Quasi-periodic solutions for generalized Kaup system with quasi-periodic forcing

Abstract

In this paper, a one-dimensional quasi-periodically forced Kaup system ∂tu=∂xv,∂tv=∂xu−13∂xxxu−εϕ(t)(u+u3)x with periodic boundary conditions is considered, where ε is a small positive parameter and ϕ(t) is a real analytic quasi-periodic function in t with frequency vector ω*=(ω1*,ω2*,…,ωm*). It is proved that, under a suitable hypothesis on ϕ(t), there are many quasi-periodic solutions that correspond to finite dimensional invariant tori. The proof relies on a Birkhoff normal form and an infinite dimensional Kolmogorov-Arnold-Moser theorem.