necklace solitary waves on bounded domainsnecklace beams soljacic, sears, segev (1998): use necklace...

1

Necklace solitary waves on bounded domains

Gadi Fibich Tel Aviv University

• with Dima Shpigelman

• Intense laser beam that propagates in a transparent medium (air, water, glass, …)

• Direction of propagation is z

2

Physical setup

r=(x,y)

z

z=0

Kerr medium

laser beam

3

NLS in nonlinear optics

Competition between diffractionΔ𝜓 = 𝜓𝑥𝑥 +𝜓𝑦𝑦 and focusingnonlinearity

z “=” t (evolution variable)

Initial value problem in z

,

diffraction Kerr nonlinearity

2 0,, ,

x yzi z x y

r=(x,y)

z

z=0

Kerr medium

laser beam

00, , ,z x y x y

4

Optical collapse

Experiments in the 1960s showed that intense laser beams become narrower with propagation (self-focusing)

Sufficiently powerful beams undergo optical collapse

Kelley (1965) : Solutions of 2D cubic NLS

can become singular (blowup) in finite time (distance) Tc

2

0, , 0, 0, , ,ti t x y x y x y

5

NEW

6

NLS solutions of the form

where

Dependence on µ through the dilation symmetry

Power (L2 norm) of Rµ is independent of µ:

3( ) 0R R R x

Solitary waves

solitary ( ), ( , ),i te R x y

x x

1( ) ( ), :R R R R x x

2 2

2 2( ) ( ), ( ) :µ µ µ µP R P R P R R R dxdy

7

Infinite number of radial solutions R(n)(r)

Of most interest is the ground state R(1) (Townes profile)

Positive, monotonically decreasing

Has exactly the critical power for collapse

Single peak at the origin

Solitary waves – cont.

R(1)

number of peaks

3( ) 0, ( , )R R R x y x x

2(1)

cr 2P R

8

Stability of solitary waves

2

0, , 0, (0, , ) ( , )ti t x y x y x y

(1)0Let ( )

If 1, then in finite time

If 0<

blows up

sc1, then asatters

cR

c

c t

Lemma x

Ground-state solitary waves have a dual instability

9

Challenge

How can we ``overcome’’ the dual instability andpropagate localized laser beams in a Kerrmedium over long distances?

Important for applications

Necklace beams

Soljacic, Sears, Segev (1998): Use necklace beams

e.g., consider the input beam

Consists of 2m identical beams (``pearls’’), located

at equal distances along a circle of radius rmax

Adjacent input beams have opposite signs (phases)

Hence, Ψ0=0 on the rays between the beams

By anti-symmetry, Ψ=0 on these rays for t>0.

These rays thus act as reflecting boundaries

Beams do not interact

Beams repel each other10

0 maxsech( )cos( )c r r m

( 1/ 2)' 1, , 2n

nn m

m

Necklace beams






Hence, Ψ0=0 on the rays separating the beams

By uniqueness, Ψ=0 on these rays for t>0

11


Necklace beams










Necklace structure preserved for t>0

12


Necklace beams










Necklace structure preserved for t>0

Beams with opposite phases repel each other

13


Idea:

Let each beam (``pearl’’) have power below Pcr

Nonlinearity weaker than diffraction, so beam expands

Out-of-phase adjacent beams repel each other

Slows down the expansion of each beam

14

Necklace beams


Simulation

15

Simulation

16

Simulation

17

Necklace expansion much slower than for a single beam

Ultimately, necklace either scatters as t∞, or collapses in finite time (distance), simultaneously at 2m points

Experiments

Grow et al. (07): First observation of a necklace beam in a Kerr medium

18

input beam

after propagating 30cm in glass

Recent application

Jhajj et al (14):

used a necklace beamconfiguration to set up athermal waveguide in air

19

Necklace beams

Vortex beams

20

Comparison with vortex beams


0 maxsech( )exp( )c r r im

Proposition (Fibich and Shpigelman, 16): The NLS

does not admit necklace solitary waves. In other words, there are no necklace solutions of

Necklace solitary waves?

