KAM tori for 1D nonlinear wave equations with periodic boundary conditions

In this paper, one-dimensional (1D) nonlinear wave equations u(tt) - u(xx) + V(x)u = f(u), with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u = 0. It is proved that for "most" potent ials V(x), the above equation admits small-amplitude periodic or quasi-peri odic solutions corresponding to finite dimensional invariant tori for an as sociated infinite dimensional dynamical system. The proof is based on an in finite dimensional KAM theorem which allows for multiple normal frequencies .