A simple criterion of transverse linear instability for solitary waves

We prove an abstract instability result for an eigenvalue problem with parameter. We apply this criterion to show the transverse linear instability of solitary waves on various examples from mathematical physics.Comment: The main result has been improved and its proof simplifie

Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation

We prove a new smoothing type property for solutions of the 1d quintic Schr\"odinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.Comment: Presentation improve

Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr\"Odinger Equations

We study nonlinear Schr\"odinger equations, posed on a three dimensional Riemannian manifold $M$. We prove global existence of strong $H^1$ solutions on $M=S^3$ and $M=S^2\times S^1$ as far as the nonlinearity is defocusing and sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space $\R^3$ and the flat torus \T^3 respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.Comment: Lemma 4.6 in the previous version was false, we made slight modifications to use only a weaker version of this lemm