1 Continuations of NLS solutions beyond the singularity Gadi Fibich Tel Aviv University ff Moran Klein - Tel Aviv University B. Shim, S.E. Schrauth, A.L. Gaeta - Cornell

2 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) r=(x,y) z z=0 Kerr Medium Input Beam

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

4  Kelley (1965) : Solutions of 2D cubic NLS can become singular in finite time (distance) T c Finite-time singularity

Beyond the singularity 5 ? No singularities in nature Laser beam propagates past T c 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 7 TcTc ? NLS

Continuation of singular NLS solutions 8 TcTc NLS no ``viscous’’ terms

Continuation of singular NLS solutions 9 TcTc NLS ``jump’’ condition no ``viscous’’ terms

Continuation of singular NLS solutions  2 key papers by Merle (1992)  Less than 10 papers 10 TcTc NLS ``jump’’ condition no ``viscous’’ terms

Talk plan 1.Review of NLS theory 2.Merle’s continuation 3.Sub-threshold power continuation 4.Nonlinear-damping continuation 5.Continuation of linear solutions 6.Phase-loss property 11

12 General NLS  d – dimension, σ – nonlinearity   Definition of singularity: T c - singularity point

13 Classification of NLS global existence (no blowup) Subcritical σd<2 blowupCritical σd=2 blowupSupercritical σd>2

 σd= 2  Physical case considered earlier (σ=1,d=2)  Since 2σ= 4/d, critical NLS can be rewritten as 14 Critical NLS (focus of this talk)

15  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 (L 2 norm) Solitary waves d=2 Townes profile

16 Critical power for collapse Thm (Weinstein, 1983): A necessary condition for collapse in the critical NLS is  P cr - critical power/mass/L 2 -norm for collapse

Explicit blowup solutions  Solution width L(t)  0 as t  T c  ψ R,α explicit becomes singular at T c  Blowup rate of L(t) is linear in t 17 t

Minimal-power blowup solutions  ψ R,α explicit has exactly the critical power  Minimal-power blowup solution  ψ R,α explicit is unstable, since any perturbation that reduces its power will lead to global existence 18 Thm (Weinstein, 86; Merle, 92) The explicit blowup solutions ψ R,α explicit are the only minimal-power solutions of the critical NLS.

Stable blowup solutions of critical NLS Fraiman (85), Papanicolaou and coworkers (87/8)  Solution splits into a singular core and a regular tail  Singular core collapses with a self-similar ψ R profile  Blowup rate is given by  Tail contains the rest of the power ( )  Rigorous proof: Perelman (01), Merle and Raphael (03) 19

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 ψ R,α explicit profile  Blowup rate is linear  ψ B-W are unstable, since they are based on ψ R,α explicit (Merle, Raphael, Szeftel; 2011)  Non-generic solutions 20

21 Continuation of NLS solutions beyond the singularity ?

Talk plan 1.Review of NLS theory 2.Merle’s continuation 3.Sub-threshold power continuation 4.Nonlinear-damping continuation 5.Continuation of linear solutions 6.Phase-loss property 22

Explicit continuation of ψ R,α explicit (Merle, 92)  Let ψ ε be the solution of the critical NLS with the ic   Ψ ε exists globally  Merle computed rigorously the limit 23

Before singularity, since After singularity Thm (Merle 92) 24

Before singularity After singularity Thm (Merle 92) 25

Before singularity After singularity Thm (Merle 92) 26

 NLS is invariant under time reversibility  Hence, solution is symmetric w.r.t. to collapse-arrest time T ε arrest  As ε  0, T ε arrest  T c  Therefore, continuation is symmetric w.r.t. T c  Jump condition Symmetry Property - motivation 27

Thm (Merle 92) 28 Symmetry property: Continuation is symmetric w.r.t. T c Phase-loss Property: Phase information is lost at/after the singularity After singularity

Phase-loss Property - motivation  Initial phase information is lost at/after the singularity  Why?  For t>T c, on-axis phase is ``beyond infinity’’ 29

 Merle’s continuation is only valid for  Critical NLS  Explicit solutions ψ R,α explicit  Unstable  Non-generic  Can this result be generalized? 30

Talk plan 1.Review of NLS theory 2.Merle’s continuation 3.Sub-threshold power continuation 4.Nonlinear-damping continuation 5.Continuation of linear solutions 6.Phase-loss property 31

Sub threshold-power continuation ( Fibich and Klein, 2011 )  Let f(x) ∊ H 1  Consider the NLS with the i.c. ψ 0 = K f(x)  Let K th be the minimal value of K for which the NLS solution becomes singular at some 0<T c <∞  Let ψ ε be the NLS solution with the i.c. ψ 0 ε = (1-ε)K th f(x)  By construction,  0<ε ≪1, no collapse  -1 ≪ ε<0, collapse  Compute the limit of ψ ε as ε  0+  Continuation of the singular solution ψ(t, x; K th )  Asymptotic calculation (non-rigorous) 32

Before singularity Core collapses with ψ R,α explicit 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 ψ 0 = K f(x) Proposition (Fibich and Klein, 2011) 33

Before singularity After singularity Symmetry w.r.t. T c (near the singularity) Hence, Proposition (Fibich and Klein, 2011) 34

Proposition (Fibich and Klein, 2011) 35 Phase information is lost at the singularity Why? After singularity

Simulations - convergence to ψ B-W  Plot solution width L(t; ε) 36

Simulations – loss of phase  How to observe numerically?  If 0<ε ≪ 1, post-collapse phase is ``almost lost’’  Small changes in ε lead to O(1) changes in the phase which is accumulated during the collapse  Initial phase information is blurred 37

