Quasi-periodic solutions for quasi-linear generalized KdV equations

TL;DR

This paper establishes the existence of small amplitude, linearly stable, quasi-periodic solutions for a broad class of quasi-linear generalized KdV equations using a Nash-Moser iterative scheme and advanced normal form techniques.

Contribution

It introduces a novel approach combining Nash-Moser, Birkhoff normal form, and KAM methods to handle quasi-linear nonlinearities in generalized KdV equations.

Findings

Proves existence of Cantor families of solutions.

Demonstrates linear stability of these solutions.

Develops a new bifurcation analysis for quasi-linear PDEs.

Abstract

We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash-Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.