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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension


Authors: J. Krieger and W. Schlag
Journal: J. Amer. Math. Soc. 19 (2006), 815-920
MSC (2000): Primary 35Q55, 35Q51, 37K40, 37K45
Published electronically: February 20, 2006
MathSciNet review: 2219305
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Abstract | References | Similar Articles | Additional Information

Abstract: Standing wave solutions of the one-dimensional nonlinear Schrödinger equation

$\displaystyle i\partial_t \psi + \partial_{x}^2 \psi = -\vert\psi\vert^{2\sigma} \psi $

with $ \sigma>2$ are well known to be unstable. In this paper we show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult $ L^2$-critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors' companion paper Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.


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Additional Information

J. Krieger
Affiliation: Department of Mathematics, Harvard University, Science Center, 1 Oxford Street, Cambridge, Massachusetts 02138
Email: jkrieger@math.harvard.edu

W. Schlag
Affiliation: Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
Email: schlag@math.uchicago.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00524-8
PII: S 0894-0347(06)00524-8
Keywords: Critical Schr\"odinger equation, stable manifolds, modulation theory, spectral theory
Received by editor(s): January 13, 2005
Published electronically: February 20, 2006
Additional Notes: The first author was partially supported by the NSF grant DMS-0401177. He also wishes to thank Caltech, where part of this work was done.
The second author was partially supported by the NSF grant DMS-0300081 and a Sloan fellowship.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.