Quasi-periodic solutions for nonlinear wave equations (announcement)

TL;DR

This paper constructs time quasi-periodic solutions for nonlinear wave equations on multi-dimensional tori, extending previous results limited to the circle and adapting methods from elliptic to hyperbolic equations using Diophantine properties.

Contribution

It generalizes the construction of quasi-periodic solutions from one-dimensional circles to higher-dimensional tori for nonlinear wave equations, introducing new Diophantine techniques.

Findings

Existence of quasi-periodic solutions in arbitrary dimensions

Extension of elliptic methods to hyperbolic equations

Use of algebraic Diophantine properties

Abstract

We construct time quasi-periodic solutions to nonlinear wave equations on the torus in arbitrary dimensions. All previously known results (in the case of zero or a multiplicative potential) seem to be limited to the circle. This generalizes the method developed in the limit-elliptic setting of NLS to the hyperbolic setting. The additional ingredient is a Diophantine property of algebraic numbers.