Polynomial upper bounds for the instability of the Nonlinear Schrödinger equation below the energy norm

Details Polynomial upper bounds for the instability of the Nonlinear Schrödinger equation below the energy norm Polynomial upper bounds for the instability of the Nonlinear Schrödinger equation below the energy norm

2002-12-09

arxiv.org/abs/math/0212113v2

Comm. Pure Appl. Anal. 2 (2003), 33-50

Mathematics

Analysis of PDEs

20 pages, no figures. Some typos corrected

Scientific paper

We continue the study (initiated in \cite{ckstt:7}) of the orbital stability of the ground state cylinder for focussing non-linear Schr\"odinger equations in the $H^s(\R^n)$ norm for $1-\eps < s < 1$, for small $\eps$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.

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