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Families of periodic solutions of resonant PDEs. (English) Zbl 0994.37040

Summary: We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation \(u_{tt}-u_{xx} \pm u^3=0\) (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the \(n\)th family converge to \(2\pi/n\) when the amplitude tends to zero.

MSC:

37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35B10 Periodic solutions to PDEs
35B20 Perturbations in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
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