Families of Periodic Solutions for Some Hamiltonian PDEs

We consider the nonlinear wave equation $u_{tt}-u_{xx}=\pm u^3$ and the beam equation $u_{tt}+u_{xxxx}=\pm u^3$ 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. In addition to 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.” Our proofs are computer-assisted.

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