Quasi-periodic travelling waves for a class of damped beams on rectangular tori

TL;DR

This paper proves the existence of small amplitude quasi-periodic travelling wave solutions with two frequencies for damped beam equations on rectangular tori, extending previous rotating wave solutions to more general geometries.

Contribution

It introduces a novel application of Lyapunov--Schmidt reduction and the implicit function theorem to establish quasi-periodic solutions on both isotropic and anisotropic tori.

Findings

Existence of small amplitude quasi-periodic travelling waves with two frequencies.

Extension of solutions from rotating waves to quasi-periodic waves on rectangular tori.

Applicability to both isotropic and anisotropic geometries.

Abstract

This article concerns a class of beam equations with damping on rectangular tori. When the generators satisfy certain relationship, by excluding some value of two model parameters, we prove that such models admit small amplitude quasi-periodic travelling wave solutions with two frequencies, which are continuations of two rotating wave solutions with one frequency. This result holds not only for an isotropic torus, but also for an anisotropic torus. The proof is mainly based on a Lyapunov--Schmidt reduction together with the implicit function theorem.