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On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. (English) Zbl 1155.35092

Summary: We consider nonlinear Schrödinger equations
\[ iu_t +\Delta u +\beta (|u|^2)u=0\quad \text{for }(t,x)\in \mathbb{R}\times \mathbb{R}^d, \]
where \(d\geq 3\) and \(\beta\) is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as \(t\to\infty\), assuming the so-called Fermi Golden Rule (FGR) hypothesis. We improve the “sign condition” required in a recent paper by Gang Zhou and I. M. Sigal.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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