HIGH FREQUENCY SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION ON SURFACES

We address the problem of describing solutions of the nonlinear Schrodinger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrodinger equation.