21

2

, , 0, ,ti t x y x y

3( , ) 0, ,R x y R x yR

Theorem (Ao, Musso, Pacard, Wei, 2012 & 2015)

There exist multi-peak, sign-changing solutions of

These solutions do not have the necklace form

Multi-peak solitary waves

22

3( , ) 0, ,R x y R x yR

Theorem (Ao, Musso, Pacard, Wei, 2012 & 2015)

There exist multi-peak, sign-changing solutions of

These solutions do not have the necklace form

As with all solitary-wave solutions of the critical NLS, they are strongly-unstable

Multi-peak solitary waves

23

3( , ) 0, ,R x y R x yR

( )ite R x

Propagation in PR crystal

Yang et al. (05): Propagation in a photorefractive (PR) crystalwith optically-induced photonic lattice

Showed theoretically and experimentally that necklacesolitary waves exist, and can be stable

Because of the need to induce a photonic lattice, notapplicable for propagation in a Kerr medium

24

0

2

2 20

, , 0,( , )1

: sin ( )( , ) sin ( )

tI x y

I x y

Ei t x y

x y x yI

How to stabilize necklace beams in a Kerr medium?

Idea: Confine beam to a hollow-core fiber

To leading-order, fiber walls are perfect reflectors

Therefore, propagation governed by the NLS on a bounded domain

Typically, D is the unit disk B1 = {x2+y2≤1}

25

2

, , 0, ( , ) ,

0, ( , )

ti t x y x y D

x y D

How to stabilize necklace beams in a Kerr medium?

Idea: Confine beam to a hollow-core fiber

To leading-order, fiber walls are perfect reflectors

Propagation governed by the NLS on a bounded domain D ⊆ ℝ2

Typically, D is the unit disk B1 = {x2+y2≤1}

26

2

, , 0, ( , ) ,

0, ( , )

ti t x y x y D

x y D

NLS on a bounded domain

Admits solitary waves where

Studied by Fibich and Merle (01), Fukuizumi, Hadj Selem and Kikuchi (12), Noris, Tavares and Verzini (15)

Mostly for radial solutions on the unit disk B1

27

2

, , 0, ( , ) ,

0, ( , )

ti t x y x y D

x y D

solitary ( ),iµt

µe R x

3 0, ( , ) ,

0, ( , )

R R x y DR

R x y D

Radial solitary waves on the unit disk B1

Unlike free-space, no dilation symmetry

28

3( ) 0, 0 1,

0, 1

r rR R R

rR

(1)(1)( ) ( )r rR R

0, 0 1,

0, 1

R R r

R r

(1)

lin( ) 0,

dP R

d

Let 𝜓 = 𝑒𝑖𝜇𝑡[𝑅𝜇 + 𝜖[𝑢 𝒙 + 𝑖𝑣 𝒙 ] 𝑒Ω𝑡]

Then the linearized equation for the perturbation reads

0 𝐿+𝐿− 0

𝑢𝑣

=Ω𝑢𝑣

,

where

𝐿+ = Δ − 𝜇 + 3 𝑅𝜇2, 𝐿− = Δ − 𝜇 + 𝑅𝜇

2

Linear instability if Re(Ω)>0

Lemma (Vakhitov and Kolokolov): If 𝑅𝜇 >0, then 𝜓 = 𝑒𝑖𝜇𝑡𝑅𝜇 is

linearly stable if and unstable if

Linear stability - review

29

( ) 0d

P Rd

( ) 0d

P Rd

VK condition

Lemma If 𝑅𝜇 >0, then 𝜓 = 𝑒𝑖𝜇𝑡𝑅𝜇 is orbitally stable if

and unstable if

• Weinstein, Grillakis, Shatah, Strauss

Orbital stability - review

30

( ) 0d

P Rd

( ) 0d

P Rd

Lemma If 𝑅𝜇 >0, then 𝜓 = 𝑒𝑖𝜇𝑡𝑅𝜇 is orbitally stable if

and unstable if

• Weinstein, Grillakis, Shatah, Strauss

• On the unit disc, VK condition is satisfied:

Orbital stability - review

31

( ) 0d

P Rd

( ) 0d

P Rd

THM (Fibich and Merle, 2001) The ground-state solitary waves on the unit disk are orbitally stable:

Unlike in free-space, where they are strongly unstable

``Reflecting boundary has a stabilizing effect’’

But,

Goal: Find stable solitary waves with P>Pcr


Idea: use necklace solitary waves!

