Small data scattering for the nonlinear Schr\"odinger equation on product spaces

We consider the cubic nonlinear Schr\"odinger equation, posed on $\R^n\times M$, where $M$ is a compact Riemannian manifold and $n\geq 2$. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.Comment: 10 pages, slightly revised version, to appear on Comm. PD

Stability and instability of the KdV solitary wave under the KP-I flow

We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation