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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Scipio Cuccagna and Nicola Visciglia
Journal: Trans. Amer. Math. Soc. 363 (2011), 2357-2391
MSC (2010): Primary 35Q55
Published electronically: November 19, 2010
MathSciNet review: 2763720
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Abstract: We consider a nonlinear Schrödinger equation

$\displaystyle iu_{t} -h_{0}u + \beta ( \vert u\vert^{2} )u=0 , (t,x)\in \mathbb{R}\times \mathbb{R}, $

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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Additional Information

Scipio Cuccagna
Affiliation: DISMI, University of Modena and Reggio Emilia, via Amendola 2, Padiglione Morselli, Reggio Emilia 42100 Italy
Email: cuccagna.scipio@unimore.it

Nicola Visciglia
Affiliation: Dipartimento di Matematica “L. Tonelli”, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy
Email: viscigli@dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05046-9
PII: S 0002-9947(2010)05046-9
Received by editor(s): September 27, 2008
Published electronically: November 19, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.