Data di Pubblicazione:
2001
Citazione:
Families of periodic solution of resonant PDEs / D. Bambusi, S. Paleari. - In: JOURNAL OF NONLINEAR SCIENCE. - ISSN 0938-8974. - 11:1(2001 Dec), pp. 69-87. [10.1007/s003320010010]
Abstract:
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 utt - uxx ± u3 = 0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the nth family converge to 2π/n when the amplitude tends to zero.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Averaging theory; Lypunov-Schmidt decomposition; Periodic solutions; Resonant PDEs
Elenco autori:
D. Bambusi, S. Paleari
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