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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension
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by J. Krieger and W. Schlag PDF
J. Amer. Math. Soc. 19 (2006), 815-920 Request permission

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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Additional 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