On the formation of singularities in solutions of the critical nonlinear Schrödinger equation

Annales Henri Poincare

Volumn 2, Issue 4, 2001, Pages 605-673

  Perelman, Galina ^a  

^a ECOLE POLYTECHNIQUE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035537074     PISSN: 14240637     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00001048     Document Type: Article

References (39)

1. H. Bateman, Higher transcendental functions, v.II, New York (1953).
2. J. Bourgain, W. Wang, Construction of blow-up solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) XXV , 197-215 (1997).
3. V.S. Buslaev, G.S. Perelman, Scattering for the nonlinear Schrödinger equation : states close to a soliton, St. Peters. Math. J. 4, 1111-1143 (1993).
4. V.S. Buslaev, G.S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equation, Amer. Math. Soc. Transl. 2 164, 75-99 (1995).
5. T. Cazenave, An introduction to nonlinear Schrödinger equation, Textos de Métodes Mateática, UFRJ, Rio de Janeiro, (1989).
6. T. Cazenave, F.B. Weissler, The structure of solutions to the pseudoconformal invariant nonlinear Schrödinger equation, Proc. Royal Soc. Edinburgh 117A, 251-273 (1991).
7. s, Nonlin. Anal., Theory, Methods & Appl. 14 , 807-836 (1990).
8. S. Dyachenko, A.C. Newell, A. Pushkarev, V.I. Zakharov, Optical turbulence : weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation, Phys. D 57, 96-160 (1992).
9. M.V. Fedoryuk, Asymptotic analysis : Linear ordinary differential equations, Berlin (1993).
10. G.M. Fraiman, Asymptotic stability of manifold of self-similar solutions in self-focusing, Sov. Phys. JETP 61, 228-233 (1985).
11. J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations I, II, J. Func. Anal. 32, 1-71 (1979).
12. J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations III, Ann. Inst. H. Poincaré Phys. Thoér. 28, 287-316 (1978).
13. R. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys. 8, 1794-1797 (1977).
14. L. Glangetas, F. Merle, Existence of self-similar blow-up solution for the Zakharov equation in dimension two, Comm. Math. Phys. 160, 173-215 (1994).
15. M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein - Gordon equations, Comm. Pure Appl. Math., 747-774 (1988).
16. N. Kosmatov, V. Schvets, V. Zakharov, Computer simulation of wave collapse in nonlinear Schrödinger equation, Phys. D 52, 16-35 (1991).
17. E.W. Laedke, R. Blaha, K.H. Spatschek, E.A. Kuznetsov, On the stability of collapse in the critical case, J. Math. Phys. 33 (3), 967-973 (1992).
18. M. Landman, G. Papanicolaou, C. Sulem, P.L. Sulem, Rate of blow-up for solutions of the nonlinear Schródinger equation at critical dimension, Phys. Rev. A 38, 3837-3843 (1988).
19. B.J. LeMesurier, G. Papanicolaou, C. Sulem, P.L. Sulem, Focusing and multi-focusing solutions of the nonlinear Schrödinger equation, Phys. D 31, 78-102 (1988).
20. B.J. LeMesurier, G. Papanicolaou, C. Sulem, P. Sulem, Local structure of the self-focusing singularity of the nonlinear Schrödinger equation, Phys. D 32, 210-226 (1988).
21. V.M. Malkin, Dynamics of wave collapse in the critical case, Phys. Lett. A 151, 285-288 (1990).
22. V.M. Malkin, On the analytical theory for stationary self-focusing of radiation, Phys. D 64, 251-266 (1993).
23. V.M. Malkin, Singularity formation for nonlinear Schrödinger equation, in "Partial Differential equations and their applications", P.C. Grenier, V. Ivrii, L.A. Seco and C. Sulem, eds, CRM Proceedings and Lecture Notes 12, 183-198 (1997).
24. Y. Martel, F. Merle, Stability of blowup profile and lower bounds for blowup rate for the critical generalized KdV equation, preprint (2000).
25. F. Merle, On uniqueness and continuation properties after blow-up time of self-similar solutions of the nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math. 15, 203-254 (1992).
26. F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69, 427-453 (1993).
27. F. Merle, Lower bounds for the blow-up rate of solutions of the Zakharov equation in dimension two, Comm. Pure Appl. Math. 49, 765-79 (1996).
28. 2 solution for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc. 111, 487-496 (1991).
29. G. Perelman, On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, Sem. EDP Ec. Polytec. (1999-2000).
30. A.I. Smirnov, G.M. Fraiman, The interaction representation in the self-focusing theory, Phys. D 51, 2-15 (1991).
31. C. Sulem, P.L. Sulem, Focusing nonlinear Schrödinger equation and wave-packet collapse, Nonlin. Anal., Theory, Meth. & Appl. 30 (2), 833-844 (1997).
32. C. Sulem, P.L. Sulem, The nonlinear Schrdinger equation. Self-focusing and wave collapse. Appl. Math. Scien. 139 (1999), Springer-Verlag, New York.
33. A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations I, Comm. Math. Phys. 133, 119-146 (1990).
34. A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering theory for nonintegrable equations II, J. Diff. Eq. 98, 376-390 (1992).
35. M.I. Weinstein, Modulation stability of ground states of nonlinear Schrödinger equation, SIAM J. Math. Anal. 16, 472-491 (1985).
36. M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39, 51-68 (1986).
37. M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87, 567-576 (1983).
38. M.I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Part. Diff. Eq. 11 (5), 545-565 (1986).
39. M.I. Weinstein, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Diff. Eq. 69, 192-203 (1987).