Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential

TL;DR

This paper establishes the existence of Sobolev regular quasi-periodic solutions for multidimensional wave equations with a multiplicative potential, using a Nash-Moser scheme and multiscale analysis to handle small divisors.

Contribution

It extends the theory of quasi-periodic solutions to wave equations with multiplicative potentials in multiple dimensions, employing novel tame estimates and weaker non-resonance conditions.

Findings

Existence of Sobolev quasi-periodic solutions for wave equations with potentials.

Development of a multiscale inductive approach for inverse linearized operators.

Achievement of separation properties under weaker non-resonance conditions.

Abstract

We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T^d, d \geq 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the "separation properties" of the small divisors assuming weaker non-resonance conditions than in [11].