00-306 Dario Bambusi, Simone Paleari

Families of periodic solutions of resonant PDE's (232K, Postscript) Jul 28, 00

Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

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

Files: 00-306.src( 00-306.keywords , risonanze.ps )