1 / 72

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..)

zlata
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related