I N D I I Springer Series in Nonlinear Dynamics

I N D I I Springer Series in Nonlinear Dynamics

Series Editors: F. Calogero, B. Fuchssteiner, G. Rowlands, M. Wadati, and V. E. Zakharov

Solitons - Introduction and Applications Editor: M. Lakshmanan

What Is Integrability? Editor: V. E. Zakharov

Rossby Vortices and Spiral Structures By M. V. Nezlin and E. N. Snezhkin

Algebro-Geometrical Approach to Nonlinear Evolution Equations By E. D. Belokolos, AI. Bobenko, V. Z. Enolsky, A R. Its and V. B. Matveev

Darboux 'fransformations and Solitons By V. B. Matveev and M. A Salle

Optical Solitons By F. Abdullaev, S. Darmanyan and P. Khabibullaev

Wave Turbulence Under Parametric Excitation Applications to Magnetics ByV.S. Vvov

Koimogorov Spectra ofThrbuience I Wave Turbulence By V. E. Zakharov, V. S. Vvov and G. Falkovich

Nonlinear Processes in Physics Editors: AS. Fokas, D. J. Kaup, A C. Newell and V. E. Zakharov

A.S. Fokas D.I Kaup A.C. Newell Y.E. Zakharov (Eds.)

Nonlinear Processes in Physics Proceedings of the III Potsdam - V Kiev Workshop at Clarkson University, Potsdam, NY, USA August 1-11, 1991

With 41 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Professor A. S. Fokas Professor D. J. Kaup Clarkson University, Potsdam, NY 13699-5815, USA

Professor A. C. Newell University of Arizona, Thcson, AZ 85721, USA

Professor V. E. Zakharov Landau Institute for Theoretical Physics, u1. Kosygina 2, 117334 Moscow, Russia and University of Arizona, Tucson, AZ 85721, USA

ISBN-13:978-3-642-77771-4 e-ISBN-13:978-3-642-77769-1 DOl: 10.1007/978-3-642-77769-1

Library of Congress Cataloging·in·Publication Data. Nonlinear processes in physics 1 A. S. Fokas ... [et al.]. p. cm. - (Springer series in nonlinear dynamics) Includes bibliographical references and index. ISBN-13:978-3-642-77771-4 1. Nonlinear theories-Congresses. 2. Soliton theory-Congresses. 3. Mathematical physics-Congresses. I. Fokas, A. S., 1952 -. II. Series. QC20.7.N6N662 1993 530.1'4-dc20 92-32811

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Preface

In 1979, a historical meeting took place at the Institute for Theoretical Physics in Kiev, USSR, where 48 American Scientists, specialists in nonlinear and turbulent processes, met for two weeks with their soviet counterparts. This meeting pro­vided the unique opportunity for USA and USSR participants to directly interact personally and scientifically with each other. This interaction was of great impor­tance not only for the individuals involved but also for the science of nonlinear phenomena in general.

At the end of the meeting, it was agreed that this exchange should continue, and it was decided to have the next meeting in the USA in 1981. Unfortunately, due to the political situation at that time, the second meeting in the USA never materialized. However, in 1983, the Soviet scientists organized in Kiev a second Workshop. This second meeting was again quite successful. Similar meetings, with growing success were organized at Kiev in 1987, and 1989. It should be noted that 405 participants from 22 countries participated at the fourth Kiev workshop on Nonlinear and Turbulent Processes. The Chainnan of this workshop was V. Zakharov, who has also been a co-chainnan of all the previous workshops.

Even earlier, in 1972, there had been a Potsdam workshop (the first Potsdam meeting) on nonlinear waves which was organized by Alan Newell. This work­shop had served as a valuable precursor for much of the work on nonlinear waves in the USA. It was then repeated in 1978 (the second Potsdam workshop) which was a time in the midst of the recent explosive growth in nonlinear waves.

Except for the above four Kiev Workshops, there had been no scientific meet­ings where a large number of Soviet and USA scientists working in nonlinear and turbulent processes had met. At the closing of the Fourth Kiev Workshop, it was decided to have the next meeting in the USA. Because of the general political climate, it appeared that such an endeavor could now be feasible.

