Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao

Research output: Contribution to journalArticle

19 Scopus citations

Abstract

We study the long-time behaviour of the focusing cubic NLS on R in the Sobolev norms Hs for 0 < s < 1. We obtain polynomial growth-type upper bounds on the Hs norms, and also limit any orbital Hs instability of the ground state to polynomial growth at worst; this is a partial analogue of the H1 orbital stability result of Weinstein [27],[26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the "I-method" from earlier papers [9]-[15] which pushes down from the energy norm, as well as an "upside-down I-method" which pushes up from the L2 norm.

Original languageEnglish (US)
Pages (from-to)31-54
Number of pages24
JournalDiscrete and Continuous Dynamical Systems
Volume9
Issue number1
StatePublished - Jan 1 2003

Keywords

  • Orbital stability
  • Schrödinger equation
  • Upper bound on sobolev norms

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