Solitary waves on the unit disk - stability

32

(1)( )i te rR

solitary(1)0 0 2If , then inf ( ) ( )it e tR

(1)cr0 ( )P R P

THM (Fibich and Merle, 2001) The ground-state solitary waves on the unit disk are orbitally stable:

Unlike in free-space, where they are strongly unstable

``Reflecting boundary has a stabilizing effect’’

But,

Goal: Find stable solitary waves with P>Pcr


Idea: use necklace solitary waves!

Solitary waves on the unit disk - stability

33

(1)( )i te rR

solitary(1)0 0 2If , then inf ( ) ( )it e tR

(1)cr0 ( )P R P

Types of necklace solitary waves

34

3 0, ( , ) ,

0, ( , )

R R x y DR

R x y D


35

circle

3 0, ( , ) ,

0, ( , )

R R x y DR

R x y D


36

circle rectangle

3 0, ( , ) ,

0, ( , )

R R x y DR

R x y D


37

circle rectangle annulus

3 0, ( , ) ,

0, ( , )

R R x y DR

R x y D

New solitary waves

Rectangular necklace solitary wave

38

Compute a positive/ground-state/single-pearl solitary waveon a square

Construct rectangular necklace using 3*2 single pearls

Single pearl on a square

39

3 0, 1 , 1,

0, , 1

0, 1 , 1,

R R x yR

R x y

x yR

Compute using non-spectral renormalization method

40


3 0, 1 , 1,

0, , 1

0, 1 , 1,

R R x yR

R x y

x yR

Compute using non-spectral renormalization method

41

0, 1 , 1,

0, , 1

R R x y

R x y


3 0, 1 , 1,

0, , 1

0, 1 , 1,

R R x yR

R x y

x yR

42

weakly nlregime

linR cR


3 0, 1 , 1,

0, , 1

0, 1 , 1,

R R x yR

R x y

x yR

43

strongly nlregime

free-space

free-space( )

R R

R

x

weakly nlregime

linR cR


3 0, 1 , 1,

0, , 1

0, 1 , 1,

R R x yR

R x y

x yR

44

𝑑

𝑑𝜇𝑃 𝑅𝜇

(1)> 0


3 0, 1 , 1,

0, , 1

0, 1 , 1,

R R x yR

R x y

x yR

45

From single pearl to a rectangular necklace

46



47

(3 2)

3

is a necklace solution of

0, 3 3, 2 2,

0, 3, 2

with 3 2 pearls

R

R R x yR

R x y

Lemma :

Rµ=0 on interfacesbetween pearls

48

Can also construct necklace necklace solitary waves on non-rectangular domains

Compute a positive/ground-state/single-pearl on a sector of a circle

Circular necklace solitary wave

49

3 0, ,1

0, 1

R R R

R

x

x

weakly nlregime

strongly nlregime

𝑑

𝑑𝜇𝑃 𝑅𝜇 > 0

Circular necklace solitary wave

50

3 0, ,1

0, 1

R R R

R

x

x

weakly nlregime

strongly nlregime

𝑑


(1)> 0

Compute a positive/ground-state/single-pearl on a sector of a circle

From single pearl to a circular necklace

51

From single pearl to a circular necklace

52


Compute a positive/ground-state/single-pearl on a sector of an annulus

Annular necklace solitary wave

53

min max

m n x

3

i ma

0,

0, ,

rR R R

R

r

r r

x

x

weakly nlregime

Strongly nlregime

𝑑


Compute a positive/ground-state/single-pearl on a sector of an annulus

Annular necklace solitary wave

54

min max

m n x

3

i ma

0,

0, ,

rR R R

R

r

r r

x

x

weakly nlregime

Strongly nlregime

𝑑


From single pearl to an annular necklace

55

From single pearl to an annular necklace

56


Stability

Single-pearl solitary waves

Necklace solitary waves

57

Single-pearl stability

Perturbed single-pearl solutions ψ0 = 1.05𝑅𝜇1

Numerical observation: All single-pearl solitary waves are stable.

