Existence and stability of quasi-periodic solutions for derivative wave equations

TL;DR

This paper proves the existence and stability of quasi-periodic solutions in derivative wave equations using a new KAM theorem, demonstrating small amplitude solutions with zero Lyapunov exponents and reducible linearized equations.

Contribution

It introduces a novel KAM theorem for infinite dimensional reversible systems, establishing the existence of stable quasi-periodic solutions for derivative wave equations.

Findings

Existence of Cantor families of small amplitude solutions.

Solutions have zero Lyapunov exponents, indicating stability.

Linearized equations are reducible to constant coefficients.

Abstract

In this note we present a new KAM result which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems.