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 ,. 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 - 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)|
|Number of pages||24|
|Journal||Discrete and Continuous Dynamical Systems|
|State||Published - Jan 1 2003|
- Orbital stability
- Schrödinger equation
- Upper bound on sobolev norms