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.


35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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