Invariant tori for 1D quintic nonlinear wave equation

Abstract

In this paper, one-dimensional (1D) nonlinear wave equationutt−uxx+mu+u5=0 on the finite x-interval [0,π] with Dirichlet boundary conditions is considered. It is proved that, for any integer b≥2, there are many b-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. This is an extension of the previous work [6] by the same authors, where b=2. The proof is based on infinite dimensional KAM theory and partial Birkhoff normal form.