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

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

2002-06-20

arxiv.org/abs/math/0206218v2

Mathematics

Analysis of PDEs

updated draft

Scientific paper

We study the long-time behaviour of the focusing cubic NLS on $\R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein. In the sequel to this paper we generalize this result to other nonlinear Schr\"odinger equations. Our arguments are based on the ``$I$-method'' from our earlier papers, which pushes down from the energy norm, as well as an ``upside-down $I$-method'' which pushes up from the $L^2$ norm.

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