Dario Bambusi, Simone Paleari
Families of periodic solutions of resonant PDE's
(232K, Postscript)

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.