pathways in a model granu-lar system

Katie Newhall

UNC Chapel Hill

[email protected]

Many intriguing dynamical properties of complex systems, such

as metastability or resistance to applied forces, emerge from the

underlying energy landscape. High-dimensional systems can have

complex energy landscapes with numerous energy-minimizing states.

Especially in randomly packed granular materials for which know-

ing the single global energy minimizing state is unimportant, un-

derstanding the interconnectivity of minimums via transition paths

through saddles allows for extracting the dominant features of the

system. The energy landscape of a jammed 2D packing of bidis-

perse disks is modeled as a collection of overlapping circles, defin-

ing an energy penalty based on the amount of overlap. I propose a

systematic approach to mapping out the transition pathways from

energy minimizer to saddle point to minimizer forming a network

of transition pathways. This computational method is based on the

climbing string method of W. Ren and E. Vanden-Eijnden that has

been successfully applied to problems in chemistry. The ultimate

goal is to relate observable phenomena like a granular material’s

rearrangements preceding failure events to dynamics on the net-

work representation of the energy landscape of the system.

Limiting behaviors of high dimensional stochasticspin ensemble

Y. Gao∗, J. Marzuola and K. Newhall

Department of Mathematics, University of North Carolina at Chapel Hill


[email protected]

K. Kirkpatrick

Department of Mathematics, University of Illinois at Urbana-Champaign

[email protected]

J. Mattingly

Department of Mathematics, Duke University

[email protected]

Lattice spin models in statistical physics are used to understand

magnetism. Their Hamiltonians are a discrete form of a version

of a Dirichlet energy, signifying a relationship to the Harmonic

map heat flow equation. The Gibbs distribution, defined with this

Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm

to generate dynamics tending towards an equilibrium state. In the

limiting situation when the inverse temperature is large, we estab-

lish the relationship between the discrete M-H dynamics and the

57

continuous Harmonic map heat flow associated with the Hamil-

tonian. We show the convergence of the M-H dynamics to the

Harmonic map heat flow equation in two steps: First, with fixed

lattice size and proper choice of proposal size in one M-H step,

the M-H dynamics acts as gradient descent and will be shown to

converge to a system of Langevin stochastic differential equations

(SDE). Second, with proper scaling of the inverse temperature in

the Gibbs distribution and taking the lattice size to infinity, it will

be shown that this SDE system converges to the deterministic Har-

monic map heat flow equation. Our results are not unexpected, but

show remarkable connections between the M-H steps and the SDE

Stratonovich formulation, as well as reveal trajectory-wise out of

equilibrium dynamics to be related to a canonical PDE system with

geometric constraints. We are currently working on introducing

spatially correlated noise to obtain the convergence to a stochastic

PDE.

Improving sampling accuracy of SG-MCMC meth-ods via non-uniform subsampling of gradients

Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao∗

Georgia Institute of Technology

[email protected]

In the training of neural networks or Bayesian inferences with big

data, additive gradients that sum a large amount of terms need to

be repeatedly evaluated. To reduce the computational cost of such

evaluations, the machine learning community relied on Stochastic-

Gradient-MCMC methods, which approximate gradients by stochas-

tic ones via uniformly subsampled data points. This, however, in-

troduces extra variance artificially. How to design scalable algo-

rithms that correctly sample the target distribution is an outstand-

ing challenge.

This talk will describe a heuristic step towards this challenge. The

core idea is to use exponentially weighted stochastic gradients

(EWSG) to replace uniform ones. A demonstration based on sec-

ond-order Langevin equation coupled with a Metropolis chain will

be provided. The improved performance will be discussed through

both theoretical evidence and numerical experiments on multiple

learning tasks. While statistical accuracy has improved, the speed

of convergence was empirically observed to be at least comparable

to the uniform version.

Averaging for systems of nonidentical molecular mo-tors

Joseph Klobusicky and Peter Kramer




John Fricks

School of Mathematical and Statistical Sciences

Arizona State University

[email protected]

The shuttling of molecular cargo across a cell is aided by the di-

rected transport of molecular motors on a microtubule network. A

cargo may be attached to several motors which can attach and de-

tach from a microtubule during a typical procession. Since motors

ensembles operate in the nanoscale, directly observing an attach-

ment state is difficult, and creates a need for developing models

which provide statistics for multiple motor ensembles based on

known parameters from one motor systems.

The effect of moderate noise on a limit cycle oscilla-tor: counterrotation and bistability

Jay Newby

Department of Mathematical and Statistical Sciences, University of Al-

berta, Edmonton, Canada

[email protected]

The effects of noise on the dynamics of nonlinear systems is known

to lead to many counterintuitive behaviors. Using simple planar

limit cycle oscillators, we show that the addition of moderate noise

leads to qualitatively different dynamics. In particular, the system

can appear bistable, rotate in the opposite direction of the deter-

ministic limit cycle, or cease oscillating altogether. Utilizing stan-

dard techniques from stochastic calculus and recently developed

stochastic phase reduction methods, we elucidate the mechanisms

underlying the different dynamics and verify our analysis with the

use of numerical simulations. Last, we show that similar bistable

behavior is found when moderate noise is applied to the FitzHugh-

Nagumo model, which is more commonly used in biological appli-

cations.

Stochastic parameterization of subgrid-scales in one-dimensional shallow water equations

Matthias Zacharuk, Stamen Dolaptchiev, Ulrich Achatz

Johann Wolfgang Goethe-Universitt Frankfurt/Main

[email protected], [email protected],

[email protected]

Ilya Timofeyev∗

University of Houston

[email protected]

We address the question of parameterizing the subgrid scales in

simulations of geophysical flows by applying stochastic mode re-

duction to the one-dimensional stochastically forced shallow water

equations. The problem is formulated in physical space by defin-

ing resolved variables as local spatial averages over finite-volume

cells and unresolved variables as corresponding residuals. Based

on the assumption of a time-scale separation between the slow spa-

tial averages and the fast residuals, the stochastic mode reduction

procedure is used to obtain a low-resolution model for the spatial

averages alone with local stochastic subgrid-scale parameterization

coupling each resolved variable only to a few neighboring cells.

The closure improves the results of the low-resolution model and

outperforms two purely empirical stochastic parameterizations. It

is shown that the largest benefit is in the representation of the en-

ergy spectrum. By adjusting only a single coefficient (the strength

of the noise) we observe that there is a potential for improving

the performance of the parameterization, if additional tuning of the

coefficients is performed. In addition, the scale-awareness of the

parameterizations is studied.

1. M. Zacharuk, S. I. Dolaptchiev, U. Achatz, I. Timofeyev, ”Stochas-

tic subgrid-scale parameterization for one-dimensional shallow water

58

dynamics using stochastic mode reduction”, Q.J.R. Meteorol. Soc.,

144(715), (2018), 1975-1990.

Coupling for Hamiltonian Monte Carlo

Nawaf Bou-Rabee


Rutgers University Camden

311 North Fifth Street

Camden, NJ 08102

[email protected]

We present a new coupling approach to study the convergence of

the Hamiltonian Monte Carlo (HMC) method. Specifically, we

prove that the transition step of HMC is contractive w.r.t. a care-

fully designed Kantorovich (L1 Wasserstein) distance. The lower

bound for the contraction rate is explicit. Global convexity of the

potential is not required, and thus multimodal target distributions

are included. Explicit quantitative bounds for the number of steps

required to approximate the stationary distribution up to a given er-

ror ǫ are a direct consequence of contractivity. These bounds show

that HMC can overcome diffusive behavior if the duration of the

Hamiltonian dynamics is adjusted appropriately. This talk is based

on joint work with Andreas Eberle and Raphael Zimmer.

N. Bou-Rabee, A. Eberle, and R. Zimmer, Coupling and convergence

for Hamiltonian Monte Carlo, arXiv preprint arXiv:1805.00452, 2018.

SESSION 22: “Modern methods for dispersive wave equations”

Singular limits of certain Hilbert-Schmidt integraloperators and applications to tomography

Marco Bertola

Department of Mathematics & Statistics, Concordia University

Montreal, Quebec H3G 1M8 Canada

[email protected]

Elliot Blackstone∗, Alexander Katsevich and Alexander Tovbis

Department of Mathematics, University of Central Florida

Orlando, Florida 32816-1364 U.S.A.

[email protected], [email protected] and Alexan-

[email protected]

In this talk we discuss the asymptotics of the spectrum of self-

adjoint Hilbert-Schmidt integral operators with the so-called inte-

grable kernels in a certain singular limit, where the limiting opera-

tor is still bounded but has a continuous spectral component. Such

operators appear when studying stability of the interior problem of

tomography. They are related to Finite Hilbert Transform (FHT)

on several intervals, when neighboring intervals are touching each

other. The case of separate intervals, when the corresponding inte-

gral operators are of Hilbert-Schmidt class, was studied in [1]. Our

work is based on the method of Riemann-Hilbert problems.

1. M. Bertola, A. Katsevich and A. Tovbis, Singular Value Decomposi-

tion of a Finite Hilbert Transform Defined on Several Intervals and the

Interior Problem of Tomography: The Riemann-Hilbert Problem Ap-

proach. Communications on Pure and Applied Mathematics, 2016.

KdV is wellposed in H−1

Rowan Killip∗ and Monica Visan

Department of Mathematics, UCLA


I will describe a proof of the well-posedness of the Korteweg–de

Vries equation in the Sobolev space H−1 that works both on the

line and on the circle. On the line, this result was previously un-

known; on the circle it was proved by Kappeler and Topalov. This

is joint work [1] with Monica Visan.

1. R. Killip and M. Visan, KdV is wellposed in H−1. Preprint

arXiv:1802.04851.

The construction and evaluation of shock wave so-lutions to the KdV equation and a linear KdV-likeequation

Thomas Trogdon∗

University of California, Irvine

[email protected]

We consider the problem of computing the inverse scattering trans-

form for the KdV equation on R when the initial data q0(x) satis-

fies limx→+∞ q0(x) = limx→−∞ q0(x). We build on the work

of Cohen and Kappeler (1985) [2] and Andreiev et al. (2016) [1].

In particular, we demonstrate how the use of both left and right

reflection coefficients is necessary, in contrast to decaying initial

data. Properties of this solution motivate a linearization that shares

non-trivial structure with its nonlinear counterpart. This is joint

work with Deniz Bilman, Dave Smith and Vishal Vasan.

1. K Andreiev, I Egorova, T L Lange, and G Teschl. Rarefaction waves

of the Korteweg–de Vries equation via nonlinear steepest descent. J.

Differ. Equ., 261(10):5371–5410, 2016.

2. A Cohen and T Kappeler. Scattering and inverse scattering for steplike

potentials in the Schrodinger equation. Indiana Univ. Math. J., 34:127–

180, 1985.

Long-time asymptotics for the massiveThirring model

Aaron Saalmann

Weyertal 86-90

50939 Cologne, Germany

[email protected]

From the analytical point of view, the massive Thirring model

(MTM), i(ut + ux) + v + u|v|2 = 0,i(vt − vx) + u + |u|2v = 0,

is of special interest, because it has a representation in terms of a

Lax pair, consisting of two linear operators L and A. Thanks to the

Lax pair, the MTM admits an exact solution by the inverse scatter-

ing transform (IST), see [1].

As it is also known from other nonlinear dispersive equations one

can create solitons for the MTM. These special solutions are waves

that move at constant speed and do not change in shape. They can

59

refuse to disperse only because of the presence of the nonlinear-

ity in the equation. It is relatively simple to characterize solitons,

based on their scattering data. Using suitable Riemann-Hilbert

techniques it is possible to analyse the interaction of two (or more)

solitons. Furthermore, it can be shown precisely that each soli-

ton will eventually enter the light cone |t| > |x|. Using the

∂–method (nonlinear steepest descent) one can show that outside

the light cone any solution (not only solitons) converges to zero

with a rate of |t|−3/4. Inside the light-cone there are basically

two different possibilities. Assuming that the initial data is free

of solitons one can use the ∂–method and some model Riemann–

Hilbert problems to show that the solution of the MTM scatters to

a linear solution modulo phase correction. This linear solution can

be computed explicitly from the scattering data and its amplitude

decays with a rate of |t|−1/2. The second possibility is that the

initial data contains finitely many solitons. Then, one can prove

that any solution breaks up into finitely many single solitons that

travel at different speeds and thus, diverge. The remainder term is

O(|t|−1/2).

