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

Krieger, J.; Schlag, W.

doi:10.1090/S0894-0347-06-00524-8

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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