 00306 Dario Bambusi, Simone Paleari
 Families of periodic solutions of resonant PDE's
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Jul 28, 00

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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
LyapunovSchmidt 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.
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