Simulations - loss of phase 38 O(10 -5 ) change in ic lead to O(1) post-collapse phase changes

Simulations - loss of phase 39 O(10 -5 ) change in ic lead to O(1) post-collapse phase changes

Talk plan 1.Review of NLS theory 2.Merle’s continuation 3.Sub-threshold power continuation 4.Nonlinear-damping continuation 5.Continuation of linear solutions 6.Phase-loss property 40

NLS continuations  So far, only within the NLS model:  Lower the power below P th, and let P  P th -  Different approach: Add an infinitesimal perturbation to the NLS  Let ψ ε be the solution of  If ψ ε exists globally for any 0<ε ≪ 1, can define the ``vanishing –viscosity continuation’’ 41

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  … 42

Nonlinear damping  ``Viscosity’’ = nonlinear damping  Physical – multi-photon absorption  Destroys Hamiltonian structure  Good! 43

Critical NLS with nonlinear damping  Vanishing nl damping continuation : Take the limit δ  0+  Consider ψ 0 is such that ψ becomes singular when δ=0  if q≥ 4/d, collapse arrested for any δ>0  If q δ c (ψ 0 )>0  Can define the continuation for q≥ 4/d 44

Explicit continuation  Critical NLS with critical nonlinear damping (q=4/d)  Compute the continuation of ψ R,α explicit as δ  0+  Use modulation theory (Fibich and Papanicolaou, 99)  Systematic derivation of reduced ODEs for L(t)  Not rigorous 45

Asymptotic analysis  Near the singularity  Reduced equations given by  Solve explicitly in the limit as δ  0+ 46

Asymptotic analysis  Near the singularity  Reduced equations given by  Solve explicitly in the limit as δ  0+  Asymmetric with respect to T c  Damping breaks reversibility in time 47

Before singularity After singularity Phase information is lost at the singularity Why? Proposition (Fibich, Klein, 2011) 48

Simulations – asymmetric continuation L= α) T c -t( L=κα(t-T c )

Simulations – loss of phase

Nonlinear-damping continuation of loglog solutions 51 Highly asymmetric Slope  ±∞

Talk plan 1.Review of NLS theory 2.Merle’s continuation 3.Sub-threshold power continuation 4.Nonlinear-damping continuation 5.Continuation of linear solutions 6.Phase-loss property 52

Collapse in linear optics  Can solve explicitly in the geometricaloptics limit (k 0  ∞)  Linear collapse at z=F 53 z=t k 0 is wave #

Continuation of singular GO solution  consider the linear Schrödinger with k 0 <∞  Global existence  Can solve explicitly (without GO approx)  Compute the limit as k 0  ∞ 54

 No phase loss after singularity  Why? Post-collapse phase loss is a nonlinear phenomena Continuation of singular GO solution 55

Talk plan 1.Review of NLS theory 2.Merle’s continuation 3.Sub-threshold power continuation 4.Nonlinear-damping continuation 5.Continuation of linear solutions 6.Phase-loss property 56

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 ``almost lost’’ 57

Simulations – NLS with nonlinear saturation 58

Simulations – NLS with nonlinear saturation 59

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 60

Simulations of propagation in water  NLS with dispersion, space-time focusing, multiphoton absorption, plasma …  Input-power randomly chosen between MW  On-axis phase after propagation of 24cm 61

Importance of loss of phase  NLS solution is invariant under multiplication by e iθ  Multiplication by e iθ does not affect the dynamics  But, relative phase of two beams does affect the dynamics 62 collapse no collapse

Importance of loss of phase  NLS solution is invariant under multiplication by e iθ  Multiplication by e iθ does not affect the dynamics  But, relative phase of two beams does affect the dynamics 63 collapse no collapse

Importance of loss of phase  NLS solution is invariant under multiplication by e iθ  Multiplication by e iθ does not affect the dynamics  But, relative phase of two beams does affect the dynamics 64 collapse no collapse

Importance of loss of phase  NLS solution is invariant under multiplication by e iθ  Multiplication by e iθ does not affect the dynamics  But, relative phase of two beams does affect the dynamics 65 collapse no collapse Post-collapse chaotic interactions

Experiments (Shim et al., 2012)  Interaction between two ``identical’’ crossing beams after propagation of 24cm in water - seven consecutive shots 66

Experiments (Shim et al., 2012)  Interaction between two ``identical’’ crossing beams after propagation of 24cm in water - seven consecutive shots 67

Experiments (Shim et al., 2012)  Interaction between two parallel beams with initial π phase difference, after propagation of 24cm in water - five consecutive shots 68

Summary 1.Sub-threshold power continuation  ψ 0 ε = (1-ε)K th f(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 ψ R,α explicit  Asymmetric w.r.t. T c 69

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 T c  Jump condition  Only holds for time-reversible continuations  Not a universal feature 70

Open problems  What is the `correct’’ continuation?  Additional properties of continuations?  ``Entropy’’ conditions?  ``Riemann Problems’’?  … 71

References G. Fibich and M. Klein Nonlinearity Continuations of the nonlinear Schrödinger equation beyond the singularity Nonlinearity 24: , 2011 G. Fibich and M. Klein Nonlinear-damping continuation of the nonlinear Schrödinger equation- a numerical study Physica D 241: , 2012 B. Shim, S.E. Schrauth,, A.L. Gaeta, M. Klein, and G. Fibich Loss of phase of collapsing beams Physical Review Letters 108: ,