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

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- by J. Krieger and W. Schlag;
- J. Amer. Math. Soc.
**19**(2006), 815-920 - DOI: https://doi.org/10.1090/S0894-0347-06-00524-8
- Published electronically: February 20, 2006
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## Abstract:

Standing wave solutions of the one-dimensional nonlinear Schrödinger equation \[ i\partial _t \psi + \partial _{x}^2 \psi = -|\psi |^{2\sigma } \psi \] 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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## Bibliographic Information

**J. Krieger**- Affiliation: Department of Mathematics, Harvard University, Science Center, 1 Oxford Street, Cambridge, Massachusetts 02138
- MR Author ID: 688045
- Email: jkrieger@math.harvard.edu
**W. Schlag**- Affiliation: Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, Illinois 60637
- MR Author ID: 313635
- Email: schlag@math.uchicago.edu
- 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. - © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 815-920 - MSC (2000): Primary 35Q55, 35Q51, 37K40, 37K45
- DOI: https://doi.org/10.1090/S0894-0347-06-00524-8
- MathSciNet review: 2219305