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On asymptotic stability of ground states of NLS. (English) Zbl 1084.35089

For even ground states of the nonlinear Schrödinger equation \[ i u_{t}+\Delta u+\beta (| u|^{2})u=0, (t,x)\in \mathbb R\times \mathbb R^3, \] the author proves an asymptotic stability result. Ground states are solutions of the form \(e^{it\omega+\gamma} \varphi_{\omega}(x),\gamma\) and \(\omega\) constants, \(\varphi_{\omega}(x)\) a positive exponentially decreasing as \(| x|\rightarrow\infty\) sperically symmetric solution to the equation \(-\Delta\varphi_{\omega} +\omega\varphi_{\omega}- \beta(\varphi^{2}_{\omega})\varphi_{\omega}=0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

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