In the talk it will be explained how the MTM can be rewritten in

terms of a Riemann-Hilbert problem and the main aspects of the

Riemann–Hilbert analysis will be discussed.

1. Dmitry E. Pelinovsky and Aaron Saalmann. Inverse Scattering for the

Massive Thirring Model. Fields Institute Communications, (2019), (ac-

cepted).

Semiclassical soliton ensembles and the three-waveresonant interaction (TWRI) equations

Robert Buckingham

University of Cincinnati

[email protected]

Robert Jenkins∗

Colorado State University

[email protected]

Peter Miller

University of Michigan

[email protected]

I’ll discuss some of our ongoing work [1] on the the TWRI equa-

tions, a universal model of the first stage of nonlinear behavior

in weakly nonlinear systems which support resonant triads. This

system is integrable with a third order Lax-Pair. The higher or-

der nature of the system complicates the scattering theory for the

Lax operator. I’ll present a scheme we’ve introduced to study the

system using a soliton ensemble approach, some numerical exper-

iments, and analytic results.

1. R. Buckingham, R. Jenkins and P. Miller, Semiclassical soliton ensem-

bles for the three wave resonant interaction equations, Comm. Math.

Phys., 354 (2017), 1015-1100.

Asymptotics of rational solutions of the defocusingnonlinear Schrodinger equation

Robert J. Buckingham and Donatius DeMarco∗

Department of Mathematical Sciences. University of Cincinnati

PO Box 210025 Cincinnati, OH 45221.


The defocusing nonlinear Schrodinger equation has a family of ra-

tional solutions that can be expressed in terms of generalized Her-

mite polynomials. These special polynomials have strong ties to

rational solutions of the fourth Painleve equation. The family of

solutions to the NLS equation can be expressed in terms of or-

thogonal polynomials. Using this, we apply the Deift-Zhou non-

linear steepest-decent method to asymptotically analyze the limit

n → ∞, where n indexes the rational solutions to the nonlinear

Schrodinger equation.

1. R. Buckingham,Large-degree asymptotics of rational Painleve-IV

functions associated to generalized Hermite polynomials, Int. Math.

Resea. Notic., (2018)

2. P. Clarkson, The fourth Painleve equation and associated special poly-

nomials, J. Math. Phys., 65 (2003), 5350–5374

Long-time behavior of solutions to the modified KdVequation in weighted sobolev space

Gong Chen and Jiaqi Liu

University of Toronto


he long time behavior of solutions to the defocusing modified Korte-

weg-de vries (MKDV) equation is established for initial conditions

in some weighted Sobolev spaces. Our approach uses the inverse

scattering transform and the nonlinear steepest descent method of

Deift and Zhou and its reformulation by Dieng, Miller and

McLaughlin through ∂-method.

Soliton resolution for the derivative NLS

Robert Jenkins

Department of Mathematics, Colorado State University

Fort Collins, Colorado 80523-1801, U. S. A.

[email protected]

Jiaqi Liu

Department of Mathematics, University of Toronto

Toronto, Ontario, Canada M5S 2E4

[email protected]

Peter A. Perry∗

Department of Mathematics, University of Kentucky

Lexington, Kentucky 40506–0027, U. S. A.

[email protected]

Catherine Sulem

Department of Mathematics, University of Toronto

Toronto, Ontario, Canada M5S 2E4

[email protected]

Kaup and Newell [4] showed that the Derivative Nonlinear Schro-

dinger equation

iut + uxx − iε(|u|2u

)x= 0, (4)

which describes the propagation of nonlinear Alfven waves in plas-

mas [5], is completely integrable. Here we’ll report on joint work

60

with Robert Jenkins, Jiaqi Liu, and Catherine Sulem [3] which ex-

ploits the complete integrability to show that, for generic decaying

initial data, the soliton resolution conjecture holds for this equa-

tion. That is, we show that the solution u(x, t) of the initial value

problem for (4) resolves into the sum of finitely many soliton so-

lutions and a radiation term. A consequence of our analysis is the

asymptotic stability of soliton solutions.

To obtain the asymptotics we use the ∂-steepest descent methods

pioneered by Dieng and McLaughlin [2] and further developed by

Borghese, Jenkins, and McLaughlin [1] to prove soliton resolution

for the cubic focusing NLS.

1. Michael Borghese, Robert Jenkins, Kenneth D. T.-R. McLaughlin.

Long-time asymptotic behavior of the focusing nonlinear Schrodinger

equation. Ann. Inst. H. Poincare Anal. Non Lineaire 35 (2018), no. 4,

887–920.

2. Momar Dieng, Kenneth D. T.-R. McLaughlin. Long-time Asymptotics

for the NLS equation via ∂-methods. arXiv:0805.2807.

3. Robert Jenkins, Jiaqi Liu, Peter Perry, Catherine Sulem. Soliton reso-

lution for the derivative nonlinear Schrodinger equation. Comm. Math.

Phys. 363 (2018), no. 3, 1003–1049.

4. David Kaup, Alan Newell. An exact solution for a derivative nonlinear

Schrodinger equation. J. Mathematical Phys. 19 (1978), no. 4, 798–

801.

5. Einar Mjolhus. Modulational instability of hydromagnetic waves par-

allel to magnetic field. J. Plasma Physics 16 (1976), 321–334.

SESSION 23: “Nonlinear waves in optics, fluids and plasma”

High-order accurate conservative finite differencesfor Vlasov equations in 2D+2V

J. W. Banks∗ and A. Gianesini Odu


110 8th street

Troy, NY USA


n this talk, we discuss numerical simulation for the Vlasov-Poisson

and Vlasov-Maxwell systems in phase space using high-order ac-

curate conservative finite difference algorithms. One significant

challenge confronting direct kinetic simulation is the significant

computational cost associated with high-dimensional phase space

descriptions. Here, we advocate the use of high-order accurate nu-

merical schemes as a means to reduce the computational cost re-

quired to deliver a given level of error in the computed solution.

We pursue a discretely conservative finite difference formulation

of the governing equations, and discuss fourth- and sixth-order

accurate schemes. In addition, we employ a minimally dissipa-

tive nonlinear scheme based on the well-known WENO approach.

These algorithms represent the core of the Eulerian-based kinetic

code LOKI [1, 2], which simulates solutions to Vlasov systems

in 2+2-dimensional phase space. To leverage large computational

resources, LOKI uses MPI parallelism, details of which are dis-

cussed here. Results of code verification studies using the method

of manufactured solutions are presented. Results are also presented

for the physically motivated scenarios including classical Landau

damping, and growth of longitudinal and transverse plasma insta-

bilities in single and multiple species plasmas.

1. J. W. BANKS AND A. GIANESINI ODU AND R. L. BERGER AND

T. CHAPMAN AND W. T. ARRIGHI AND S. BRUNNER, High-Order

Accurate Conservative Finite Difference Methods for Vlasov Equations

in 2D+2V, SIAM J. Sci. Comput. (submitted).

2. J. W. BANKS AND J. A. F. HITTINGER, A new class of non-linear,

finite-volume methods for Vlasov simulation, IEEE T. Plasma. Sci., 38

(2010), pp. 2198–2207.

Non-canonical Hamiltonian structure and integra-bility for 2D fluid surface dynamics

A. I. Dyachenko1, S. A. Dyachenko2, P. M. Lushnikov3,∗, V. E.

Zakharov1,4, and N. M. Zubarev5,6

1Landau Institute For Theoretical Physics, Russia, 2Department of Mathe-

matics, University of Illinois at Urbana-Champaign, USA, 3Department of

Mathematics and Statistics, University of New Mexico, USA, 4Department

of Mathematics, University of Arizona, USA, 5Institute for Electrophysics,

Yekaterinburg, Russia, 6Lebedev Physical Institute, Moscow, Russia

[email protected]

We consider 2D fluid surface dynamics. A time-dependent con-

formal transformation maps a fluid domain into the lower complex

half-plane of a new spatial variable [1]. The fluid dynamics is fully

characterized by the complex singularities in the upper complex

half-plane of the conformal map and the complex velocity. Both a

single ideal fluid dynamics (corresponds e.g. to oceanic waves dy-

namics) and a dynamics of superfluid Helium 4 with two fluid com-

ponents are considered. A superfluid Helium case is shown to be

completely integrable for the zero gravity and surface tension limit

with the exact reduction to the Laplace growth equation which is

completely integrable through the connection to the dispersionless

limit of the integrable Toda hierarchy and existence of the infinite

set of complex pole solutions [2]. A single fluid case with nonzero

gravity and surface tension turns more complicated with the infinite

set of new moving poles solutions found [3,4] which are however

unavoidably coupled with the emerging moving branch points in

the upper half-plane. Residues of poles are the constants of motion.

These constants commute with each other in the sense of underly-

ing non-canonical Hamiltonian dynamics [5]. It suggests that the

existence of these extra constants of motion provides an argument

in support of the conjecture of complete Hamiltonian integrability

of 2D free surface hydrodynamics [4,6].

[1] P.M. Lushnikov, S.A. Dyachenko and D.A. Silantyev, Proc. Roy. Soc.

A 473, 20170198 (2017).

[2] P.M. Lushnikov and N.M. Zubarev, Phys. Rev. Lett. 120, 204504

(2018).

[3] A. I. Dyachenko and V. E. Zakharov, Free-Surface Hydrodynamics in

the conformal variables, arXiv:1206.2046.

[4] A. I. Dyachenko, S. A. Dyachenko, P. M. Lushnikov and V. E. Za-

kharov, Dynamics of Poles in 2D Hydrodynamics with Free Surface: New

Constants of Motion. arXiv:1809.09584

[5] A. I. Dyachenko, P. M. Lushnikov and V. E. Zakharov, Non-Canonical

Hamiltonian Structure and Poisson Bracket for 2D Hydrodynamics with

Free Surface, Submitted to Journal of Fluid Mechanics (2018).

arXiv:1711.02841

61

[6] A. I. Dyachenko and V. E. Zakharov, Phys. Lett. A 190, 144-148

(1994).

Well-balanced discontinuous Galerkin methods forblood flow simulation with moving equilibrium

Jolene Britton and Yulong Xing

Department of Mathematics, University of California, Riverside

Department of Mathematics, The Ohio State University


The simulation of blood flow in arteries can be modeled by a sys-

tem of conservation laws and have a range of applications in med-

ical contexts. This system of partial differential equations is in

the same vein as the shallow water equations. We present well-

balanced discontinuous Galerkin methods for the blood flow model

which preserve the general moving equilibrium. Schemes for sys-

tems with zero-velocity have been recently been addressed, how-

ever we focus on the development of schemes that consider general

moving equilibrium. Recovery of well-balanced states via appro-

priate source term approximations and approximations of the nu-

merical fluxes are the key ideas. Numerical examples will be pre-

sented to verify the well-balanced property, high order accuracy,

and good resolution for both smooth and discontinuous solutions.

Invariant conserving local discontinuous Galerkinmethods for the modified Camassa-Holm equation

Zheng Sun and Yulong Xing

Department of Mathematics, Ohio State University, Columbus OH 43210


In this presentation, we design, analyze, and numerically test an

invariant preserving local discontinuous Galerkin method for solv-

ing the nonlinear modified Camassa-Holm equation. This model

is integrable and admits peakon solitons. The proposed numerical

method is high order accurate, and preserves two invariants, mo-

mentum and energy, of this nonlinear equation. The L2-stability

of the scheme for general solutions is a consequence of the energy

preserving property. The numerical simulation results for differ-

ent types of solutions of the modified Camassa-Holm equation are

provided to illustrate the optimal convergence rate, energy conser-

vation and other capability of the proposed method.

Non-uniqueness of Leray-Hopf weak solutions forthe 3D Hall-MHD system

Mimi Dai

University of Illinois at Chicago

[email protected]

We will talk about the non-uniqueness of weak solutions in Leray-

Hopf space for the three dimensional magneto-hydrodynamics with

Hall effect. We adapt the widely appreciated convex integration

framework developed in a recent work of Buckmaster and Vicol

for the Navier-Stokes equation, and with deep roots in a sequence

of breakthrough papers for the Euler equation.

