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

Buslaev, Vladimir S.; Sulem, Catherine

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].

MSC:

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

Keywords:

long-time behavior; nonlinear Schrödinger equations; solitary wave; dispersive part