Indeed, they satisfy the VK condition 𝑑


1 > 058

t=0

t=10

Necklace stability

Necklace is stable under perturbations that preserve its anti-symmetries, since then it inherits the single-pearl stability

e.g. ψ0 = 1.05𝑅𝜇𝑛

To excite the instability numerically, use perturbations that break the anti-symmetries, such as

ψ0 = [1 + 𝑛𝑜𝑖𝑠𝑒 𝑥, 𝑦 ]𝑅𝜇𝑛

Multiply bottom-left pearl of 𝑅𝜇𝑛

by 1.0559

Since , necklace satisfies the VK condition 𝑑


𝑛 > 0

This does not imply stability, however, since the VK condition implies stability only for positive solitary waves

Go back to the original linearized eigenvalue problem:

Let 𝜓 = 𝑒𝑖𝜇𝑡[𝑅𝜇𝑛+ 𝜖[𝑢 𝒙 + 𝑖𝑣 𝒙 ] 𝑒Ω𝑡]

Then

0 𝐿+𝐿− 0

𝑢𝑣

=Ω𝑢𝑣

,

where

𝐿+ = Δ − 𝜇 + 3 𝑅𝜇𝑛

2, 𝐿− = Δ − 𝜇 + 𝑅𝜇

𝑛2

Instability if Re(Ω)>0

Necklace stability

60

( ) (1)nP nPR R

Circular necklace with 4 pearls

61

lin cr

cr

Stable for

Unstable for

Instability not related to collapse!

` L̀ess stable'' than radial solitary waves

cr

(4)cr0.24P R P

µ

lin cr

cr

cr

Stable for

Unstable for

Necklace becomes unstable when the power of each

pearl is 0.08P

Necklace instability not related to collapse!

Necklace `̀ less stable'' than radial soli

tary waves


62

cr

(4)necklaceth cr: 0.24PP R P

µ

lin cr

cr

cr

Stable for

Unstable for


pearl is 0.08P



tary waves


63

cr


µ


64

lin cr

cr

cr

Stable for

Unstable for


pearl is 0.08P



tary waves

cr

(4)necklaceth : ?PP R

µ


65

lin cr

cr

cr

Stable for

Unstable for


pearl is 0.08P



tary waves

cr


µ

Dynamics

Multiply bottom-left pearl of 𝑅𝜇4 by 1.05

66

cr1

cr1

Dynamics

Multiply bottom-right pearl of 𝑅𝜇4 by 1.05

67

cr1

cr1

Instability but no

collapse!

Stable necklace propagation above Pcr?

Saw that

Idea 1: take more pearls

68

cr

(4)cr0.24P R P


Saw that


Threshold power for necklace instability decreases with number of pearls

69

cr

(4)cr0.24P R P

cr cr

(6) (8)cr cr0.23 , 0.2P PR P R P


Saw that



70

cr

(4)cr0.24P R P

cr cr

(6) (8)cr cr0.23 , 0.2P PR P R P


Saw that



Idea 2 : Use rectangular domain

71

cr

(4)cr0.24P R P

cr cr

(6) (8)cr cr0.23 , 0.2P PR P R P


Saw that




Surprising, but not good enough

72

cr

(4)cr0.24P R P

cr cr

(6) (8)cr cr0.23 , 0.2P PR P R P

cr

(2*2)cr0.55P R P


Saw that





73

cr

(4)cr0.24P R P

cr cr

(6) (8)cr cr0.23 , 0.2P PR P R P

cr

(2*2)cr0.55P R P

To raise the threshold power for necklace stability, need to understand the instability dynamics

linearized eigenvalue problem:

Instability if Re(Ω)>0 Yes/No approach

Fibich, Ilan, Sivan, Weinstein (06-08): Deduce instability dynamics from the unstable eigenfunctions

Instability dynamics

74

0 𝐿+𝐿− 0

𝑢𝑣

=Ω𝑢𝑣


75

++

++

++++

--

--


76

++

++

++++

--

--


77

++

++

++++

--

--

e.f. pert.


78

++

++

++++

--

--

e.f. pert.


79

++

++

++++

--

--


80

++

++

++++

--

--


81

++

++

++++

--

--


82

++

++

++++

--

--


Saw that





Idea 3: Use annular domain, since hole reduces interactions between pearls

83

cr

(4)cr0.24P R P

cr cr

(6) (8)cr cr0.23 , 0.2P PR P R P

cr

(2*2)cr0.55P R P

Annular necklace with 4 pearls

84

Stable for

Hole does not help


85

lin cr

cr

(4)cr0.24P R P

Stable for

Hole does not help

Second stability regime!