A hydrodynamic formulation for solid-state ferro-magnetism

Ezio Iacocca*

University of Colorado at Boulder, Department of Applied Mathematics,

Colorado, USA

[email protected]

Ferromagnetic materials have been known to humanity for over

4 millennia, yet its properties continue to challenge our physical

understanding. Part of the difficulty (and beauty) of magnetism

is that quantum mechanical effects at sub-nanometer scales man-

ifest at macroscopic scales. Microscopic magnetization dynam-

ics are described by a vectorial partial differential equation known

as the Landau-Lifshitz equation (LLE). While numerical methods

are regularly utilized to solve the LLE, analytical approaches are

typically limited to linearized or weakly nonlinear regimes. In

this talk, I will present a hydrodynamic formulation for the LLE

equation that is amenable to analytical study in the nonlinear, dis-

persive regime[1]. I will discuss the paradigm of interpreting a

solid-state material in the context of a fluid and its relation to well-

known systems such as Bose-Einstein condensates and other fa-

miliar concepts such as sub and supersonic flow [2]. In the context

of effectively defocusing media, I will present a matched asymp-

totic solution for a spin channel with arbitrary injection strength

that sustain nonlinear waves in effectively one-dimensional chan-

nels [3, 4]. For weak, subsonic injection strength, the solution ex-

hibits an algebraic spatial profile. At large, supersonic injection

strength, a stationary soliton is formed within a narrow, bound-

ary layer near the injection site that is asymptotically matched to

an algebraic profile. The soliton effectively reduces the efficiency

of spin transport in the channel, a dispersive, coherent counterpart

to the onset of turbulence in pipe flow of a viscous fluid at high

Reynolds numbers.

1. E. Iacocca, T. J. Silva and M. A. Hoefer, Breaking of Galilean Invari-

ance in the Hydrodynamic Formulation of Ferromagnetic Thin Films,

Phys. Rev. Lett., 118 (2017), 017203.

2. E. Iacocca and M. A. Hoefer, Vortex-antivortex proliferation from an

obstacle in thin film ferromagnets, Phys. Rev. B, 95 (2017), 134409.

3. E. Iacocca, T. J. Silva and M. A. Hoefer, Symmetry-broken dissipa-

tive exchange flows in thin-film ferromagnets with in-plane anisotropy,

Phys. Rev. B, 96 (2017), 134434.

4. E. Iacocca and M. A. Hoefer, Hydrodynamic description of long-

distance spin transport through noncollinear magnetization states: the

role of dispersion, nonlinearity, and damping, arXiv:1812.10438

Weak solutions for the 3D Navier-Stokes equationswith discontinuous energy

Alexey Cheskidov and Xiaoyutao Luo

Department of Mathematics, Statistics and Computer Science,

University of Illinois At Chicago, Chicago, Illinois 60607


Since the classical work of Leray it is known that for any diver-

gence free initial data with finite energy there exists a weak solu-

tion to the 3D Navier-Stokes equations. We construct finite energy

wild solutions with various properties. First, we show that there

exists a weak solution whose jump discontinuities of the energy

profile are dense and of positive Lebesgue measure in time. The

proof relies on a family of approximate solutions to the stationary

Navier-Stokes equations and a new convex integration scheme. As

62

a byproduct, we also obtain finite energy nontrivial stationary weak

solutions to the unforced 3D Navier-Stokes equations.

Inverse cascade of gravity waves in the presence ofcondensate: numerical results and analytical expla-nation

Alexander O. Korotkevich


University of New Mexico, Albuquerque, NM, USA

and Landau Institute for Theoretical Physics, Moscow, Russia.

[email protected]

We consider direct numerical simulation of isotropic turbulence of

surface gravity waves in the framework of the primordial dynam-

ical equations. We use approximation of a potential flow of ideal

incompressible fluid. System is described in terms of weakly non-

linear equations [1] for surface elevation η(~r, t) and velocity po-

tential at the surface ψ(~r, t) (~r =−−−→(x, y))

η = kψ − (∇(η∇ψ))− k[ηkψ] + k(ηk[ηkψ])

+1

2∆[η2kψ] +

1

2k[η2∆ψ] + F−1[γkηk],

ψ = −gη − 1

2

[(∇ψ)2 − (kψ)2

]

− [kψ]k[ηkψ]− [ηkψ]∆ψ + F−1[γkψk] + P~r.

Here dot means time-derivative, ∆ — Laplace operator, k is a lin-

ear integral operator(

k =√−∆

), F−1 is an inverse Fourier trans-

form, γk is a dissipation rate, P~r is the driving term which simu-

lates pumping on small scales. These equations were derived as a

results of Hamiltonian expansion in terms of kη up to the fourth

order terms.

Like in works [2, 3] formation of long waves background (con-

densate) and inverse cascade was observed. This time all inver-

tial interval (range of scales where there is no pumping or damp-

ing, only nonlinear interaction of waves) in the inverse cascade re-

gion. Currently observed slopes of the inverse cascade are close to

nk ∼ k−3.15, which differ significantly from theoretically predicted

nk ∼ k−23/6 ≃ k−3.83. In our work we propose some analytical

analysis of results, which is in part based on recent works [4, 5].

1. V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of

Turbulence I (Springer-Verlag, Berlin, 1992).

2. A. O. Korotkevich, Phys. Rev. Lett., 101, 074504 (2008), 0805.0445.

3. A. O. Korotkevich, Math. Comput. Simul., 82, 1228 (2012),

0911.0741.

4. A. O. Korotkevich, JETP Lett., 97, 3 (2013), 126-130.

5. A. O. Korotkevich and V. E. Zakharov, Nonlin. Process. Geophys., 22,

(2015), 325-335.

SESSION 24: “Mathematical perspectives in quantum mechanics

and quantum chemistry”

Rigorous derivation of nonlinear Dirac equations forwave propagation in honeycomb structures

Jack Arbunich and Christof Sparber

Department of Mathematics, Statistics, and Computer Science, M/C 249,

University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607,

USA


We show how to rigorously obtain nonlinear equations of Dirac

type as an effective description for slowly modulated, weakly

nonlinear waves in honeycomb lattices. Both, local and nonlocal

Hartree-nonlinearities are discussed and connections to closely re-

lated earlier results in semiclassical analysis are pointed out. Our

results have recently been published in [1].

1. J. Arbunich and C. Sparber, Rigorous derivation of nonlinear Dirac

equations for wave propagation in honeycomb structures, J. Math.

Phys., 59 (2018), 011509, 19pp.

On the excited states of the interacting boson sys-tem: A non-Hermitian view

Dionisios Margetis and Stephen Sorokanich

Department of Mathematics, University of Maryland, College Park, MD

20742 USA


In this talk, we focus on modeling aspects of the weakly interacting

Boson system under periodic boundary conditions, as well as in

the presence of a trapping external potential. The main goal is to

provide an analytical description of the excited many-body states

of this system. A central theme is a non-unitary transformation

of the system Hamiltonian, which results in investigating a non-

Hermitian operator. This view is examined as an alternative to the

Bogoliubov transformation.

Nonlinear Schrodinger equations with a potential indimension 3

Avraham Soffer∗

Rutgers University

[email protected]

Fabio Pusateri


[email protected]

We present recent results and ongoing work on the long-time dy-

namics of small solutions of nonlinear Schrodinger equations with

potentials in 3 dimensions.

Inspired by problems related to the stability of (topological) soli-

tons, our general goal is to understand the global dynamics of dis-

persive and wave equations of the form

i∂tu + L(|∇|)u + V(x)u = N(u, u), u(t = 0) = u0,

for an unknown u : (t, x) ∈ R × Rd −→ C with small initial

condition u0, where L is the linear dispersion relation, V is a real

potential, and N is a nonlinear function vanishing quadratically

when u = 0.

63

In this talk we will give a global existence and pointwise decay

result in the case of the Schrodinger equation, L = −∆, in dimen-

sion 3 with a sufficiently smooth and decaying potential V with

no bound states, and a nonlinearity N = u2. Despite its apparent

simplicity, this model presents several difficulties since a quadratic

nonlinearity in 3d is critical with respect to the Strauss exponent;

moreover, even in the case V = 0, the nonlinearity u2 creates non-

trivial fully coherent interactions (unlike the case of N = u2, see

[1]). Using the Fourier transform adapted to the Schrodinger opera-

tor −∆+V, we are able to prove integrable-in-time decay through

a distorted Fourier analogue of weighted estimates. A key aspect

of our analysis is the development of novel multilinear harmonic

analysis techniques in this setting, which rely on a precise under-

standing of the “nonlinear spectral measure” and its singularities,

and extend the more manageable 1d analysis of [2].

1. P. Germain, Z. Hani and S. Walsh. Nonlinear resonances with a poten-

tial: multilinear estimates and an application to NLS. Int. Math. Res.

Not., IMRN (2015), 8484-8544.

2. P. Germain, F. Pusateri and F. Rousset. The nonlinear Schrodinger

equation with a potential in dimension 1. Ann. Inst. H. Poincare C,

(2018), 1477-1530.

A central limit theorem for integrals of random waves

Matthew de Courcy-Ireland


Princeton and EPF Lausanne

[email protected]

Marius Lemm∗


Harvard University

[email protected]

It is known from work of Han and Tacy that the mean-square of

random waves on Riemannian manifolds converges to a constant in

the high-frequency limit over shrinking balls. We establish a cen-

tral limit theorem for the appropriately normalized mean-square.

Concentration properties of Majorana spinors in theJackiw–Rossi theory

Akos Nagy


Duke University

[email protected]

Following the works of Jackiw et al. on the plane [?, ?], I will in-

troduce an Abelian gauge theory on Riemann surfaces. Physically,

the theory describes the surface excitations of a TI-SC interface.

Solutions of the corresponding variational equations are Majorana

spinors over Ginzburg–Landau vortices.

I will present my results on closed surfaces. The solutions posses

an interesting “concentration” property, which is in accordance

with the physical expectations. Using this concentration property I

will describe the solutions in the large coupling limit.

Edge states in honeycomb structures

Michael I Weinstein

Department of Applied Physics and Applied Mathematics

and Department of Mathematics

Columbia University

New York, NY

[email protected]

This talk concerns recent progress on the mathematical theory of

graphene and its artificial analogues with a focus on edge states, the

localization of energy about spatially extended line-defects. Two

types of line-defects are discussed: a) the interpolation between

deformed honeycomb media across a domain wall and b) honey-

comb media sharply terminated and interfaced with a vacuum. I’ll

discuss the roles played by the spectral properties of the single elec-

tron model for the bulk honeycomb structure, and the orientation

of the line-defect. Collaborations with A Drouot, CL Fefferman,

JP Lee-Thorp, J Lu, A Watson and Y Zhu.

1. A. Drouot , C. L. Fefferman and M. I. Weinstein,

Defect modes for dislocated periodic media,


2. C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein,

Bifurcations of edge states – topologically protected and non-protected

– in continuous 2D honeycomb structures, 2D Materials, 3 014008

(2015)

3. C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein, Edge states in

honeycomb structures, Annals of PDE, 2 #12 (2016)

4. C. L. Fefferman, J. P. Lee-Thorp and M. I. Weinstein, Honeycomb

Schroedinger operators in the strong-binding regime, Comm. Pure

Appl. Math., 71 #6 (2017)

5. C. L. Fefferman and M. I. Weinstein, Edge States of continuum

Schroedinger operators for sharply terminated honeycomb structures,


6. J. P. Lee-Thorp, M.I. Weinstein and Y. Zhu, Elliptic operators with

honeycomb symmetry; Dirac points, edge states and applications to

photonic graphene,

Arch. Rational Mech. Anal., https://doi.org/10.1007/s00205-018-

1315-4 (2018)

7. J. Lu, A. Watson and M. I. Weinstein, Dirac operators and domain

walls, https://arxiv.org/abs/1808.01378 (2018)

Boltzmann equations via Wigner transform and dis-persive methods

Thomas Chen∗, Ryan Denlinger, Natasa Pavlovic




[email protected]

In this talk, we present some of our recent work on the analysis

of Boltzmann equations with tools of nonlinear dispersive PDEs.

