KAM for autonomous quasi-linear perturbations of mKdV

TL;DR

This paper proves the existence of small amplitude, linearly stable, quasi-periodic solutions for strongly nonlinear perturbations of the mKdV equation using a combination of Birkhoff normal form, Nash-Moser iteration, and KAM reducibility techniques.

Contribution

It introduces a novel approach combining weak Birkhoff normal form and linear KAM reducibility to handle quasi-linear perturbations of mKdV.

Findings

Existence of Cantor families of solutions proven.

Solutions are small amplitude and linearly stable.

Method extends KAM theory to quasi-linear PDEs.

Abstract

We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear (also called strongly nonlinear) autonomous Hamiltonian differentiable perturbations of the mKdV equation. The proof is based on a weak version of the Birkhoff normal form algorithm and a nonlinear Nash-Moser iteration. The analysis of the linearized operators at each step of the iteration is achieved by pseudo-differential operator techniques and a linear KAM reducibility scheme.