KAM for Reversible Derivative Wave Equations

Abstract

This research paper investigates the existence and stability of small-amplitude, analytic, and linearly stable quasi-periodic solutions for a class of reversible derivative wave equations. The study employs KAM theory and focuses on the conditions necessary for time and space reversibility in nonlinear partial differential equations.

Introduction

The paper addresses a key question in KAM theory concerning equations with derivatives in their nonlinearity. It builds upon previous work on Hamiltonian perturbations of KdV and extends the analysis to derivative nonlinear wave equations (DNLW). The authors explore the conditions required for nonlinearities, such as time and space reversibility, to ensure the existence of quasi-periodic solutions, contrasting them with cases where such solutions do not exist.

Key concepts discussed include the role of reversibility in replacing Hamiltonian structure for quasi-periodic solutions, the impact of nonlinearities like yx and y³ terms, and the development of analytical tools to handle x-dependent nonlinearities. The research presents a theorem establishing the existence of analytic quasi-periodic solutions under specific assumptions on the equation's parameters and nonlinearity.