The starting point of our approach is to map the Boltzmann equa-

tion, by use of the Wigner transform, to an equation similar to a

Schrodinger equation in density matrix formulation with a nonlin-

ear self-interaction. We prove local well-posedness, propagation of

moments, and small data global well-posedness in spaces defined

by weighted space-time norms of Sobolev type.

64

1. T. Chen, R. Denlinger, N. Pavlovic, Local well-posedness for Boltz-

mann’s equation and the Boltzmann hierarchy via Wigner transform,

Commun. Math. Phys., to appear.

https://arxiv.org/abs/1703.00751

2. T. Chen, R. Denlinger, N. Pavlovic, Moments and Regularity for a

Boltzmann Equation via Wigner Transform, submitted.

https://arxiv.org/abs/1804.04019

New developments in quantum chemistry on a quan-tum computer

Artur F. Izmaylov∗

Department of Physical and Environmental Sciences, University of Toronto

Scarborough, Toronto, Ontario M1C 1A4, Canada

[email protected]

Quantum computers are an emerging technology intended to ad-

dress computational problems that are exponentially hard for clas-

sical computers. The electronic structure problem of quantum chem-

istry is one of such problems. One of the most practical approaches

to engaging currently available universal-gate quantum comput-

ers to this problem is the variational quantum eigensolver (VQE)

method. In this talk I will discuss two recent improvements of the

VQE method: 1) introducing symmetry constraints and 2) improv-

ing projective measurement process.

To create a robust and computationally efficient VQE approach that

would be able to access any electronic state of interest it is essential

to introduce symmetry constraints. Two approaches to introducing

symmetry constraints were considered: 1) the penalty functions

[1] and 2) constructing projectors on irreducible representations

of symmetry operators. It was found that even though the lat-

ter approach is more rigorous, its hardware resource requirements

make it practically infeasible. On the other hand, constrained VQE

through application of penalty functions can obtain electronic states

with a certain number of electrons and spin without significant ad-

ditional quantum resources.

Current implementations of the VQE technique involve splitting

the system qubit Hamiltonian into parts whose elements commute

within their single qubit subspaces. The number of such parts

rapidly grows with the size of the molecule, this increases the un-

certainty in the measurement of the energy expectation value be-

cause elements from different parts need to be measured indepen-

dently. To address this problem we introduce a more efficient par-

titioning of the qubit Hamiltonian using fewer parts that need to be

measured separately [2].

1. I.G. Ryabinkin, S.N. Genin, and A.F. Izmaylov, Constrained variational

quantum eigensolver: Quantum computer search engine in the Fock

space, J. Chem. Theory Comp., 15 (2019), 249-255.

2. A.F. Izmaylov, T.C. Yen, and I.G. Ryabinkin, Revising measurement

process in the variational quantum eigensolver: Is it possible to re-

duce the number of separately measured operators? arXiv preprint,

arXiv:1810.11602

A perturbation-method-based post-processing ofplanewave approximations for nonlinearSchrodinger equations

Benjamin Stamm

MathCCES, Schinkelstr. 2, 52062 Aachen, Germany

[email protected]

In this talk we consider a post-processing of planewave approxima-

tions for nonlinear Schrodinger equations by considering the exact

solution as a perturbation of the discrete, computable solution. Ap-

plying then Katos perturbation theory leads to computable correc-

tions with a provable increase of the convergence rate in the asymp-

totic range for a very little computational overhead. We illustrate

the key-features of this post-processing for the Gross-Pitaevskii

equation that serves as a toy problem for DFT Kohn-Sham mod-

els. Finally some numerical illustrations in the context of DFT

Kohn-Sham models are presented.

1. E. Cances, G. Dusson, Y. Maday, B. Stamm, M. Vohral180k, Post-

processing of the planewave approximation of Schrodinger equations.

Part I: linear operators, submitted, HAL preprint hal-01908039

2. E. Cances, G. Dusson, Y. Maday, B. Stamm, M. Vohral180k, A

perturbation-method-based post-processing for the planewave dis-

cretization of Kohn-Sham models, J. Comput. Phys., Vol. 307, pp.

446459 (2016)

3. E. Cances, G. Dusson, Y. Maday, B. Stamm, M. Vohral180k, A

perturbation-method-based a posteriori estimator for the planewave

discretization of nonlinear Schrodinger equations, C. R. Acad. Sci.

Paris., Vol. 352, No. 11, pp. 941-946 (2014)

Spinning Landau-Lifshitz solitons - a quantum me-chanical analogy

Christof Melcher

RWTH Aachen

[email protected]

In this talk we shall discuss dynamic excitations of topological

solitons in two-dimensional ferromagnets. We shall focus on sys-

tems without individual rotational symmetry in spin and coordinate

space, respectively. Examples include chiral skyrmions in magnets

without inversion symmetry and curvature stabilized vortices on a

spherical shell. As a consequence of reduced rotational symmetry,

the Hamiltonian dynamics governed by the Landau-Lifshitz equa-

tion lacks conservation of individual angular momenta, which may

be interpreted as an emerging spin-orbit phenomenon generating

joint rotations in spin and coordinate space. We shall examine vari-

ational formulations and existence of spinning solitons on spherical

shells by means of concentration-compactness methods combining

joint work with S. Komineas and Z. N. Sakellaris, respectively.

SESSION 25: “Nonlinear waves, singularities, vortices, and turbu-

lence in hydrodynamics, physical, and biological systems”

Powerful conformal maps for adaptive resolving ofthe complex singularities of Stokes wave

Denis A. Silantyev∗

Courant Institute, University of New York, New York, NY

[email protected]

Pavel M. Lushnikov

University of New Mexico, Albuquerque, NM

65

A new highly efficient method is developed for computation of

traveling periodic waves (Stokes waves) on the free surface of deep

water. The convergence rate of the numerical approximation is de-

termined by the complex singularities of the travelling wave in the

complex plane above the free surface [1]. An auxiliary conformal

mapping is introduced which moves the singularities away from

the free surface thus dramatically speeding up Fourier series con-

vergence of the solution by adapting the numerical grid for resolv-

ing singularities [2]. Three options for the auxiliary conformal map

are described with their advantages and disadvantages for numer-

ics. Their efficiency is demonstrated for computing Stokes waves

near the limiting Stokes wave (the wave of the greatest height) with

100-digit precision. Drastically improved convergence rate signif-

icantly expands the family of numerically accessible solutions and

allowing to study the oscillatory approach of these solutions to the

limiting wave in great detail.

1. Sergey A. Dyachenko, Pavel M. Lushnikov, Aleksander O. Korotke-

vich, The complex singularity of a Stokes wave, Pis’ma v ZhETF, vol.

98, iss. 11, pp. 767-771 (2013).

2. Pavel M. Lushnikov, Sergey A. Dyachenko, Denis A. Silantyev, New

conformal mapping for adaptive resolving of the complex singularities

of Stokes wave, Proc. Roy. Soc. A, vol. 473, 2202, (2017).

The Zakharov-Dyachenko conjecture on the integra-bility of gravity water waves

Massimiliano Berti

SISSA

[email protected]

Roberto Feola

Laboratoire de Mathematiques Jean Leray, Universite de Nantes

[email protected]

Fabio Pusateri∗

Mathematics Department, University of Toronto

[email protected]

We consider the gravity water waves system with a periodic one-

dimensional interface in infinite depth, and prove a rigorous reduc-

tion of these equations to Birkhoff normal form up to degree four.

This proves a conjecture of Zakharov-Dyachenko [3] based on the

formal Birkhoff integrability of the water waves Hamiltonian trun-

cated at order four. As a consequence, we also obtain a long-time

stability result: periodic perturbations of a flat interface that are of

size ǫ in a sufficiently smooth Sobolev space lead to solutions that

remain regular and small up to times of order ǫ−3. This is the first

such long-time existence result for quasilinear PDEs in the absence

of external parameters.

Some of the main difficulties in the proof are the quasilinear na-

ture of the equations, the presence of small divisors arising from

near-resonances, and non-trivial resonant four-waves interactions,

the so-called Benjamin-Feir resonances. The main ingredients that

we use are: (1) various reductions to constant coefficient opera-

tors through flow conjugation techniques; (2) the verification of

key algebraic properties of the gravity water waves system which

imply the integrability of the equations at non-negative orders; (3)

smoothing procedures and Poincare-Birkhoff normal form trans-

formations; (4) a normal form identification argument that allows

us to handle Benajamin-Feir resonances by comparing with the for-

mal computations of [3] and Craig-Worfolk [2] Craig-Sulem [1].

1. W. Craig and C. Sulem. Mapping properties of normal forms transfor-

mations for water waves. Boll. Unione Mat. Ital., 9 (2016), 289-318.

2. W. Craig and P. Worfolk. An integrable normal form for water waves

in infinite depth. Phys. D, 84 (1995), 3-4, 513-531.

3. V.E. Zakharov and A.I. Dyachenko. Is free-surface hydrodynamics an

integrable system? Physics Letters A, 190 (1994), 144-148.

Stability and noise in frequency combs:harnessing the music of the spheres

Curtis R. Menyuk, Zhen Qi, and Shaokang Wang

CSEE Dept., University of Maryland Baltimore County

1000 Hilltop Circle, Baltimore, MD 21250

[email protected]

Frequency combs have revolutionized the measurement of time

and frequency and impacted a wide range of applications spanning

basic physics, astrophysics, medicine, and defense. Frequency

combs are modeled mathematically at lowest order by the nonlin-

ear Schrodinger equation (NLSE), as is the case for many other

physical systems. Although the NLSE can yield important qualita-

tive insights, it is too simplistic to be useful for quantitative mod-

eling.

The key theoretical issues in understanding and designing frequency

combs are finding regions in the adjustable parameter space where

combs operate stably, determining their noise performance, and

optimizing them for high power, low noise, and/or large band-

width. Similar issues arise in many of the physical systems that are

modeled at lowest order by the NLSE. To date, these issues have

been studied either by using brute-force evolutionary simulations

or by using dynamical systems methods in nearly-analytical limits,

where the equations are too simplified to model the experimental

systems accurately.

In recent work, we have shown that these issues can be efficiently

and accurately addressed by combining 400-year-old dynamical

systems methods with modern computational techniques. Our com-

putational tools are 3–5 orders of magnitude faster than standard

evolutionary methods and provide important physical insight. We

have applied these tools to frequency combs from passively mod-

elocked lasers with fast and with slow saturable absorbers and to

frequency combs from microresonators. Our methods predict im-

proved operating regimes for combs that are produced from both

the passively modelocked lasers and the microresonators.

Despite our progress to date, there is much that remains to be done

to put the computational tools that we have developed on a firm

theoretical foundation and to make them sufficiently robust so that

they can be used on a broad range of modern-day experimental

frequency comb systems. We discuss the open questions, as well

as our progress.

Higher-order Runge–Kutta-type schemes based on

66

the method of characteristics for hyperbolic equa-tions with crossing characteristics

Taras I. Lakoba and Jeffrey S. Jewell

Department of Mathematics and Statistics, University of Vermont, Burling-

ton, VT 05401

[email protected]

The numerical Method of Characteristics (MoC) is widely used to

solve hyperbolic evolution equations. For example, for a system

w1, t + c w1, x = f1(w1, w2), w2, t − c w2, x = f2(w1, w2),

a change of independent variables: (x, t) → (ξi, t) for the ithequation (i = 1, 2), where ξ1 = x − c t and ξ2 = x + c t, re-

duces these partial differential equations to ordinary differential

equations (ODEs) along characteristics:

w1, t = f1(w1, w2) along ξ1 = const, (5a)

w2, t = f2(w1, w2) along ξ2 = const. (5b)

Each of the ODEs is then solved by an ODE numerical solver. One

of the main advantages of the MoC is that it preserves the linear

dispersion relation of the hyperbolic equations (), while allowing to

specify arbitrary (i.e., not only periodic) boundary conditions. One

of the main disadvantages of the MoC so far has been the fact that

among explicit ODE solvers of (5), only first- and second-order

accurate ones have been known. In this talk I will explain how one

can construct MoC schemes based on higher-order Runge–Kutta

(RK)-type ODE solvers.

