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

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

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

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  1. Continuations of NLS solutionsbeyond the singularityGadi Fibich Tel Aviv Universityff • 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) Input Beam z Kerr Medium z=0

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

  4. Finite-time singularity • Kelley (1965) : Solutions of 2D cubic NLS can become singular in finite time (distance) Tc

  5. Beyond the singularity ? • 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 • …

  6. 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

  7. Continuation of singular NLS solutions ? NLS Tc

  8. Continuation of singular NLS solutions no ``viscous’’ terms NLS NLS Tc

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

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

  11. Talk plan • Review of NLS theory • Merle’s continuation • Sub-threshold power continuation • Nonlinear-damping continuation • Continuation of linear solutions • Phase-loss property

  12. General NLS • d – dimension, σ – nonlinearity • Definition of singularity: Tc - singularity point

  13. Classification of NLS

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

  15. Solitary waves • 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) d=2 Townes profile

  16. 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

  17. Explicit blowup solutions • Solution width L(t)0 as tTc • ψR,αexplicitbecomes singular at Tc • Blowup rate of L(t) is linear in t t

  18. Minimal-power blowup solutions • ψR,αexplicithas exactly the critical power • Minimal-power blowup solution • ψR,αexplicitis unstable, since any perturbation that reduces its power will lead to global existence Thm(Weinstein, 86; Merle, 92) The explicit blowup solutions ψR,αexplicitare the only minimal-power solutions of the critical NLS.

  19. 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 ψR profile • Blowup rate is given by • Tail contains the rest of the power ( ) • Rigorous proof: Perelman (01), Merle and Raphael (03)

  20. 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,αexplicit profile • Blowup rate is linear • ψB-Ware unstable, since they are based on ψR,αexplicit(Merle, Raphael, Szeftel; 2011) • Non-generic solutions

  21. Continuation of NLS solutions beyond the singularity ?

  22. Talk plan • Review of NLS theory • Merle’s continuation • Sub-threshold power continuation • Nonlinear-damping continuation • Continuation of linear solutions • Phase-loss property

  23. Explicit continuation of ψR,αexplicit (Merle, 92) • Let ψεbe the solution of the critical NLS with the ic • Ψε exists globally • Merle computed rigorously the limit

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

  25. Thm (Merle 92) • Before singularity • After singularity

  26. Thm (Merle 92) • Before singularity • After singularity

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

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

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

  30. Merle’s continuation is only valid for • Critical NLS • Explicit solutions ψR,αexplicit • Unstable • Non-generic • Can this result be generalized?

  31. Talk plan • Review of NLS theory • Merle’s continuation • Sub-threshold power continuation • Nonlinear-damping continuation • Continuation of linear solutions • Phase-loss property

  32. Sub threshold-power continuation (Fibichand 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 ψεbe the NLS solution with the i.c. ψ0ε= (1-ε)Kth f(x) • By construction, • 0<ε≪1, no collapse • -1≪ε<0, collapse • Compute the limit of ψεasε0+ • Continuation of the singular solution ψ(t, x; Kth) • Asymptotic calculation (non-rigorous)

  33. Proposition (Fibichand Klein, 2011) • Before singularity • 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)

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

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

  36. Simulations - convergence to ψB-W • Plot solution width L(t; ε)

  37. 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

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

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

  40. Talk plan • Review of NLS theory • Merle’s continuation • Sub-threshold power continuation • Nonlinear-damping continuation • Continuation of linear solutions • Phase-loss property

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

  42. 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

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

  44. Critical NLS with nonlinear damping • Vanishing nl damping continuation : Take the limit δ0+ • Consider ψ0is 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

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

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

  47. Asymptotic analysis • Near the singularity • Reduced equations given by • Solve explicitly in the limit as δ0+ • Asymmetricwith respect to Tc • Damping breaks reversibility in time

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

  49. Simulations – asymmetric continuation L=κα(t-Tc) L=α)Tc-t(

  50. Simulations – loss of phase

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