Continuations of NLS solutions beyond the singularity Gadi Fibich Tel Aviv University ff

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Continuations of NLS solutionsbeyond the singularity

Gadi Fibich Tel Aviv University

ff

• Moran Klein - Tel Aviv University• B. Shim, S.E. Schrauth, A.L. Gaeta - Cornell

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NLS in nonlinear optics

Models the propagation of intense laser beams in Kerr medium (air, glass, water..)

Competition between focusing Kerr nonlinearity and diffraction

z“=”t (evolution variable)

focusing nonlinearitydiffraction

2 0, ,z xx yyi z x y

r=(x,y)

z

z=0Kerr Medium

Input Beam

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Self Focusing

Experiments in the 1960’s showed that intense laser beams undergo catastrophic self-focusing

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Kelley (1965) : Solutions of 2D cubic NLS can become singular in finite time (distance) Tc

2 2

0 4 x ye

Finite-time singularity

20, , 0, 0, , ,t xx yyi t x y x y x y

Beyond the singularity

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?• No singularities in nature

• Laser beam propagates past Tc

• NLS is only an approximate model• Common approach: Retain effects that were neglected in NLS

model: Plasma, nonparaxiality, dispersion, Raman, …• Many studies• …

Compare with hyperbolic conservation laws Solutions can become singular (shock waves) Singularity arrested in the presence of viscosity

Huge literature on continuation of the singular inviscid solutions: Riemann problem Vanishing-viscosity solutions Entropy conditions Rankine-Hugoniot jump conditions Specialized numerical methods …

Goal – develop a similar theory for the NLS

Continuation of singular NLS solutions

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Tc

?NLS


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Tc

NLSNLSno ``viscous’’ terms


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Tc

NLSNLS

``jump’’ condition

no ``viscous’’ terms


2 key papers by Merle (1992) Less than 10 papers

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Tc

NLSNLS

``jump’’ condition

no ``viscous’’ terms

Talk plan

1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property

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General NLS

d – dimension, σ – nonlinearity

2, 0ti t x

( )1 1

1,

...d d

d

x x x x

x xx ==¶ + +¶D

Definition of singularity:

Tc - singularity point

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0 , limc

Ht THy y

®Î =¥

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Classification of NLS

global existence

(no blowup)

Subcritical σd<2

blowup Critical σd=2

blowup Supercritical

σd>2

( )1, dx xx =

2, 0ti t x

σd= 2 Physical case considered earlier (σ=1,d=2) Since 2σ= 4/d, critical NLS can be rewritten as

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4/, 0dti t x

Critical NLS (focus of this talk)

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Solutions of the form

The profile R is the solution of

Enumerable number of solutions Of most interest is the ground state:

Solution with minimal power (L2 norm)

4/ 11 0, 0, 00ddR R R R R Rrr

Solitary waves

,ite R r r x

d=2Townes profile

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Critical power for collapse

Thm (Weinstein, 1983): A necessary condition for collapse in the critical NLS is

Pcr - critical power/mass/L2-norm for collapse

2 20 2 2

,cr crP P R

Explicit blowup solutions

Solution width L(t)0 as tTc

ψR,αexplicit becomes singular at Tc

Blowup rate of L(t) is linear in t

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t

2( )explicit 4

R, /2

2

0

1( , ) ,( ) ( )

( ) ), ( )(

tL ri t iL

d

t

c

rt r RL t L t

L t t L s dsT

e

Minimal-power blowup solutions

ψR,αexplicit has exactly the critical power

Minimal-power blowup solution

ψR,αexplicit is unstable, since any perturbation that

reduces its power will lead to global existence 18

22explicit

R,

1( ) ( )d

rR

L t L t

2

2crP R

Thm (Weinstein, 86; Merle, 92) The explicit blowup solutions ψR,α

explicit are the only minimal-power solutions of the critical NLS.

