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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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