Long time dynamics of Schrödinger and wave equations on flat tori

Abstract

We consider a class of linear time dependent Schrödinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schrödinger equations on arbitrary flat tori. For the linear Schrödinger equation, we prove a $t^{\epsilon }$ $(\forall \epsilon >0)$ upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on "clusterization properties" of the eigenvalues of the Laplacian on a flat torus and on suitable "separation properties" of the singular sites of Schrödinger and wave operators, which are integers, in space-time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [Del10] to control the speed of growth of the Sobolev norms, and Berti-Corsi-Procesi abstract Nash-Moser theorem [BCP15] to construct quasi-periodic solutions.

We extended the previous results to the case of arbitrary flat tori