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Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. (English) Zbl 1038.35128

The author considers a nonlinear Schrödinger equation with a smooth localized real potential \(V\), \(i\partial_t\psi=(-\Delta+V)\psi+\lambda | \psi| ^2\psi\), where \(-\Delta+V\) is assumed to have three or more bound states. It is shown, under certain conditions on the initial data, that the solution will converge locally to a nonlinear ground state. This extends the author’s previous work with H.-T. Yau [Commun. Pure Appl. Math. 55, 153–216 (2002; Zbl 1031.35137), Int. Math. Res. Notes 31, 1629–1673 (2002; Zbl 1011.35120)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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