Hole does help!


86

lin cr

cr

(4)cr0.24P R P

(4)cr cr3.1 3.7PP R P

Dynamics

ψ0 = [1 + 0.02 𝑛𝑜𝑖𝑠𝑒 𝑥, 𝑦 ]𝑅𝜇4

87

10

30

Dynamics

ψ0 = [1 + 0.02 𝑛𝑜𝑖𝑠𝑒 𝑥, 𝑦 ]𝑅𝜇4

88

10

30

Dynamics

ψ0 = [1 + 0.02 𝑛𝑜𝑖𝑠𝑒 𝑥, 𝑦 ]𝑅𝜇4

89

10

30

Dynamics

ψ0 = [1 + 0.02 𝑛𝑜𝑖𝑠𝑒 𝑥, 𝑦 ]𝑅𝜇4

90

10

30

stable solitary wave with P>3Pcr

Positive solitary waves on the annulus

Can look for radial solutions

Are these the ground-state (minimal power) solutions?

Compute ground-state solutions numerically using the non-spectral renormalization method

91

3min max

min max

maxmin

( ) 0,

0, ,

0,

R R R r r

R r r

R r r

x x

x

x

3min max

min max

min max

( ) 0,

0, ,

0,

R r R rR r r

R r r r

R rr r

Ground states on the annulus

92

Radial for - - - - -

Non-radial for _____

3min max

min max

maxmin

( ) 0,

0, ,

0,

R R R r r

R r r

R r r

x x

x

x

lin cr

cr

Symmetry breaking

Related to variational characterization of the ground state

2

2

inf ( )H UPU

R equation on the annulus

93

3min max

min max

( ) 0,

0, ,

R R R r r

R r r

x x

x

First example of

1. Non-uniqueness of positive solutions of the R equation

2. A positive solution of the R equation which is not a ground-state

Previously observed for the inhomogeneous NLS (Kirr et al., 08)

Stability on the annulus

94

cr0.3

cr0.3

cr0.3

Radial, stable

Radial, unstable

Non-radial, stable

Computation of necklace solitary waves

95

3 0, ( , ) ,

0, ( , )

, ( , )0

R R x y DR

R x y D

x yR D

Nonlinear elliptic problem

Multidimensional

cannot use radial symmetry

Non-positive solution with necklace structure

Bounded domain

Plan:

Compute a single pearl (positive)

Extend into a necklace using the anti-symmetries

Computation of a single pearl

96

3 0, ( , ) ,

0, ( , )

, ( , )0

R R x y DR

R x y D

x yR D

Nonlinear elliptic problem

Multidimensional

Non-positive solution

Bounded domain

When D = ℝ2, can use Petviashvili’s spectral renormalization method

Cannot apply on a bounded domain

Introduce a non-spectral version

Non-spectral renormalization method

Rewrite equation for Rµ as

Solve using fixed point iterations

L discretized using finite differences

Problem: Iterations diverge to zero or to infinity

To prevent divergence, renormalize at each stage Rk ckRk

Determine ck from the requirement that ckRk satisfies an integral relation, such as

97

3, :LR LR

3

11 , 0,1,...kkk kR L Rc

1 31 , 0,1,...k k kR L R

1 3, ,R R R L R

1D NLS on bounded domain

Fukuizumi, Hadj Selem, and Kikuchi (12) studied multi-peak solitary waves of 1D NLS

On B1:

2D radial analog are R (n) (r)

2D non-radial analog are necklace solutions

Theory in 1D and 2D cases is very similar

98

2

, 0, 1 1,

0, 1

t xxi t x x

x

Summary

New type of solitary waves of the 2D NLS on bounded domains

Stable at low powers

Threshold for necklace instability < critical power for collapse

Instability unrelated to collapse

Second stability region for necklace solitary waves on the annulus

Stable solitary waves with P>Pcr

Symmetry breaking of the ground state on the annulus

Non-spectral renormalization method for computing solitary waves on bounded domains

G. Fibich and D. Shpigelman

Positive and necklace solitary waves on bounded domains

Physica D 315: 13-32, 201699

necklace solitary waves on bounded domainsnecklace beams soljacic, sears, segev (1998): use necklace...

Documents