Stable blowup solutions of critical NLSFraiman (85), Papanicolaou and coworkers (87/8) Solution splits into a singular core and a regular tail Singular core collapses with a self-similar ψR profile

Blowup rate is given by

Tail contains the rest of the power ( ) Rigorous proof: Perelman (01), Merle and Raphael (03)

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2( ) ( )

log logc

c

tTL ttT

loglog law

R / 2

1( , ) ~ ( , )( ) ( )d

rt r t r Rt L tL

20 2 crP

Bourgain-Wang solutions (1997)

Another type of singular solutions of the critical NLS Solution splits into a singular core and a regular tail Singular core collapses with ψR,α

explicit profile Blowup rate is linear

ψB-W are unstable, since they are based on ψR,αexplicit

(Merle, Raphael, Szeftel; 2011) Non-generic solutions

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Continuation of NLS solutions beyond the

singularity

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Talk plan


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Explicit continuation of ψR,αexplicit (Merle, 92)

Let ψε be the solution of the critical NLS with the ic

Ψε exists globally Merle computed rigorously the limit

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explicit0 R,( ) (1 ) ( 0, ), 0 1r t r

0lim ( , ), 0t r t

2 2 2

220 (1 ) crPR

• Before singularity, since

• After singularity

Thm (Merle 92)

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explicit

0lim ( , ) ( , ) , 0c ct r t r tT T

explicitR,( 0) (1 ) ( 0),t t

explicitR,0

lim ( , ) ( , ), 0 ct r t r t T

• Before singularity


Thm (Merle 92)

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explicitR,0

lim ( , ) ( , ) , 0c ct r t r tT T

explicitR,0

lim ( , ) ( , ), 0 ct r t r t T



Thm (Merle 92)

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explicitR,0

lim ( , ) ( , ) , 0c ct r t r tT T

0lim ( )

0

cL t tT

t

explicitR,0

lim ( , ) ( , ), 0 ct r t r t T

NLS is invariant under time reversibility

Hence, solution is symmetric w.r.t. to collapse-arrest time Tε

arrest

As ε 0, Tεarrest Tc

Therefore, continuation is symmetric w.r.t. Tc

Jump condition

Symmetry Property - motivation

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*andt t

Thm (Merle 92)

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• Symmetry property: Continuation is symmetric w.r.t. Tc

• Phase-loss Property: Phase information is lost at/after the singularity


explicit *R,0

For any , there exists a sequence 0 such that

lim ( , ) ( , ) , 0n

n

n

ic ct r t r tT Te

Phase-loss Property - motivation

Initial phase information is lost at/after the singularity

Why?

For t>Tc, on-axis phase is ``beyond infinity’’

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explicit 2R,

0

arg ( ,0) ( ) ( ) , ( ) )(

lim ( )

t

c

t Tc

t t L s ds L t tT

t

Merle’s continuation is only valid for Critical NLS Explicit solutions ψR,α

explicit

Unstable Non-generic

Can this result be generalized?

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Talk plan


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Sub threshold-power continuation (Fibich and Klein, 2011)

Let f(x) ∊H1

Consider the NLS with the i.c. ψ0 = K f(x) Let Kth be the minimal value of K for which the NLS solution

becomes singular at some 0<Tc<∞ Let ψε be the NLS solution with the i.c. ψ0

ε= (1-ε)Kth f(x) By construction,

0<ε≪1, no collapse -1≪ε<0, collapse

Compute the limit of ψε as ε0+ Continuation of the singular solution ψ(t, x; Kth) Asymptotic calculation (non-rigorous)

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• Core collapses with ψR,αexplicit profile

• Blowup rate is linear• Solution also has a nontrivial tail

Conclusion: Bourgain-Wang solutions are ``generic’’, since they are the ``minimal-power’’ blowup solutions of ψ0 = K f(x)

Proposition (Fibich and Klein, 2011)

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0( , ) ( , ), 0lim cB Wt r t r t T



• Symmetry w.r.t. Tc (near the singularity)

• Hence,


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0( , ) ( , ), 0lim cB Wt r t r t T

