Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities

Abstract

We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of $b$ frequencies, $b\le d+2$, in arbitrary dimension $d$ and for arbitrary non integrable algebraic nonlinearity $p$. This reflects the conservation of $d$ momenta, energy and $L^2$ norm. In 1d, we prove the existence of quasi-periodic solutions with arbitrary $b$ and for arbitrary $p$, solving a problem that started Hamiltonian PDE.

19 pp, slightly more details on proofs of Theorems 2 and 3

References in corpus (2)

• Bounded Sobolev Norms for Linear Schrödinger Equations Under Resonant Perturbations
• Quasi-periodic solutions of nonlinear random Schrödinger equations