Large KAM tori for quasi-linear perturbations of KdV

TL;DR

This paper proves the persistence of large, space-periodic multi-solitons in KdV under small, quasi-linear Hamiltonian perturbations, demonstrating the applicability of KAM techniques to strongly nonlinear PDEs.

Contribution

It extends KAM theory to show the existence of large quasi-periodic solutions in nonlinear PDEs like KdV, addressing a longstanding open question.

Findings

Persistence of multi-solitons under perturbations

Applicability of KAM techniques to nonlinear PDEs

Existence of large quasi-periodic solutions

Abstract

In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.