Nicolas Camps

This talk is devoted to a statistical approach to nonlinear waves on surfaces. Specifically, we consider the cubic Schrödinger equation (NLS) posed on the 2 dimensional sphere and we study the collective dynamics of initial data distributed according to the Gibbs measure. First, we give a brief overview of the Cauchy problem for NLS posed on compact surfaces. Then, we introduce the Gibbs measure problem and a general resolution scheme developed by J. Bourgain in 1996, in the particular case of the torus. In the case of the sphere, however, we uncover strong instabilities due to concentrated spherical harmonics around a great circle, which prevent a direct extension of Bourgain’s method to the sphere. In the second part of the talk, we present a probabilistic quasi-linear resolution scheme with a critical flavor that is tuned to overcome the instabilities and to solve the Gibbs measure problem for the sphere. These are ongoing joint works with Nicolas Burq, Chenmin Sun and Nikolay Tzvetkov.