0lim ( , ) ( , ) , 0 1c cB Wt r t r tT T

0lim ( ) , 1c cL t t t TT


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*B-W0


lim ( , ) ( , ) , 0 1n

n

n

c cit r t r tT Te

• Phase information is lost at the singularity

• Why?


explicitR,lim arg lim argB Wt Tc t Tc

Simulations - convergence to ψB-W

Plot solution width L(t; ε)

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0lim ( ) cL t tT

2

40, 0, (1 )

1.481395tht xx

th

xi t x KK

e

Simulations – loss of phase

How to observe numerically? If 0<ε≪1, post-collapse phase is `àlmost lost’’ Small changes in ε lead to O(1) changes in the phase

which is accumulated during the collapse Initial phase information is blurred

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Simulations - loss of phase

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O(10-5) change in ic lead to O(1) post-collapse phase changes

Simulations - loss of phase

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0lim arg( ( ) )ct T

O(10-5) change in ic lead to O(1) post-collapse phase changes

Talk plan


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NLS continuations So far, only within the NLS model:

Lower the power below Pth , and let PPth- Different approach: Add an infinitesimal perturbation to

the NLS Let ψε be the solution of

If ψε exists globally for any 0<ε≪1, can define the ``vanishing –viscosity continuation’’

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2

``viscosity''

, 0[ ]ti t F x

0

, : lim , , 0continuation t t t

x x

NLS continuations via vanishing -``viscosity’’ solutions

What is the `viscosity’? Should arrest collapse even when it is infinitesimally small Plenty of candidates:

Nonlinear saturation (Merle 92) Non-paraxiality Dispersion …

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Nonlinear damping

``Viscosity’’ = nonlinear damping Physical – multi-photon absorption Destroys Hamiltonian structure

Good!

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Critical NLS with nonlinear damping

Vanishing nl damping continuation : Take the limit δ0+

Consider ψ0 is such that ψ becomes singular when δ=0 if q≥ 4/d, collapse arrested for any δ>0 If q< 4/d, collapse arrested only for δ> δc(ψ0)>0

Can define the continuation for q≥ 4/d

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4/

10

, 0, 0 1

0,

d qti t i

H

x

x x

0

, : lim , , 0continuation t t t

x x

Explicit continuation

Critical NLS with critical nonlinear damping (q=4/d) Compute the continuation of ψR,α

explicit as δ0+

Use modulation theory (Fibich and Papanicolaou, 99) Systematic derivation of reduced ODEs for L(t) Not rigorous

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4/

explicit0 R,

, (1 ) 0

0,

dti t i

t r

x

Asymptotic analysis Near the singularity

Reduced equations given by

Solve explicitly in the limit as δ0+

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~

3 2( ) , ( )tt tt tL

L L

R / 2

1( , ) ~ ( , )( ) ( )d

rt r t r Rt L tL

Asymptotic analysis Near the singularity

Reduced equations given by

Solve explicitly in the limit as δ0+

Asymmetric with respect to Tc Damping breaks reversibility in time

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~

3 2( ) , ( )tt tt tL

L L

R / 2

1( , ) ~ ( , )( ) ( )d

rt r t r Rt L tL

0

( ) 0lim ( ; ) 1.614

( ),c c

c c

t tT TL tt tT T



• Phase information is lost at the singularity• Why?

Proposition (Fibich, Klein, 2011)

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explicitR,0

lim

* explicitR,0


lim 1.61( , ) , ), 4(n

n

ic ct r t rT Te

explicitR,lim arg

t Tc

Simulations – asymmetric continuation

L=α)Tc-t( L=κα(t-Tc)

Simulations – loss of phase

Nonlinear-damping continuation of loglog solutions

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240

6

Example:

, 0, 1.6 xt xxi t x i e

5 4......... 2.5 4____ 1 5

ee

e

• Highly asymmetric• Slope ±∞

0lim ( ; ) c

tc

t TtL t T

Talk plan


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Collapse in linear optics

Can solve explicitly in the geometricaloptics limit (k0∞)

