×

On asymptotic stability of solitary waves for nonlinear Schrödinger equations. (English) Zbl 1028.35139

The authors analyse the long-time behaviour of solutions to the nonlinear Schrödinger equation in 1D space dimension for initial conditions in a small neighbourhood of a stable solitary wave. They use the spectral decomposition of the solution on the eigenspaces associated to the discrete and continuous spectrum of the linearized operator near the solitary wave. Using some hypothesis on the structure of the spectrum of the linearized operator, the authors prove that, asymptotically in time, the solution decomposes into a solitary wave with slightly modified parameters and a dispersive part described by the Schrödinger equation.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Cazenave, T.; Lions, P.L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. math. phys., 85, 549-561, (1982) · Zbl 0513.35007
[2] Buslaev, V.; Perelman, G., Scattering for the nonlinear Schrödinger equation: states close to a soliton, Saint |St. Petersburg math. J., 4, 1111-1142, (1993)
[3] Buslaev, V.; Perelman, G., On the stability of solitary waves for nonlinear Schrödinger equations, Amer. math. soc. transl., 164, 75-98, (1995) · Zbl 0841.35108
[4] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., LIV, 1110-1145, (2001) · Zbl 1031.35129
[5] P. Deift, X. Zhou, Perturbation theory for infinite dimensional integrable systems on the line, Preprint · Zbl 1006.35089
[6] Ginibre, J.; Velo, G., On a class of Schrödinger equations. I: the Cauchy problem, general case; II: scattering theory, general case, J. funct. anal., 32, 1-32, (1979), 33-71 · Zbl 0396.35028
[7] Grikurov, V.E., Perturbation of unstable solitons for generalized NLS with saturating nonlinearity, (), 170-179
[8] Grillakis, M.; Shatah, J.; Strauss, W.; Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, part II, J. funct. anal., J. funct. anal., 94, 308-348, (1990) · Zbl 0711.58013
[9] McKean, H.; Shatah, J., The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. pure appl. math., 44, 1067-1083, (1991) · Zbl 0773.35075
[10] Perelman, G., On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. inst. Henri Poincaré, 2, 605-673, (2001) · Zbl 1007.35087
[11] Pelinovsky, D.; Kivshar, Yu.; Afanasjev, V.V., Internal modes of envelope solitons, Phys. D, 116, 121-142, (1998) · Zbl 0934.35175
[12] Soffer, A.; Weinstein, M.; Soffer, A.; Weinstein, M., The case of anisotropic potentials and data, Comm. math. phys., J. differential equations, 98, 376-390, (1992) · Zbl 0795.35073
[13] Soffer, A.; Weinstein, M., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136, 9-74, (1999) · Zbl 0910.35107
[14] Strauss, W., Nonlinear scattering theory at low energy, J. funct. anal., J. funct. anal., 43, 281-293, (1981) · Zbl 0494.35068
[15] C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse, in: Applied Mathematical Sciences, Vol. 139, Springer · Zbl 0928.35157
[16] Weinstein, M., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. pure appl. math., 39, 51-68, (1986) · Zbl 0594.35005
[17] Yau, H.-T.; Tsai, T.-P., Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and radiation dominated solutions, Comm. pure appl. math., LV, 1-64, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.