The III Potsdam-V Kiev Workshop on Nonlinear Processes in Physics was held at Clarkson University, Potsdam, NY from August 1-11, 1991. It was spon­sored by the Clarkson School of Science and was funded by the National Science Foundation, the Department of Energy, the US Air Force of Scientific Research, the US Office of Naval Research, the Sloan Foundation and the School of Science of Clarkson University. The organizing committee was A.S. Fokas, D.J. Kaup, A.C. Newell and V.E. Zakharov. '

This was the first major scientific workshop in the USA where the Soviet scientists fonned a major contingent of the participants. There were 31 USSR participants out of a total of 106 pa¢cipants. The emphasis of the workshop was on the interaction between mathematical techniques and problems of physical

v

interest. It was particularly successful in mixing plasma physicists, fluid physicists and soliton theorists. The exchanges between these groups, particularly with the participation of the Soviet delegation, were quite stimulating.

The lectures ranged from algebraic features of integrable systems and vortex dynamics to applications in plasma physics, ionospheric physics, nonlinear op­tics, oceanic studies and solid state. It is not possible to give full justice to all the excellent lectures here, but we shall mention some results. Lax presented a survey of the work on the zero dispersion limit for several types of dispersive systems and Levermore, Tian, and Venakides presented new important developments and applications of the Lax-Levermore theory and the associated Whitham's equa­tion. Explicit solutions of this equation, using algebraic-geometric techniques, were presented by Krichever and Dubrovin. Krichever also reviewed the recent appearance of Whitham's equation in the minimal models of 2D quantum grav­ity. Another connection between soliton theory and 2D quantum gravity was presented by Its who discussed the role of discrete Painleve equations and gave a rigorous description of their continuous limit Deift described a rigorous method­ology for studying the long time behavior of the Riemann-Hilbert problems arising in the inverse spectral theory. Santini discussed the possibility of solving purely algebraic equations by the algebraic-geometric techniques developed in soliton theory. New results in the inverse spectral theory of evolution equations in two spatial dimensions were described by Boiti (dromions for DSn, Zhou (KPI), Sung (DSII), and Pogrebkov (KPI). McKean described the spectral theory associated with bi-Hamiltonian structures in classical mechanics, and Dorfman reviewed Hamiltonian and symplectic structures for evolution equations in one and two spatial variables. McLaughlin described algebraic-geometric aspects of the perturbation theory of certain soliton equations. Applications of the dressing method to nonlocal nonlinear evolution equations and to nonlinear evolution equa­tions in multidimensions were discussed by Degasperis and Sabatier respectively. Takhtajan introduced a reversible soliton cellular automaton. Korepin derived and analyzed the integrable PDE's satisfied by quantum correlation functions. Beals described the action angle formulation of the Gel'fand-Dikii hierarchies. Shulman talked about new results regarding degenerate dispersion law. Alber analyzed cer­tain complicated but integrable Hamiltonian systems. Bogoyanlenskij analyzed a certain 2 + 1 version of the KdV and showed that it exhibits the phenomena of breaking of solitons. Conte and Fordy explained how to implement the Painleve test in the presence of negative resonances.

Grunbaum discussed a novel inverse problem arising in low energy medical imaging (diffuse tomography) and Monk described analytical and numerical as­pects of a certain inverse scattering problem. Hasegawa described some of the technical points involved with using solitons as pulses in the proposed trans­atlantic optical cable and Rupasov described new rigorous results ,in the quantum theory of stimulated Raman scattering. Bona showed how a rough wave model could give a respectable explanation of the appearance of underwater sand ridges seen on sloping beaches.

Important new results, in vortex dynamics were presented by Zabusky, Majda, Ichikawa, Horton and Petviashvili. Zabusky showed excellent detailed visual de-

VI

scriptions of the collisions and reconnection of vortex filaments while Majda and Ichikawa each independently described the stretching and kinking of vortex fila­ments in irrotational fluids. Both Horton and Petviashvili discussed the rotational case and in particular, showed that the strength of the vortex determines whether the vortex behaves as a point vortex or a KdV soliton in collisions. Langmuir tur­bulence and collapsing cavitons in the ionosphere were discussed by Don DuBois while Rao and Kaup described how the mode conversion into electron Bernstein waves could cause the observed quenching of the downshifted peaks seen in the Tromsj21 and Arecibo ionospheric modification experiments. Morales discussed the nonlinear refraction of an rf wave in the ionosphere while Hada and Hamilton both presented studies of nonlinear Alfven waves in space plasmas. One of the major presentations in collapse physics was made by Malkin who presented new analytical results for the self-focusing problem of the two-dimensional nonlinear Schrodinger equation. Zakharov and Rubenchik each discussed aspects of weak turbulence.