To begin, I will show how the standard RK solver can be modified

for a system like (5), where each equation is solved along its own

characteristic. However, it turns out that such a modified algorithm

can become strongly numerically unstable. To overcome this insta-

bility, I will explain how the above modification can be applied to

a so-called pseudo-RK solver, which has not been found to suffer

from the instability problem. (A pseudo-RK solver is a hybrid be-

tween an RK and a multi-step solver.) Finally, I will explain how

non-periodic boundary conditions can be implemented for an MoC

scheme based on a higher-order pseudo-RK solver.

Efficient numerical methods for nonlinear dynamicswith random parameters

Adi Ditkowski, Gadi Fibich, and Amir Sagiv∗

Department of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel

[email protected], [email protected], and [email protected]

We present a novel numerical approach for the study of nonlinear

PDEs with random initial conditions or parameters. The naive ap-

proach to compute the statistics of these random dynamics, e.g., the

Monte-Carlo and histogram methods, might be prohibitively inef-

ficient. This problem has spurred the growth in recent years of the

field of uncertainty quantification. Specifically, the Polynomial-

Chaos Expansion (gPC), a spectrally-accurate algorithm for the

computation of statistical moments, has become widely popular.

Nevertheless, and perhaps surprisingly, we show that the gPC ap-

proach might fail to compute efficiently the probability density

function (PDF) of the model output.

Our newly developed spline-based method offer a good approxima-

tion of PDF, with theoretical guarantees [1]. Therefore, the method

may open a new road to the study of noise and randomness in non-

linear wave equations. We apply our numerical approach to pre-

dict the emergence of phase randomness in the Nonlinear Schro-

dinger equation (NLS) [2], random solitary waves interactions, the

emergence of polarization randomness in the coupled NLS [3], and

shock formation in the Burgers equation.

1. A. Ditkowski, G. Fibich, and A. Sagiv A spline-based approach

to uncertainty-quantification and density estimation. arXiv preprint,

arXiv:1803:10991 (2018).

2. A. Sagiv, A. Ditkowski, and G. Fibich. Loss of phase and universality

of stochastic interactions between laser beams. Opt. Exp., 25:24387–

24399, 2017.

3. G. Patwardhan, X. Gao, A. Sagiv, A. Dutt, J. Ginsberg, A. Ditkowski,

G. Fibich, and A. Gaeta. Loss of polarization in collapsing beams.

arXiv preprint, arXiv:1808.07019 (2018).

On density functional theory

Israel Michael Sigal

Dept. of Mathematics


[email protected]

In this talk I will review some recent results in the density func-

tional theory including the time-dependent one and the one cou-

pled to the electro-magnetic field. I will also formulate some open

problems. The talk is based on the joint results with Ilias Chenn.

Rogue waves in the nonlocal PT-symmetric nonlin-ear Schrodinger equation

Bo Yang and Jianke Yang

Department of Mathematics and Statistics, University of Vermont, Burling-

ton, VT 05405, USA

[email protected]; [email protected]

Rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger

(NLS) equation are studied. These waves are derived by the Dar-

boux transformation and bilinear KP reduction methods, and ex-

pressed as determinants in terms of Schur polynomials. Unlike

rogue waves in the local NLS equation, the present rogue waves

show a much wider variety. For instance, the polynomial degrees

of their denominators can be not only n(n + 1), but also n(n −1) + 1, n2 and other integer values, where n is an arbitrary posi-

tive integer. Dynamics of these rogue waves is also examined. It

is shown that these rogue waves can be bounded for all space and

time or develop collapsing singularities, depending on their types

as well as values of their free parameters. In addition, the solution

dynamics exhibits rich patterns, most of which have no counter-

parts in the local NLS equation.

1. B. Yang and J. Yang, “Rogue waves in the PT-symmetric nonlinear

Schrodinger equation”, Lett. Math. Phys. DOI: 10.1007/s11005-018-

1133-5 (2018).

2. B. Yang and J. Yang, “On general rogue waves in the parity-time-

symmetric nonlinear Schrodinger equation”, preprint.

67

Family of potentials with power-law kink tails

Avadh Saxena

Los Alamos National Lab, USA

[email protected]

Avinash Khare∗

Savitribai Phule Pune University, India

[email protected]

We provide examples of a large class of one dimensional higher

order field theories with kink solutions which asymptotically have

a power-law tail either at one end or at both ends. We provide

analytic solutions for the kinks in a few cases but mostly provide

implicit solutions. We also provide examples of a family of poten-

tials with two kinks, both of which have power law tails either at

both ends or at one end. In addition, we show that for kinks with a

power law tail at one end or both the ends, there is no gap between

the zero mode and the continuum of the corresponding stability

equation. This is in contrast to the kinks with exponential tail at

both the ends in which case there is always a gap between the zero

mode and the continuum [1].

1. A. Khare and A. Saxena, Family of potentials with power-law kink

tails, arXiv:1810.12907

Dynamical problems arising in blood flow:nonlinear waves on trees

Jerry Bona∗

Address: Department of Mathematics, Statistics and Computer Science

The University of Illinois at Chicago

[email protected]

Pulmonary arterial hypertension is a pernicious disease whose only

curative treatment at the moment is lung or heart-lung transplant.

One of the characteristics of this disease is the right-ventricle re-

modeling that occurs because the heart is asked to work harder due

to the pressure overload imposed by the pulmonary vasculature.

In this lecture, we will discuss an ongoing project aimed at ob-

taining a better understanding of this disease. Mathematically, this

comes down to a large coupled system of nonlinear wave equations

whose spatial domain is a rooted tree. Preliminary analysis of the

system is put forth and some comparisons with real data provided.

Singularities in the 2D fluids with free surface

Sergey Dyachenko∗

Department of Mathematics, University of Illinois at Urbana–Champaign

[email protected]

Alexander Dyachenko

Landau Institute for Theoretical Physics

[email protected]

Pavel Lushnikov

Department of Mathematics and Statistics, University of New Mexico

[email protected]

Vladimir Zakharov

Department of Mathematics, University of Arizona

[email protected]

We explore the singularities in the analytic continuation of the ve-

locity potential to the exterior of the fluid domain enclosed under

the free boundary. We demonstrate that certain classes of singu-

larities are persistent under the evolution in Euler equations [1].

Moreover, these singularities are associated with new, previously

undiscovered nontrivial constants of motion. Some of these motion

constants have been shown to commute under the Poisson bracket,

and suggest that free–surface hydrodynamics may have more hid-

den structure then previously discovered.

We demonstrate the results of the numerical simulations and illus-

trate with reconstruction of analytical structure of the fluid poten-

tial outside of fluid.

1. A. I. Dyachenko, S. A. Dyachenko, P. M. Lushnikov, V. E. Zakharov,

Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Con-

stants of Motion, JFM submitted (2018)

Nonlinear waves acting like linear waves in NLS

Katelyn (Plaisier) Leisman∗

University of Illinois

Dept. of Mathematics, Altgeld Hall, 1409 Green Street, Urbana, IL 61801

[email protected]

Gregor Kovacic


110 8th Street, Amos Eaton, Troy, NY 12180

[email protected]

David Cai

Shanghai Jiao Tong University, China

Courant Institute of Mathematical Sciences, New York University, USA

[email protected]

The linear part of the Nonlinear Schrodinger Equation (NLS) (iqt =qxx) has dispersion relation ω = k2. We don’t expect solutions to

the fully nonlinear equation to behave nicely or have any kind of

effective dispersion relation like this. However, I have seen that so-

lutions to the NLS are actually weakly coupled and are often nearly

sinusoidal in time with a dominant frequency, often behaving sim-

ilarly to modulated plane waves.

Instantons and fluctuations in complex systems

Tobias Schafer

City University of New York

[email protected]

After a short overview of path integral techniques and their rela-

tionship to large deviation theory, I will present recently devel-

oped methods to compute instantons (minimizers of the Freidlin-

Wentzell functional) in complex stochastic systems. The stochas-

tically driven Burgers equation [1] and the stochastic nonlinear

Schrodinger equation [2] will serve as examples. In addition to

the instanton, it is often desirable to also take into account fluc-

tuations in order to compute the prefactor. I will discuss recently

developed computational methods involving the solution of the as-

sociated matrix-Riccati equation.

68

1. T. Grafke, R. Grauer, T.Schafer, and E. Vanden-Eijnden, Relevance

of instantons in Burgers turbulence. Eurphysics Letters, 109 (2015)

34003.

2. G. Poppe and T.Schafer: Computation of minimum action paths of the

stochastic nonlinear Schrodinger equation with dissipation. J. Phys. A:

Math. Theor., 51, (2018) 335102.

Clebsch variables for stratified compressible fluids

Benno Rumpf

Mathematics Department, Southern Methodist University, Dallas, Texas

[email protected]

Clebsch variables provide a canonical Hamiltonian representation

of the Euler equation. While this is desirable from a theoretical per-

spective, Clebsch variables have practical disadvantages: Firstly,

it is often difficult to compute the initial conditions for the Cleb-

sch variables from the initial conditions of the velocity field. Sec-

ondly, Clebsch variables usually show ’non-physical’ divergences

that pose difficulties on perturbation expansions. In my talk, I will

discuss strategies to overcome these difficulties.

1. R. Salmon, Hamiltonian fluid mechanics, Ann. Rev. Fluid Mech., 20

(1988), 225-256.

Appearance of Stokes waves in deep water

Anastassiya Semenova∗, Alexander Korotkevich, Pavel Lush-

nikov


University of New Mexico


[email protected]

We study evolution of a finite amplitude monochromatic wave in

deep ocean taking into account gravity but not capillary effects. We

simulate one period of Stokes wave to allow for superharmonics,

and avoiding modulational instability at wavelengths longer than

the initial spatial period of monochromatic wave. We investigate

the possibility of generation of Stokes waves in Euler equations in

the long time limit.

Chiral magnetic skyrmions for 2D Landau-Lifshitzequations

Stephen Gustafson∗

University of British Columbia

[email protected]

Landau-Lifshitz equations are the basic dynamical equations in a

micromagnetic description of a ferromagnet. They are naturally

viewed as geometric evolution PDE of dispersive (“Schrodinger

map”) or mixed dispersive-diffusive type, which scale critically

with respect to the physical energy in two dimensions. We de-

scribe recent results on existence and stability of important topo-

logical soliton solutions known as “chiral magnetic skyrmions”.

Joint work with Li Wang.

Stable blow-up dynamics in the critical and super-critical NLS and Hartree equations

Svetlana Roudenko∗ and Kai Yang

Department of Mathematics and Statistics, DM430

Florida International University, Miami, FL 33199


Yanxiang Zhao


George Washington University, Washington DC 20052

[email protected]

We study stable blow-up dynamics in the nonlinear Schrodinger

(NLS) equation and generalized Hartree equation with radial sym-

metry in the L2-critical and supercritical regimes. The NLS equa-

tion is with pure power nonlinearity iut + ∆u + |u|2σu = 0, and

the Hartree equation is a Schrodinger-type equation with a non-

local, convolution-type nonlinearity in dimension d: iut + ∆u +(|x|−(d−2) ∗ |u|p

)|u|p−2u = 0, p ≥ 2.

First, we consider the L2-critical case for the NLS equation in di-

mensions 4 ≤ d ≤ 12 and for the Hartree in dimensions d =3, 4, 5, 6, 7. We show that a generic blow-up in both equations ex-

hibits not only the rate ‖∇u(t)‖L2 ∼ (T − t)−12 , but also the

“log-log” correction, thus, behaving similarly to the stable collapse

in the lower dimensional NLS (such as the 2d cubic NLS). In this

setting we also study blow-up profiles and show that generic blow-

up solutions converge to the rescaled Q, the ground state solution

of the elliptic equations, which is well-known in the NLS case:

−∆Q + Q − Q2σ+1 = 0, and for the Hartree it is −∆Q + Q −(|x|−(d−2) ∗ |Q|p

)|Q|p−2Q = 0.

Next, we examine the L2-supercritical cases for both equations.

For the self-similar blow-up solutions we study the profile equa-

tions and discuss the existence and local uniqueness theory of the

solutions. We then show that our numerical simulations indicate

that the solutions Q to such profile equations exhibit a multi-bump

structure, and thus, in a sense, not unique. Direct numerical sim-

ulations of the NLS and generalized Hartree equations by the dy-

namic rescaling method indicate that only one of those multi-bump

profile solutions serves as the stable blow-up profile. We also in-

vestigate the rate of the blow-up and obtain the square root blow-up

rate without any corrections. Our findings indicate that the nonlin-

earity type in the Schrodinger-type equations is not essential for

the stable collapse formation.

