Bambusi, Dario; Paleari, Simone Families of periodic solutions of resonant PDEs. (English) Zbl 0994.37040 J. Nonlinear Sci. 11, No. 1, 69-87 (2001). 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. Cited in 31 Documents 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 Keywords:small amplitude periodic solutions; completely resonant equilibrium point; semilinear reversible partial differential equation; averaging methods; Lyapunov-Schmidt decomposition; nonlinear string equation PDFBibTeX XMLCite \textit{D. Bambusi} and \textit{S. Paleari}, J. Nonlinear Sci. 11, No. 1, 69--87 (2001; Zbl 0994.37040) Full Text: DOI