On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D

Cuccagna, Scipio; Visciglia, Nicola

doi:10.1090/S0002-9947-2010-05046-9

On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D
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by Scipio Cuccagna and Nicola Visciglia PDF

Trans. Amer. Math. Soc. 363 (2011), 2357-2391 Request permission

Abstract:

We consider a nonlinear Schrödinger equation \begin{equation*} iu_{t} -h_{0}u + \beta ( |u|^{2} )u=0 , (t,x)\in \mathbb {R}\times \mathbb {R}, \end{equation*} with $h_{0}= -\frac {d^{2}}{dx^{2}} +P(x)$ a Schrödinger operator with finitely many spectral bands. We assume the existence of an orbitally stable family of ground states. Exploiting dispersive estimates in Cuccagna (2008), Cuccagna and Visciglia (2009), and following the argument in Cuccagna (to appear) we prove that under appropriate hypotheses the ground states are asymptotically stable.

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