Quasi-periodic solutions for fully nonlinear forced reversible Schroedinger equations

TL;DR

This paper proves the existence and stability of quasi-periodic solutions for a class of fully nonlinear, forced, reversible Schrödinger equations using advanced analytical techniques like Nash-Moser and pseudo-differential operators.

Contribution

It introduces a novel approach combining Nash-Moser iteration with a reducibility theorem to handle the highest order derivatives in nonlinear Schrödinger equations.

Findings

Existence of quasi-periodic solutions established.

Proved linear stability of these solutions.

Developed a new change of variables technique for diagonalization.

Abstract

In this paper we consider a class of fully nonlinear forced and reversible Schroedinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such has diffeomorphisms of the torus and pseudo-differential operators. This procedure automtically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability.