Asymptotic stability of solitary waves for nonlinear Schrödinger equations. (English) Zbl 1017.35104

Bona, Jerry (ed.) et al., The legacy of the inverse scattering transform in applied mathematics. Proceedings of an AMS-IMS-SIAM joint summer research conference, South Hadley, MA, USA, June 17-21, 2001. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 301, 163-181 (2002).
Summary: This review deals with long-time behavior of solutions of nonlinear Schrödinger equations for initial conditions in a small neighborhood of a stable solitary wave. Under some hypothesis on the structure of the spectrum of the linearized operator near the soliton, the solution decomposes, asymptotically in time, into a solitary wave with slightly modified parameters and a dispersive part described by the free Schrödinger equation. Time behavior of the correction is explicitly calculated.
For the entire collection see [Zbl 1001.00024].


35Q55 NLS equations (nonlinear Schrödinger equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35B40 Asymptotic behavior of solutions to PDEs