Families of Periodic Solutions for Some Hamiltonian PDEs

Abstract. We consider the nonlinear wave equation utt − uxx = ±u and the beam equation utt+uxxxx = ±u on an interval. Numerical observations indicate that time-periodic solutions for these equations are organized into structures that resemble branches and seem to undergo bifurcations. Besides describing our observations, we prove the existence of time-periodic solutions for various periods (a set of positive measure in the case of the beam equation) along the main nontrivial “branch”.