Renormalization Group and the Melnikov Problem for PDE's

TL;DR

This paper introduces a novel renormalization group approach to prove the persistence of quasi-periodic tori in infinite-dimensional PDE systems, extending KAM theory to cases with multiple normal frequencies.

Contribution

It provides a new proof technique for invariant tori in PDEs using renormalization group methods, including cases with multiple normal frequencies.

Findings

Proves persistence of quasi-periodic elliptic tori in infinite-dimensional systems.

Establishes existence of small-amplitude, quasi-periodic solutions for nonlinear wave equations.

Extends KAM theory to systems with multiple normal frequencies.

Abstract

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions.