Quasi-periodic solutions for 1D wave equation with the nonlinearity u2p+1u2p+1

In this paper, one-dimensional (1D) nonlinear wave equationutt−uxx+mu+u2p+1=0,p∈N, on the finite x  -interval [0,π][0,π] with Dirichlet boundary conditions is considered. It is proved that there are many 2-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.