## Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves.(English)Zbl 1072.35167

Small in $$H^1$$ solutions are decomposed into solitary and dispersive waves. It is shown that as $$t \rightarrow \infty$$, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges strongly in $$H^1$$ to a solution of the free Schrödinger equation.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems 35Q51 Soliton equations
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