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On asymptotic stability of moving ground states of the nonlinear Schrödinger equation


Author: Scipio Cuccagna
Journal: Trans. Amer. Math. Soc. 366 (2014), 2827-2888
MSC (2010): Primary 35Q55
DOI: https://doi.org/10.1090/S0002-9947-2014-05770-X
Published electronically: February 6, 2014
MathSciNet review: 3180733
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Abstract: We extend to the case of moving solitons, the result on asymptotic stability of ground states of the NLS obtained by the author in an earlier paper. For technical reasons we consider only smooth solutions. The proof is similar to the earlier paper. However now the flows required for the Darboux Theorem and the Birkhoff normal forms, instead of falling within the framework of standard theory of ODE's, are related to quasilinear hyperbolic symmetric systems. It is also not obvious that the Darboux Theorem can be applied, since we need to compare two symplectic forms in a neighborhood of the ground states not in $ H^{1}(\mathbb{R}^3)$, but rather in the space $ \Sigma $ where also the variance is bounded. But the NLS does not preserve small neighborhoods of the ground states in $ \Sigma $.


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Additional Information

Scipio Cuccagna
Affiliation: Department of Mathematics and Computer Sciences, University of Trieste, via Valerio 12/1 Trieste, 34127 Italy
Email: scuccagna@units.it

DOI: https://doi.org/10.1090/S0002-9947-2014-05770-X
Received by editor(s): July 25, 2011
Received by editor(s) in revised form: October 22, 2011, and December 5, 2011
Published electronically: February 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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