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 ǫ p@ǫ ą 0q 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.

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

“…Recently, there are some important progresses on long time stability and possible growth of Sobolev norm for high dimensional Hamiltonian PDEs on irrational tori. See ( [7,21,22,26]) for example. It is natural to ask whether our method can be applied to this case.…”

Analytic solutions of nonlinear elliptic equations on rectangular tori

“…So far, there are also many results about Sobolev norms growth for equations on rectangular tori, see e.g. [3,17,20,21]. Remark 1.3.…”

“…Berti-Bolle made use of a modified Nash-Moser iterations together with the multi-scale analysis. In a latest work by Berti-Maspero [3], they proved the existence of Sobolev regular quasi-periodic solutions for the NLW and NLS on arbitrary rectangular tori. We should remak that all existence solutions mentioned above are at most Gevrey regular in time or space variables.…”

“…Very recently, Wang [34] proved the existence of analytic quasi-periodic Floquet-Bloch solutions for NLS on R d . In fact, the present paper is also motivated by work of Berti-Maspero [3], Wang [34] and Yuan [38].…”

KAM Theory for Partial Differential Equations

“…which is an irrationality condition on the slopes of the "singular cone" in (2.48), see [133] to avoid this condition. For NLS this condition is not required.…”