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

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.