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 language | English (US) |
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Pages (from-to) | 31-54 |
Number of pages | 24 |
Journal | Discrete and Continuous Dynamical Systems |
Volume | 9 |
Issue number | 1 |
DOIs | |
State | Published - Jan 2003 |
Keywords
- Orbital stability
- Schrödinger equation
- Upper bound on sobolev norms