Proof of a Spectral Property related to the singularity formation for the L2 critical nonlinear Schrödinger equation

Physica D: Nonlinear Phenomena

Volumn 220, Issue 1, 2006, Pages 1-13

  Fibich, Gadi ^a   Merle, Frank ^b   Raphaël, Pierre ^c  

^a TEL AVIV UNIVERSITY   (Israel)

^b UNIVERSITÉ DE CERGY PONTOISE   (France)

^c UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Collapse; Log log law; Nonlinear Schr dinger equation; Self similar solution; Singularity

Indexed keywords

COERCIVE FORCE; COMPUTER AIDED ANALYSIS; NONLINEAR EQUATIONS; SPECTRUM ANALYSIS; THEOREM PROVING; WAVE EQUATIONS;

COLLAPSE; LOG-LOG LAW; LOG-LOG SPEED; NONLINEAR SCHRODINGER EQUATION; SELF-SIMILAR SOLUTION; SINGULARITY;

HIGH ENERGY PHYSICS;

EID: 33746745308     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2006.06.010     Document Type: Article

References (28)

1. Berestycki H., and Lions P.-L. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82 4 (1983) 313-345
2. Bourgain J., and Wang W. Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scer. Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) ; (1-2) (1998) 197-215
3. Dyachenko S., Newell A.C., Pushkarev A., and Zakharov V.E. Optical turbulence: Weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D 57 1-2 (1992) 96-160
4. Fibich G., and Papanicolaou G. Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension. SIAM J. Appl. Math. 60 (1999) 183-240
5. Fraiman G. Asymptotic stability of manifold of self-similar solutions in self-focusing. Sov. Phys. JETP 61 2 (1985) 228-233
6. Gidas B., Ni W.M., and Nirenberg L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979) 209-243
7. Ginibre J., and Velo G. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32 1 (1979) 1-32
8. n. Arch. Ration. Mech. Anal. 105 3 (1989) 243-266
9. Landman M.J., Papanicolaou G.C., Sulem C., and Sulem P.-L. Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A (3) 38 8 (1988) 3837-3843
10. LeMesurier B., Papanicolaou G., Sulem C., and Sulem P. Local structure of the self-focusing singularity of the nonlinear Schrödinger equation. Physica D 32 (1988) 210-226
11. de la Llave R. Computer assisted proofs of stability of matter. Computer Aided Proofs in Analysis (Cincinnati, OH, 1989). IMA Vol. Math. Appl. vol. 28 (1991), Springer, New York 116-126
12. Matano H., and Merle F. On nonexistence of type II blowup for a supercritical nonlinear heat equation. Comm. Pure Appl. Math. 57 11 (2004) 1494-1541
13. Merle F. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69 2 (1993) 427-454
14. Merle F., and Raphaël P. Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. Ann. of Math. 161 1 (2005) 157-222
15. Merle F., and Raphaël P. Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13 (2003) 591-642
16. 2 critical nonlinear Schrödinger equation. Invent. Math. 156 (2004) 565-672
17. Merle F., and Raphaël P. Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation. J. Amer. Math. Soc. 19 1 (2006) 37-90
18. Merle F., and Raphaël P. Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Comm. Math. Phys. 253 3 (2004) 675-704
19. Merle F., and Raphaël P. On one blow up point solutions to the critical nonlinear Schrödinger equation. J. Hyperbolic Differ. Equ. 2 4 (2006)
20. Pelinovsky D. Radiative effects to the adiabatic dynamics of envelope-wave solitons. Physica D 119 3-4 (1998) 301-313
21. Perelman G. On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D. Ann. Henri Poincaré 2 (2001) 605-673
22. Raphaël P. Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation. Math. Ann. 331 3 (2005) 577-609
23. Reed M., and Simon B. Methods of Modern Mathematical Physics vol. I-IV (1972-1979), Academic Press, New York
24. Seco, and Luis A. Computer assisted lower bounds for atomic energies. Computer Aided Proofs in Analysis (Cincinnati, OH, 1989). IMA Vol. Math. Appl. vol. 28 (1991), Springer, New York 241-251
25. Sulem C., and Sulem P.L. The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse. Applied Mathematical Sciences vol. 139 (1999), Springer-Verlag, New York
26. Weinstein M.I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983) 567-576
27. Weinstein M.I. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985) 472-491
28. Zakharov V.E., and Shabat A.B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media. Sov. Phys. JETP 34 (1972) 62-69