Time periodic solutions for the nonlinear wave equation with long minimal period

We prove existence and multiplicity of small amplitude periodic solutions for the wave equation with small "mass" and odd nonlinearity. Such solutions bifurcate from resonant finite dimensional invariant tori of the fourth order Birkhoff normal form of the associated Hamiltonian system. The number of geometrically distinct solutions and their minimal periods go to infinity when the "mass" goes to zero. This is the first result about long minimal period for the autonomous wave equation. © 2006 Society for Industrial and Applied Mathematics.