Quasi-periodic solutions of nonlinear wave equations on the d-dimensional torus

TL;DR

This research proves the existence of small amplitude quasi-periodic solutions for nonlinear wave equations on a d-dimensional torus using a Nash-Moser scheme and multiscale analysis, applicable in any space dimension.

Contribution

It introduces a novel multiscale inductive approach to handle resonance issues in constructing quasi-periodic solutions for nonlinear wave equations.

Findings

Existence of small amplitude quasi-periodic solutions in any dimension.

Development of a multiscale approach to overcome resonance conditions.

Implementation of a Nash-Moser scheme with an approximate inverse construction.

Abstract

The main result of this research Monograph is the existence of small amplitude time quasi-periodic solutions for autonomous nonlinear wave equations $$ u_{tt} - \Delta u + V(x) u + g(x, u) = 0 \, , \quad x \in T^d \, , \quad g (x,u) = a(x) u^3 + O(u^4 ) , $$ in any space dimension and with a multiplicative potential. The proof is based on a Nash-Moser implicit function scheme. A key step is the construction of an approximate right inverse of the linearized operators obtained at any step of the iteration. In order to avoid the difficulty posed by the violation of the second order Melnikov non-resonance conditions we develop a multiscale inductive approach \'a la Bourgain. A feature of the Monograph is to present the proofs, techniques and ideas developed in a self-contained and expanded manner.