Most of these lectures are summarized in these proceedings.

Potsdam, NY Moscow Summer 1992

A.S. Fokas D.J. Kaup

A.C. Newell V.E. Zakharov

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Contents

Part I Nonlinear Equations

Multisoliton-like Solutions of Wave Propagation in Periodic Nonlinear Structures By AB. Aceves and S. Wabnitz ............................ 3

Complex Deformation of Integrable Hamiltonians over Generalized Jacobi Varieties By S.J. Alber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Collective Coordinates by a Variational Approach: Problems for Sine Gordon and tJ4 Models By J.G. Caputo and N. Flytzanis . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21

Spatiotemporal Chaos in the Nonlinear Three Wave Interaction By C.C. Chow, A Bers, and AK. Ram (With 1 Figure) . . . . . . . . . . .. 25

On the Instability of the Static Soliton-like "Bubbles" By A. de Bouard ...................................... 29

Symplectic and Hamiltonian Structures of Nonlinear Evolution Equations By lYa. Dorftnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

A Fuchs Extension to the Painleve Test By R. Conte, AP. Fordy, and A Pickering .................... 35

Modulation Equations for Nearly Integrable PDEs with Periodic Boundary Conditions By M.G. Forest ....................................... 45

Spectrum of Domain Wall Excitations in YIG By AV. Mikhailov and lA Shimokhin (With 4 Figures)

Interaction of Defects in Nonlinear Dissipative Fields By L.M. Pismen, J. Rubinstein, AA Nepomnyashchy,

47

and J.D. Rodriguez ..................................... 53

On the Analytic Degenerate Dispersion Laws By E.I. Schulman and D.D. Tskhakaya ....................... 60

XI

Part II Inverse Scattering Transforms

Breaking Solitons. Systems of Hydrodynamic Type By 0.1. Bogoyavlenskij .................................. 67

Real and Virtual Multidimensional Solitons By M. Boiti, L. Martina, O.K. Pashaev, and F. Pempinelli .......... 77

Resolvent Approach for the Nonstationary Schrodinger Equation with Line-Type Potential * By M. Boiti, F. Pempinelli, A.K. Pogrebkov, and M.C. Polivanov 82

Lattice Construction of Quantum Integrable Systems By D.A. Coker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87

Combining Dressing and Nonlocality By A. Degasperis .....•................................ 91

Initial Boundary-Value Problems for Soliton Equations By A.S. Fokas ........................................ 96

Non-Perturbative Two-Dimensional Quantum Gravity and the lsomonodromy Method By A.S. Fokas, A.R. Its, and A.V. Kitaev ..................... 102

The Action of the Vrrasoro Nonisospectral KdV Symmetries of the Whitham Equations By P.G. Grinevich ..................................... 108

A Generalized Sato's Equation of the KP Theory and Weyl Algebra By Y. Kodama. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113

Quasirelativistic Analogs of Lax Equations By B.A. Kupershmidt ................................... 114

n x n Zakharov-Shabat System of the Form (d#dx)(z2 -1Iz2)J'lj; + (zQ + P + Riz)'lj; By J.-H. Lee ..... ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 118

Homoclinic Orbits and Backlund Transformations for the Doubly Periodic Davey-Stewartson Equation By Y. Li and D.W. McLaughlin (With 2 Figures) ................ 122

Volterra Operator Algebra for Zero Curvature Representation. Universality of KP By A.Yu. Orlov ....................................... 126

Calculation of All Commutation Relations Among Scattering Data Without Using the R-Matrix Approach By G.D. Pang and A.S. Fokas ............................. 132

Fluctuating Solitons of the KdV Hierarchy By L. Trlifaj ......................................... 136

XII

The Periodic Fixed Points of Backlund Transfonnations By J. Weiss (With 3 Figures) .............................. 139

1ST for KPI By Xin Zhou 148

Part III Plasmas

A Physical Model for Nonlinear, Supersonic Equatorial Bubbles By W.J. Burke and T.L. Aggson (With 3 Figures) . . . . . . . . . . . . . . .. 153

Nonlinear Dynamics of Electron Cyclotron Heated Plasmas By G.E. Guest ........................................ 166

Nonlinear Evolution of Alfven Waves in Space Plasmas By T. Hada .......................................... 169

Alfven Solitons and the DNLS Equation By R.L. Hamilton, C.F. Kennel, and E. Mj!2Slhus ................. 175

On the Analytical Theory for Self-Focusing of Radiation By V.M. Malkin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179