Slow light propagation in two-level active media

Katelyn Plaisier-Leisman

University of Illinois at Urbana Champaign

[email protected]

Gino Biondini

University at Buffalo

[email protected]

Gregor Kovacic∗


[email protected]

69

In ruby crystals, slow light pulses were observed, and described

using two-level Maxwell-Bloch equations with high polarizability

damping. We compute that two regimes exist, depending on the

ratio of medium-polarizability and level-inversion damping. When

this ratio is moderate, soliton-like pulses exist. Damping decreases

their amplitudes and speed. A precursor of radiation coexists, and

dominates for strong damping and large damping ratio. Starting

slowly, it accelerates to the speed of light.

Expansion of the strongly interacting superfluidFermi gas: symmetry and self-similar regimes

E.A. Kuznetsov(a),(b), M.Yu. Kagan(c) and A.V. Turlapov(d)

(a) P.N. Lebedev Physical Institute RAS, Moscow, Russia(b) L.D. Landau Institute for Theoretical Physics RAS, Chernogolovka,

Moscow region, Russia(c) P.L. Kapitza Institute of Physical Problems RAS, Moscow, Russia(d) Institute of Applied Physics RAS, Nizhnii Novgorod, Russia

We consider an expansion of the strongly interacting superfluid

Fermi gas in the vacuum in the so-called unitary regime when

the chemical potential µ ∝ h2/mn−2/3 where n is the density of

the Bose-Einstein condensate of Cooper pairs of fermionic atoms.

Such expansion can be described in the framework of the Gross-

Pitaevskii equation (GPE) [1]. Because of the chemical potential

dependence on the density ∼ n−2/3 the GPE has additional sym-

metries resulting in existence of the virial theorem connected the

mean size of the gas blob and its Hamiltonian. It leads asymptot-

ically at t → ∞ to the ballistic expansion of the gas. We care-

fully study such asymptotics and reveal a perfect matching be-

tween the quasi-classical self-similar solution and the ballistic ex-

pansion of the non-interacting gas. This matching is governed by

the virial theorem derived in [2] utilizing the Talanov transforma-

tion [3] which was first obtained for the stationary self-focusing

of light in the media with cubic nonlinearity due to the Kerr ef-

fect. In the quasi-classical limit the equations of motion coincide

with 3D hydrodynamics for the perfect gas with γ = 5/3. Their

self-similar solution describes, on the background of the gas ex-

pansion, the angular oscillations of the gas shape in the framework

of the Ermakov-Ray-Reid type system.

[1] L.P.Pitaevskii, Superfluid Fermi liquid in a unitary regime, Physics

Uspekhi , v.51, pp.603-608, 2008.

[2] E.A.Kuznetsov, S.K. Turitsyn, Talanov transformation in self-focusing

problems and instability of stationary waveguides, Phys.Lett., v.112 A, pp.

273-276, 1985.

[3] V.I. Talanov, On the self-focusing of light in the cubic media, Pis’ma

Zh.Eksp.Teor.Fiz., v.11, p.303, 1970.

Anomalous correlators, ghost waves and nonlinearstanding waves in the β-FPUT system

Joseph Zaleski

Rensselaer Polytechnic Institute Troy NY 12203

[email protected]

Miguel Onorato

Dip. di Fisica, Universit di Torino and INFN, Sezione di Torino, Via P.

Giuria, 1, Torino, 10125, Italy

[email protected]

Yuri V Lvov∗

Rensselaer Polytechnic Institute Troy NY 12203

[email protected]

We investigate the celebrated β-Fermi-Pasta-Ulam-Tsingou

(FPUT) chain and establish numerically and theoretically the ex-

istence of the second order anomalous correlator. The anomalous

correlator manifests in the frequency-wave number Fourier spec-

trum as a presence of “ghost” waves with negative frequency, in

addition to the waves with positive frequencies predicted by the

linear dispersion relationship. We explain theoretically the exis-

tence of anomalous correlator and the ghost waves by nonlinear

interactions between waves. Namely, we generalize the classical

Wick’s decomposition by including the second order anomalous

correlator and show that the latter is responsible for the presence

of such “ghost” waves. From a physical point of view, the develop-

ment of the anomalous correlator is related to formation of nonlin-

ear standing waves. Indeed, we show numerically in the nonlinear

regime a transition from pure travelling waves to standing waves.

We predict that similar phenomenon might occur in nonlinear sys-

tem dominated by nonlinear interactions, including surface gravity

waves.

Excitation of interfacial waves via near-resonant sur-face — interfacial wave interactions

Joseph Zaleski∗

Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180

[email protected]

Philip Zaleski

New Jersey Institute of Technology, 323 Dr Martin Luther King Jr Blvd,

Newark, NJ 07102

[email protected]

Yuri Lvov

Rensselaer Polytechnic Institute,110 8th Street, Troy, NY 12180

[email protected]

The term “ocean waves” typically evokes images of surface waves

shaking ships during storms in the open ocean, or breaking rhyth-

mically near the shore. However, much of the ocean wave action

takes place far underneath the surface, and consists of surfaces of

constant density being disturbed and modulated. The relationship

between surface and interfacial waves provides a mechanism for

coupling of the atmosphere and the ocean—wind creates surface

waves, which in turn distribute energy to the lower bulk of the

ocean.

We consider interactions between surface and interfacial waves in

a two layer system, based on the novel Hamiltonian discovered by

Choi [2]. Our approach includes the general procedure for diag-

onalization of the quadratic part of the Hamiltonian. This allows

us to derive from first principles the coupled kinetic equations de-

scribing spectral energy transfers in this system and analyze the

interaction crossection between surface and interfacial waves. No-

tably, interactions are not limited to resonant wavenumbers. The

kinetic equations include the effects of “near”—resonant interac-

tions, physically motivated by observed changes in the shape of

70

the spectra along nonresonant wavenumbers. We find that the en-

ergy transfers are dominated by the generalization of the class III

resonances described in Alam [1]. We apply our formalism to cal-

culate the rate of growth of interfacial waves for different values of

the wind velocity and simulate the system of kinetic equations for

the case describing the evolution of coupled 1-D spectra.

1. Mohammad-Reza Alam, Journal of fluid mechanics, 691 (2012), 267-

278.

2. Wooyoung Choi, private communications.

Optical phase modulated nonlinear waves in agraphene waveguide

G. T. Adamashvili

Technical University of Georgia,

Kostava str.77, Tbilisi, 0179, Georgia.

[email protected]

D. J. Kaup∗

Department of Mathematics & Institute for Simulation and Training

University of Central Florida,

Orlando, Florida, 32816-1364, USA.

[email protected]

The different mechanisms that bring about the creation of opti-

cal nonlinear waves in a waveguide containing a graphene mono-

layer (or graphene-like two-dimensional material) are studied in

the general case when resonant and nonresonant nonlinearities are

simultaneously included. The conditions for the formation of op-

tical hybrid, nonresonant and resonant phase modulated breathers

in graphene, for waveguide TE-modes, are presented. It is shown

that the characteristic parameters of these optical nonlinear waves,

depends on the graphene Kerr-type third-order susceptibility, the

graphene conductivity, the reciprocal of Beers absorption length,

as well as the initial values of the ensemble of the atomic system

and/or the semiconductor quantum dots that are embedded in the

transition layer. In the case of the amplifier (active atomic system)

transition layer, the conditions for the existence of a dark (topo-

logical) breather, as well as the conditions when a nonlinear wave

cannot be formed, are determined and given. An explicit analyti-

cal expression for the profile of an optical nonlinear wave is also

presented.

SESSION 26: “Physical applied mathematics”

Linear instability of the Peregrine breather:Numerical and analytical investigations

Constance Schober∗ and Maria Strawn

Dept. of Mathematics, University of Central Florida, FL


Anna Calini

Dept. of Mathematics, College of Charleston, SC

[email protected]

We study the linear stability of the Peregrine breather both nu-

merically and with analytical arguments based on its derivation

as the singular limit of a single-mode spatially periodic breather

as the spatial period becomes infinite. By constructing solutions

of the linearization of the nonlinear Schrodinger equation in terms

of quadratic products of components of the eigenfunctions of the

Zakharov-Shabat system, we show that the Peregrine breather is

linearly unstable. A numerical study employing a highly accurate

Chebychev pseudo-spectral integrator confirms exponential growth

of random initial perturbations of the Peregrine breather.

1. A. Calini, C.M. Schober and M. Strawn, Linear Instability

of the Peregrine Breather: Numerical and Analytical Investi-

gations, APNUM, online publication (2018). DOI information

10.1016/j.apnum.2018.11.005

Oblique dispersive shock waves in steady shallow wa-ter flows

Adam Binswanger∗, Patrick Sprenger

Department of Applied Mathematics 526 UCB, University of Colorado,

Boulder, CO 80309-0526, USA


Mark Hoefer

Department of Applied Mathematics 526 UCB, University of Colorado,

Boulder, CO 80309-0526, USA

[email protected]

Steady shallow water flows are studied for a boundary value prob-

lem that corresponds to the deflection of a supercritical flow of a

thin sheet of water past a slender wedge. Due to surface wave

dispersion, the ensuing steady structure is a spatially extended, os-

cillatory pattern referred to as an oblique dispersive shock wave

(DSW), which can be approximated as a modulated nonlinear wave-

train limiting to an oblique solitary wave at one edge and small

amplitude harmonic waves at the other edge. This corner wedge

boundary value problem is modeled by a weakly nonlinear model

of KdV-type that incorporates higher order dispersion. Asymptotic

analysis, numerical simulations, and an in-house shallow water ex-

periment demonstrate evidence of a bifurcation in the flow pattern

as a control parameter (the wedge angle) is varied. The Bond num-

ber, B, measuring the effects of surface tension relative to grav-

ity, characterizes the bifurcation and is controlled by appropriate

variation of water depth. The bifurcation, a result of higher order

dispersion, occurs near B = 1/3, corresponding to a fluid depth

of approximately 5 mm, and is a transition between classical and

non-classical DSW profiles. They are differentiated by the mono-

tonicity or lack thereof of the solitary wave edge as well as the

structure of the modulated nonlinear wavetrain that ensues.

Solitons and pseudo-solitons in the Korteweg-deVries equation with step-up boundary conditions

Mark Ablowitz1, Xu-Dan Luo2, Justin Cole1,∗1 Department of Applied Mathematics

University of Colorado, Boulder2 Department of Mathematics


[email protected]

71

The Korteweg-deVries (KdV) equation with step-up boundary con-

ditions is considered, with an emphasis on soliton dynamics. When

an initial soliton is of sufficient size, it can propagate through the

step; in this case, the phase shift is calculated via the inverse scat-

tering transform. On the other hand, when the amplitude is not

large enough, the wave becomes “trapped” inside the ramp region.

In the trapped case, the transmission coefficient of the associated

linear Schrodinger equation can become large at a point exponen-

tially close to the continuous spectrum. This point is referred to as

a pseudo-embedded eigenvalue. Employing the inverse problem, it

is shown that the continuous spectrum associated with a branch cut

in the neighborhood of the pseudo-embedded eigenvalue plays the

role of discrete spectra, which in turn leads to a trapped soliton or

“pseudo-solitons” in the KdV equation.

Various dynamical regimes, and transitions from ho-mogeneous to inhomogeneous steady states in oscil-lators with delays and diverse couplings

Ryan Roopnarain and S. Roy Choudhury


University of Central Florida


This talk will involve coupled oscillators with multiple delays, and

dynamic phenomena including synchronization at large coupling,

and a variety of behaviors in other parameter ranges including tran-

sitions between Amplitude Death and Oscillation Death. Both an-

alytic multiple scale and energy methods, as well as numerical re-

sults will be presented. Behaviors in both limit cycle and chaotic

oscillators will be compared for various couplings. Finally, the ef-

fects of distributed delays will be considered for systems already

treated using discrete delays, including bifurcation theory results

not available in the latter case.

