Quasi-periodic solutions of the Schr\"odinger equation with arbitrary algebraic nonlinearities

TL;DR

This paper establishes the existence of quasi-periodic solutions for the Schrödinger equation with arbitrary algebraic nonlinearities across various dimensions, extending previous results and solving longstanding problems in Hamiltonian PDEs.

Contribution

It introduces a geometric approach to prove the existence of quasi-periodic solutions with arbitrary frequencies and nonlinearities, generalizing prior work in the field.

Findings

Existence of quasi-periodic solutions with up to d+2 frequencies in arbitrary dimensions.

Solutions exist for arbitrary algebraic nonlinearities p.

In 1D, solutions exist for any number of frequencies and any p, addressing a longstanding problem.

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, $b\leq d+2$, 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.