Linear collapse at z=F

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2

0

0

2

2

2 , 0,

( 0, )

z

ikF

i zk

z e e

x

x

xx

2

2( )/ 2

1 , ( ) 1( )

zLd

zL zz FL e

x

GO

• z=t• k0 is wave #

Continuation of singular GO solution consider the linear Schrödinger with k0<∞ Global existence Can solve explicitly (without GO approx) Compute the limit as k0∞

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lim ( ) 1 , 0k

zL z zF

GO

*/2

GO

( , ),lim ( , )

(2 , ),idk

z z Fz

F z F ze

xx

x

No phase loss after singularity Why?

Post-collapse phase loss is a nonlinear phenomena

Continuation of singular GO solution

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GOlim arg 0 ( )z F

GO

*/2

GO

( , ),lim ( , )

(2 , ),idk

z z Fz

F z F ze

xx

x

Talk plan


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Universality of loss of phase

All NLS continuations have the Phase-loss Property Phase of all known singular NLS solutions becomes

infinite at the singularity Hence, any continuation of singular NLS solutions will

have the Phase-loss Property When collapse-arresting mechanism is small but not

zero, post-collapse phase is unique. But, the initial phase information is blurred by the large sensitivity to small perturbations of the phase accumulated during the collapse. Initial phase is `àlmost lost’’

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Simulations – NLS with nonlinear saturation

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42, , 0, 5 4ti t x y e

Simulations – NLS with nonlinear saturation

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2 10, , 0, 0.3 8ti t x y e

Experiments (Shim et al., 2012) Laser beam after propagation of 24cm in water ``Correct’’ physical continuation is not known

Post-collapse loss-of-phase is observed

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Simulations of propagation in water NLS with dispersion, space-time focusing, multiphoton

absorption, plasma … Input-power randomly chosen between 240 -260 MW On-axis phase after propagation of 24cm

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Importance of loss of phase NLS solution is invariant under multiplication by eiθ

Multiplication by eiθ does not affect the dynamics But, relative phase of two beams does affect the dynamics

62collapse no

collapse



63collapse no

collapse



64collapse no

collapse



65collapse no

collapse

Post-collapse chaotic

interactions

Experiments (Shim et al., 2012) Interaction between two `ìdentical’’ crossing beams after

propagation of 24cm in water - seven consecutive shots

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Experiments (Shim et al., 2012) Interaction between two `ìdentical’’ crossing beams after

propagation of 24cm in water - seven consecutive shots

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Experiments (Shim et al., 2012) Interaction between two parallel beams with initial

π phase difference, after propagation of 24cm in water - five consecutive shots

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Summary

1. Sub-threshold power continuation ψ0

ε= (1-ε)Kthf(x) Generalization of Merle (92) Limiting solution is a Bourgain-Wang sol., before and after

the singularity

2. Vanishing nonlinear-damping continuation Vanishing-viscosity approach Viscosity = nonlinear damping Explicit continuation of ψR,α

explicit Asymmetric w.r.t. Tc

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Summary: Properties of continuations

Loss of phase at/after singularity Universal feature Leads to post-collapse chaotic interactions Observed numerically and experimentally

Symmetry with respect to Tc

Jump condition Only holds for time-reversible continuations Not a universal feature

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Open problems

What is the `correct’’ continuation? Additional properties of continuations?

`Èntropy’’ conditions? ``Riemann Problems’’? …

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References

G. Fibich and M. Klein Continuations of the nonlinear Schrödinger equation beyond the singularity Nonlinearity 24: 2003-2045, 2011

G. Fibich and M. Klein Nonlinear-damping continuation of the nonlinear Schrödinger equation- a numerical study Physica D 241: 519-527, 2012

B. Shim, S.E. Schrauth, , A.L. Gaeta, M. Klein, and G. FibichLoss of phase of collapsing beams Physical Review Letters 108: 043902, 2012

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Documents

Continuations of NLS solutions beyond the singularity Gadi Fibich Tel Aviv University ff