Spectral renormalization algorithm applied to solv-ing initial-boundary value problems

Sathyanarayanan Chandramouli and Ziad Musslimani

MCH 221 (c), Florida State University and LOV 218, Florida State Uni-

versity


The Spectral Renormalization algorithm was developed as a novel

numerical scheme for soliton solutions. [1] The main theme of

the work was to transform the equation governing the soliton into

Fourier space and determine a non-linear, nonlocal integral equa-

tion coupled to an algebraic equation. The coupling was seen to

enforce the convergence of the constructed fixed point iteration

scheme. The method was envisioned to have wide applications

in diverse areas, including Bose-Einstein condensation and fluid

mechanics. The work was extended to the time domain in or-

der to solve Initial-Boundary value problems (IBVP) using a time-

dependent spectral renormalization algorithm.[1] Here, a conver-

gent fixed point iteration scheme was constructed by introducing a

time-dependent renormalization factor. This renormalization fac-

tor is computed either from equation(s) expressing conservation

of a physically relevant quantity, or physically relevant dissipation

rate equation(s). Thus, besides facilitating the convergence of the

fixed point iteration scheme, the physics underlying the problem is

incorporated by the simulator at every iteration, helping the scheme

mimic the behaviour of the original IBVP.

The present work aims at studying the application of the spectral

re-normalization method to the proto-typical dissipative, Burger’s

equation subject to periodic boundary conditions. Having repro-

duced the numerical experiments for the case presented in [1], the

work explores the incorporation of higher order Cauchy-Filon in-

tegration methods into the fixed point iteration. The rationale for

the incorporation is to test the robustness of the algorithm (con-

vergence of the iteration with the use of large time steps), coupled

to high accuracy. Comparisons between the different integration

strategies will be laid out, in order to explore the potential of the

algorithm.

1. M. J. Ablowitz and Z. H. Musslimani, Spectral renormalization method

for computing self-localized solutions to nonlinear systems, OPTICS

LETTERS, Vol. 30, No.16 (2005), 2140-2142.

2

2. J. T. Cole and Z. H. Musslimani, Time-dependent spectral renormal-

ization method, arXiv:1702.06851v2, (2 Aug 2017), 2140-2142.

Time-dependent spectral renormalization method ap-plied to conservative PDEs

Abdullah Aurko∗ and Ziad H. Musslimani

Department of Mathematics, Florida State University, Tallahassee, FL,

32306-4510


The time-dependent spectral renormalization method was first in-

troduced by Cole and Musslimani as a numerical means to simu-

late linear and nonlinear evolution equations [1]. The essence of

the method is to convert the underlying evolution equation from its

partial or ordinary differential form (using Duhamel’s principle)

into an integral equation. The solution sought is then viewed as a

fixed point in both space and time. The resulting integral equation

is then numerically solved using a simple renormalized fixed-point

iteration method. Convergence is achieved by introducing a time-

dependent renormalization factor which is numerically computed

from the physical properties of the governing evolution equation.

The most profound feature of the method is that it has the ability

to incorporate physics into the simulations in the form of conser-

vation laws.

In this paper, we apply this novel scheme on the classical nonlinear

Schrodinger (NLS) equation- a benchmark evolution equation, and

a prototypical example of a conservative dynamical system. We

consider the classical NLS equation as a test bed for the perfor-

mance of the time-dependent spectral renormalization scheme be-

cause: 1) It has wide physical applications, such as in optics, con-

densed matter physics, and fluid mechanics (deep water waves). 2)

The classical NLS is an integrable evolution equation that admits

an infinite number of conserved quantities. The second property

is what we aim to explore. For the NLS equation, we have the

following three conserved quantities: power, momentum, and the

Hamiltonian (energy). We first incorporate each conserved phys-

ical quantity separately, using the method, as outlined in [1], but

using a higher order integration technique for evaluating the time

integral. After that, we proceed to successfully incorporate more

72

than one physically conserved quantity simultaneously, using the

time-dependent spectral renormalization method. Future work in-

volves repeating the same procedure for the Korteweg-de Vries

(KdV) equation.

1. J. T. Cole, and Z. H. Musslimani, Time-dependent spectral renormal-

ization method, Physica D: Nonlinear Phenomena, 358 (2017), 15-24.

On N-soliton interactions: Effects of local and non-local potentials

V.S. Gerdjikov1 and M.D. Todorov2

1Institute of Mathematics and Informatics and Institute for Nuclear Re-

search and Nuclear Energy, BAS, Sofia, Bulgaria2Dept of Applied Mathematics and Computer Science, Technical Univer-

sity of Sofia, 1000 Sofia, Bulgaria


We study the dynamical behavior of the N-soliton trains of nonlin-

ear Schrodinger equation (NLSE) perturbed by local and nonlocal

potential terms:

i∂u

∂t+

1

2

∂2u

∂x2+ |u|2u(x, t) + V(x)u(x, t)

+ gu(x, t)

∞∫

−∞

R(|x − y|)|u(y, t)|2dy = 0. (6)

The effects of several types of local potentials V(x) have been ana-

lyzed earlier both for the NLSE and the Manakov model, see [1, 2].

Recently Salerno and Baizakov pointed out [3] that specific nonlo-

cal potentials with R(z) = (1/(√

2πw)) exp(−z2/(2w2)) may

lead to formations of bound state of solitons with molecular-like

interactions, i.e., attractive at long distances and repulsive at short

distances. Our aim is to check whether their results are compatible

with the adiabatic approximation. We derive perturbed complex

Toda chain like in [1, 2] with additional terms accounting for the

nonlocal potential R(z). We show that the soliton interactions dy-

namic compares favorably to full numerical results of the original

NLSE, Eq. (6).

1. M. D. Todorov, V. S. Gerdjikov and A. V. Kyuldjiev, Multi-soliton in-

teractions for the Manakov system under composite external potentials,

Proceedings of the Estonian Academy of Sciences, Phys.-Math. Series,

64, No. 3 (2015), 368-378.

2. V. S. Gerdjikov and M. D. Todorov, Manakov model with

gain/loss terms and N-soliton interactions: Effects of pe-

riodic potentials, Journal Applied Numerical Mathematics,

https://doi.org/10.1016/j.apnum.2018.05.015, arXiv:1801.04897v1

[nlin.SI].

3. M. Salerno and B. B. Baizakov, Normal mode oscillations of a nonlocal

composite matter wave soliton, Phys. Rev. E, 98 (2018), 062220.

Anomalous waves induced by abrupt changes in to-pography

Nick Moore

Florida State University

[email protected]

I will discuss laboratory experiments on randomized surface waves

propagating over variable bathymetry. The experiments show that

an abrupt depth change can qualitatively alter wave statistics, trans-

forming an initially Gaussian wave field into a highly skewed one.

The altered wave field conforms closely to a gamma distribution,

which offers a simple way to estimate statistical quantities such as

skewness or kurtosis. Compared to Gaussian, the relatively slow

decay of the gamma distribution indicates an elevated level of ex-

treme events, i.e. rogue waves. In our experiments, the probability

of a rogue wave can be up to 50 times greater than would be ex-

pected from normal statistics.

SESSION 27: Canceled

SESSION 28: “Recent advances in analytical and computational

methods for nonlinear partial differential equations”

Optimal control of HPV infection and cervical can-cer with HPV vaccine

Kinza Mumtaz, Mudassar Imran, Adnan Khan

Lahore University of Management Sciences


In this paper, we develop an HPV epidemic model and transmis-

sion dynamics from susceptible population infected by Human Pa-

pilloma Virus into cervical cancer. For ideal control under vaccina-

tion program, we have utilized one type of vaccination: a bivalent

vaccine that objectives two HPV composes (16 and 18). To portray

the cooperation of vaccinated and the other four classes (suscepti-

ble, infected, precancerous and cancerous), we built up a system of

five ODEs. Under constant vaccination controls, the basic repro-

duction number R0 and the disease-free equilibrium for the given

model are calculated in terms of related parameters. Also the sta-

bility of the disease-free equilibrium of the given model in terms

of R0 is established which is locally asymptotically stable when

R0 < 1 and unstable when R0 > 1 and globally stability occurs

when R0 ≤ 1. Using PRCC technique sensitivity analysis is ad-

ditionally investigated to review the influence of model parameters

on the Human Papilloma Virus infection widespread. Expecting

infection predominance below the consistent control, ideal control

hypothesis is utilized to detail vaccination methodologies for the

given model once the vaccination rate is performed of your time.

The result of those techniques on the infected population and there-

fore the accrued price is assessed and contrasted with the consistent

control case.

Keywords: HPV; Vaccine; Mathematical Model; Stability Analy-

sis; Sensitivity Analysis; Optimal Control

Applications of fixed point theorems to integral anddifferential equations

Muhammad Arshad Zia

International Islamic University, Islamabad Pakistan

[email protected]

73

The fixed point theory is one of the most rapidly growing topic of

nonlinear functional analysis. It is a vast and interdisciplinary sub-

ject whose study belongs to several mathematical domains such

as: classical analysis, functional analysis, operator theory, topol-

ogy and algebraic topology, etc. This topic has grown very rapidly

perhaps due to its interesting applications in various fields within

and out side the mathematics such as: integral equations, initial

and boundary value problems for ordinary and partial differential

equations, many existence theorems for the solution of differential

equations are proved by means of fixed point theorems.

Inspired by the fact that the famous Banach contraction princi-

ple has a lot of applications in theory of integral and differential

equations and looking into the applications of fixed point theory

in various domains, we have introduced a new concept of Fixed

point Theory to solve the Differential and Integral Equations. Us-

ing fixed point theory, we have verified the existence and unique-

ness of solutions for differential and integral equation. We have

also focused ourselves to establish a new fixed point theorem for

generalized contraction mappings in complete metric spaces. We

have illustrated examples to advocate the usability of our results.

A collocation method for a class of a nonlinear par-tial differential equations

Muhammad Usman

University of Dayton, 300 College Park, Dayton OH 45469-2316, USA

[email protected]

Collocation methods have attracted the attention of computational

mathematicians during the last decade. In this talk, we will discuss

some analytical results on an initial and boundary value problems

of the Korteweg-de Vries type equation. Numerical results are pre-

sented to show the verification of analytical results using sinc col-

location methods.

CONTRIBUTED PAPERS

Cardiac conductivity estimation by a variational dataassimilation procedure: Analysis and validation

Alessandro Barone and Alessandro Veneziani

Emory University, Department of Mathematics

400 Dowman Dr, Atlanta, GA 30322 USA


Flavio Fenton

School of Physics, Georgia Institute of Technology

837 State St NW, Atlanta, GA 30332 USA

[email protected]

Alessio Gizzi

Department of Engineering, University Campus Biomedico of Rome

Via Alvaro del Portillo, 21, 00128 Roma RM, Italy

[email protected]

An accurate patient-specific parameter estimation is crucial for ex-

tending computational tools from medical research to clinical prac-

tice. In cardiac electrophysiology, critical parameters are the con-

ductivity tensors and their quantification is quite troublesome in

living organisms, as witnessed by different discordant data in the

literature.

We consider a variational data assimilation approach for the es-

timation of the cardiac conductivity parameters able to combine

available patient-specific measures with mathematical models. In

particular, it relies on the least-square minimization of the misfit

between experiments and simulations, constrained by the underly-

ing mathematical model. Operating on the conductivity tensors as

control variables of the minimization, we obtain a parameter es-

timation procedure. The methodology significantly improves the

numerical approaches present in literature. Moreover, we present

an extensive numerical simulation campaign reproducing experi-

mental and realistic settings in presence of noisy data [1]. We will

discuss the interplay between the estimation of Monodomain and

Bidomain conductivities as well as experimental validation with

ex-vivo animal tissues.

This work has been supported by the NSF under grant number

DMS 1412973/1413037.

1. A. Barone, F. Fenton and A. Veneziani, Numerical sensitivity analysis

of a variational data assimilation procedure for cardiac conductivities,

Chaos, 27(9), 2017, 093930.

Effective integration of some integrable NLS equa-tions

Otis C. Wright, III

Department of Science and Mathematics

Cedarville University

251 N. Main St.

Cedarville, OH 45314

[email protected]

Some recent results are presented for the effective integration of

finite-gap solutions of integrable nonlinear Schrodinger equations [1,

2]. In particular, simple formulas are derived for critical values of

the amplitude of the solution.

