On asymptotic stability of moving ground states of the nonlinear Schrödinger equation
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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 $\Sigma$.References
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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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 2827-2888
- MSC (2010): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9947-2014-05770-X
- MathSciNet review: 3180733