Ionospheric Langmuir Turbulence Driven by an Electromagnetic Pump Below the Upper-Hybrid Frequency By D.L. Newman and M.V. Goldman (With 3 Figures) ............ 180

Mode Conversions in Ionospheric Modification Experiments By N.N. Rao and D.J. Kaup (With 2 Figures) ................... 185

On the Superstrong Wave Collapse By V.F. Shvets (With 2 Figures) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191

A Moving Boundary Problem in Plasma Physics By N. Sternberg and V.A. Godyak .......................... 195

Proton-Whistler Interactions in the Radiation Belts By E. Villalon and W.J. Burke (With 1 Figure) ............ . . . . .. 199

Part IV Nonlinear Optics

Optical Turbulence in Semiconductor Lasers By E. Abraham, H. Adachihara, O. Hess, R.A. Indik, P. Jacobsen, J.V. Moloney, and P. Ru (With 2 Figures) ..................... 213

Chaotic Dynamics Due to Competition Among Degenerate Modes in a Ring-Cavity Laser By A. Aceves, D.D. Holm, and G. Kovacic (With 2 Figures) ........ 218

Quantum Self Phase Modulation in Optical Fibres By K.J. Blow, R. Loudon, and S.J.D. Phoenix (With 1 Figure) ....... 228

XIII

Quantum Groups: Q-Boson Theories of Integrable Models and Applications in Non-Linear Optics By RK. Bullough and N.M. Bogoliubov (With 1 Figure)

Perturbation Method and Optical Solitons

232

By Y. Kodama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . .. 241

Birefringent Optical Fibers: Modulational Instability in a Near-Integrable System By DJ. Muraki, O.C. Wright, and D.W. McLaughlin

Rigorous Results in Quantum Theory of Stimulated Raman Scattering

242

By V.1. Rupasov ...................................... 247

Backlund Transformations as Physical Equations By H. Steudel (With 1 Figure) ............................. 252

Level Splitting and Band Formation of Dark Soliton Eigenvalues By G.A. Swartzlander Jr. (With 2 Figures) . . . . . . . . . . . . . . . . . . . .. 256

Soliton Propagation in a Random Medium By P.K.A. Wai, C.R Menyuk, and H.H. Chen (With 1 Figure) ....... 261

Part V Hydrodynamics and Thrbulence

Length Scales and the Navier-Stokes Equations By M. Bartuccelli, C.D. Doering, J.D. Gibbon, and S.J.A. Malham 267

Local and Nonlocal Transfer of Motion Integrals in Wave Turbulence By G.E. Falkovich and M.D. Spector ........................ 271

Weak and Strong Turbulence in the CGL Equation By J.D. Gibbon, M.V. Bartuccelli, and C.R Doering .............. 275

Numerical Test of a Weak Turbulence Approximation for an Electromagnetically Driven Langmuir Turbulence By A. Hanssen and E. Mjj1Slhus . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 279

Drift Wave Vortices in Inhomogeneous Plasmas By W. Horton, X. Su, and P.J. Morrison (With 3 Figures) .......... 281

Solitons on a Vortex Filament with Axial Flow By Y.H. Ichikawa, K. Konno, and H. Ohno (With 1 Figure) 291

Lagrangian Statistics of Turbulence By Y. Kaneda .................................. " . . . .. 299

Pattern Formation Via Resonant Interactions in Fluid Flows By S. Leibovich and A. Mahalov ........................... 303

XIV

Length Scale of Vortices and Mode Competition in Quasi 2D Shear Flows By D.Yu. Manin ........•.•............................ 305

Dynamics of Vortex-Current Filaments in MHO Plasma By V.I. Petviashvili (With 1 Figure) ......................... 306

Success of Arnol'd's Method in a Hierarchy of Ocean Models By P. Ripa •..........•...•..........•............... 310

Wavelets and Two Dimensional Turbulence By J. Weiss (With 3 Figures) .•.•.......................... 315

Part VI Inverse Scattering Problems

Spectral Theory of Linear A-Matrices and the Solution of Certain Nonlinear Algebraic and Functional Equations By B.A. Dubrovin, A.S. Fokas, and P.M. Santini ................ 329

Nonlinear Acoustic Tomography By P. Monk (With 2 Figures) .............................. 334

Spectral Transform for Nonlinear Evolution Equations with N Space Dimensions By P.C. Sabatier . . . . . . • . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . .. 339

Index of Contributors ....•............................. 343

xv