1. Wright, III, O.C., Effective integration of ultra-elliptic solutions of the

focusing nonlinear Schrodinger equation, Physica D, 321-322 (2016)

16-38.

2. Wright, III, O.C., Bounded ultra-elliptic solutions of the defocusing

nonlinear Schrodinger equation, Physica D, 360 (2017) 1-16.

Advanced dispersion engineering for wideband on-chip optical frequency comb generation

Ali Eshaghian Dorche, Ali Asghar Eftekhar, Ali Adibi

School of Electrical and Computer Engineering, Georgia Institute of Tech-

nology,

778 Atlantic Drive NW, Atlanta, GA 30332, USA


Optical frequency combs, which are the equidistant narrow-

linewidth optical signals in the frequency domain, provide a unique

platform for a variety of applications ranging from precise mea-

surements to enhanced optical signal processing and wideband in-

terconnection. To generate wideband optical frequency combs

through efficient power transfer from a pump signal to other comb

74

lines in an optical microresonator anomalous dispersion is required

to balance the Kerr nonlinearity dispersion with the cold cavity dis-

persion. Considering the versatile application of this technology,

it is of much interest to make these optical signals in a miniatur-

ized chip-scale platform; however, having anomalous dispersion in

this platform is more challenging due to limitations imposed by

both materials and fabrication processes. Silicon nitride (SiN) is

the dominant CMOS-compatible material platform for on-chip op-

tical frequency comb generation. However, achieving anomalous

dispersion in the SiN-on-oxide (SiO2) requires complicated fab-

rication processes to ameliorate cracks formed at SiN thicknesses

above 450 nm which is necessary in conventional dispersion engi-

neering techniques.

Here we report a new dispersion-engineering approach to achieve

the necessary anomalous dispersion based on optimized coupled

optical microresonators formed by bending an optimized air-clad,

over-etched, dispersion-engineered thin-film SiN waveguide. This

session will focus on advanced dispersion engineered for efficient

optical frequency comb generation on a chip, including mathemat-

ical modeling to extract the eigenmodes of our proposed structure,

numerical approach to solve generalized Lugiato-Lefever equation

solving the nonlinear dynamic of optical signal propagating inside

a microresonator.

Early stage of integrable turbulence in 1D NLS equa-tion: the semi-classical approach to statistics

Giacomo Roberti and Gennady El

Northumbria University, NE1 8ST - Department of Mathematics, Physics

and Electrical Engeneering, Newcastle upon Tyne, UK


Stephane Randoux and Pierre Suret

Univ. Lille, CNRS, UMR 8523 - Physique des Lasers Atomes et Molecules

(PHLAM),

F-59000 Lille, France


The concept of integrable turbulence introduced by Zakharov [1]

has been recently recognised as a novel theoretical paradigm of ma-

jor importance for a broad range of physical applications from pho-

tonics to oceanography. One of the applications of the integrable

turbulence theory is the statistical description of the appearance of

rogue waves.

We consider the evolution of an initial partially coherent wave

field with Gaussian statistics in the framework of the 1D Nonlinear

Schrodinger equation (1D-NLSE), and we analyse the normalized

fourth order moment of the field’s amplitude, which characterises

the “tailedness” of the probability density function (PDF) of the

field. The relation between this statistical quantiity and the spec-

tral width of the field has been recently provided in Onorato et al.

[2], however, it requires the spectral width knowledge at each step

in time. In our work, thanks to the combination of tools from the

wave turbulence theory and the semi-classical theory of 1D-NLSE,

we derive for the first time an analytical formula for the short time

evolution of the fourth order moment as a function of the statisti-

cal characteristics of the initial condition. This formula provides a

quantitative description of the appearance of the ”heavy” (”low”)

tail of the PDF in the focusing (defocusing) regime of the 1D-NLS

at the initial stage of the development of integrable turbulence, and

our theoretical predictions exhibit a good agreement with the nu-

merical simulations.

1. V. E. Zakharov. Turbulence in integrable systems. Stud. Appl. Math.,

122(3):219–234, 2009.

2. M. Onorato, D. Proment, G. El, S. Randoux, and P. Suret. On the origin

of heavy-tail statistics in equations of the nonlinear Schrodinger type.

Physics Letters A, 380(39):173–3177, 2016.

Spectral stability of ideal-gas shock layers in the strongshock limit

Bryn Balls-Barker∗ and Blake Barker

Department of Mathematics, Brigham Young

University, Provo, UT 84602, USA


Olivier Lafitte

LAGA, Institut Galilee, Universite Paris 13, 93 430

Villetaneuse and CEA Saclay, DM2S/DIR, 91 191

Gif sur Yvette Cedex, France

[email protected]

An open question in gas dynamics is the stability of viscous shock

layers, or traveling-wave solutions of the compressible Navier-

Stokes equations. In general, the Evans function, which is typi-

cally computed numerically, plays a key role in determining the

stability of these traveling wave solutions.

The goal of this research is to analytically describe the spectral sta-

bility of ideal-gas shock layers in the strong shock limit using the

Evans function. The numerical stability of this system has been

previously demonstrated [1] and we seek to make this stability

more rigorous with an analytic proof. We do this by analytically

solving for a basis of the unstable and stable manifolds and then by

using these solutions to create the Evans function. Due to numeri-

cal instability in the Evans system associated with the compressible

Navier-Stokes equations, we utilize the compound matrix method

and a change of variables to find the bases. With the resulting an-

alytic approximation to the Evans function, we are able to study

meaningful bounds on the stability of the shock layers.

1. J. Humpherys, G. Lyng and K Zumbrun, Spectral stability of ideal-gas

shock layers, Arch Rational Mech Anal, 194 (2009), 1029-1079.

Frequency downshift in the ocean

Camille R. Zaug∗ and John D. Carter

Mathematics Department

Seattle University


Frequency downshift occurs when a measure of a waves frequency

(typically its spectral peak or spectral mean) decreases monotoni-

cally. Carter and Govan (2016) derived a viscous generalization of

the Dysthe equation that successfully models frequency downshift

in wave tank experiments for certain initial conditions. The classi-

cal paper by Snodgrass et al. (1966) shows evidence that narrow-

banded swell traveling across the Pacific Ocean also display fre-

quency downshift. In this work, we test the viscous Dysthe equa-

tion against the Dysthe equation, nonlinear Schrodinger equation,

75

and the dissipative nonlinear Schrodinger equation to see which

generalization best models the ocean data reported in Snodgrass et

al. We do so by comparing the Fourier amplitudes, the change in

the spectral peak and spectral mean, and conserved quantities rep-

resenting mass and momentum between the ocean measurements

and numerical simulations.

1. J. D. Carter, A. Govan. Frequency downshift in a viscous fluid. Euro-

pean Journal of Mechanics - B/Fluids, 59 (2016), 177-185.

2. J. D. Carter, D. Henderson, and I. Butterfield. A comparison of fre-

quency downshift models of wave trains on deep water. Physics of Flu-

ids, 31 (2019), 013103.

3. F. E. Snodgrass, K. F. Hasselmann, G. R. Miller, W. H. Munk,

W. H. Powers, Propagation of ocean swell across the Pacific, Philo-

sophical Transactions of the Royal Society of London. Series A, Math-

ematical and Physical Sciences, 259 (1966), 431-497.

POSTERS

Shock formation in finite time for the 1D compress-ible Euler equations

Lucas Schauer and Geng Chen


Lawrence, KS 66049

[email protected]

The majority of physical models in science and engineering are for-

mulated as partial differential equations (PDEs). My research fo-

cuses on the analysis of fundamental properties on many important

nonlinear PDE models, especially existence, uniqueness, and sta-

bility of solutions. This also includes singularity formation, such

as the shock wave in gas dynamics. These solutions exhibiting

singularities give way to very important, exciting, and challeng-

ing research topics. The study on the formation and propagation

of these singularities, which is notoriously difficult due to the lack

of regularity, is one of the central topics in the field of nonlinear

PDEs.

Compressible Euler equations, governing compressible inviscid

flow, have been widely used for gas dynamics and engineering such

as aircraft designs, and are one of the most fundamental PDE sys-

tems. The Euler equations were first found by Leonhard Euler in

1757, and then were studied by many great mathematicians, in-

cluding Riemann, Lagrange, Stokes, Courant, Von Neumann, Lax,

etc. This system is a natural model to capture the formation and

propagation of shock waves in the gas.

For the isentropic 1-D solutions for Euler equations, I will discuss a

research project on the shock formation theory for the Euler equa-

tions with damping. One key thing I will reference is Peter Lax’s

celebrated work on shock formation in 1964. Honing his clever

technique as it pertains to this system, I can show the existence of

an optimal density lower bound. Hence, showing existence of a

blow up in finite time follows from this bound.

Stability of traveling waves in compressible Navier-Stokes

Taylor Paskett and Blake Barker

Brigham Young University

Provo, UT

[email protected]

We develop a method for proving stability of traveling waves in

compressible Navier-Stokes using rigorous numerical verification.

We use interval arithmetic to obtain complete error bounds on all

computations, including machine truncation error. We explain sev-

eral novel methods that we employed to reduce numerical error in

the computer-assisted computations.

The narrow-capture problem in a unit sphere:Global optimization of volume trap arrangements

Alexei Cheviakov∗ and Jason Gilbert

Department of Mathematics and Statistics, University of Saskatchewan

Saskatoon, SK, Canada S7N 5E6


The determination of statistical characteristics for particles under-

going Brownian motion in constrained domains have multiple ap-

plications in various areas of research. This work presents a first at-

tempt to systematically compute globally optimal configurations of

traps inside a three-dimensional domain that minimize the average

mean first passage (MFPT) time for the narrow capture problem –

the average time it takes a particle to be captured by any trap.

For a given domain, the mean first passage time satisfies a linear

Poisson problem with Dirichlet-Neumann boundary conditions.

While no closed-form general solution of such problems is known,

approximate asymptotic MFPT expressions for small traps in a unit

sphere have been found. These solutions explicitly depend on trap

parameters, including locations, through a pairwise potential func-

tion.

After probing the applicability limits of asymptotic formulas

through comparisons with numerical and available exact solutions

of the narrow capture problem, full three-dimensional global opti-

mization was performed to find optimal trap positions in the unit

sphere for 2 ≤ N ≤ 100 identical traps. The interaction en-

ergy values and geometrical features of the putative optimal trap

arrangements are presented.

1. J. Gilbert and A. Cheviakov, Globally optimal volume-trap arrange-

ments for the narrow-capture problem inside a unit sphere, Phys. Rev.

E 99 (2019), 012109.

2. A. Cheviakov and M. J. Ward, Optimizing the principal eigenvalue of

the Laplacian in a sphere with interior traps, Math. Comp. Mod. 53

(2011), 1394-1409.

Theory and observation of interacting linear wavesand nonlinear mean flows in a viscous fluid conduit

Ryan Marizza, Jessica Harris, Michelle Maiden, and

Mark A. Hoefer

Department of Applied Mathematics, University of Colorado, Boulder

[email protected]

76

A theoretical and experimental analysis is described for the inter-

actions of linear, small amplitude, dispersive waves with evolv-

ing, nonlinear mean flows that include oscillatory, compressive

dispersive shock waves and smooth expansion waves in a viscous

fluid conduit. Analysis of such interactions has been developed for

waves described by the Kortweg-de Vries (KdV) equation in the

context of shallow water waves [1]. In this poster, a similar analy-

sis is applied to linear wave-mean flow interactions for the conduit

equation that models a viscous fluid conduit—the cylindrical, free

interface between two miscible, Stokes fluids with high viscosity

contrast. A condition on the linear wave’s wave-number pre and

post interaction determines whether the linear wave will be trans-

mitted through or be trapped by the mean flow. This analysis is

complemented by direct numerical solutions of the conduit equa-

tion and preliminary experimental results.

[1] T. Congy, G. A. El, and M. A. Hoefer, Interaction of linear modulated

waves with unsteady dispersive hydrodynamic states, arXiv:1812.06593

(2018).

nonlinear evolution equations and wave phenomena ...cobweb.cs.uga.edu/~thiab/book_waves2019.pdf ·...

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