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

In this paper, one-dimensional (1D) nonlinear wave equation u t t − u x x + m u + u 2 p + 1 = 0 , p ∈ N , on the finite x-interval [ 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.