02-141 Bambusi, D.

BIRKHOFF NORMAL FORM FOR SOME NONLINEAR PDEs (438K, Postscript) Mar 21, 02

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Abstract. We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation $$ u_{tt}-u_{xx}+\pert(x,u)=0\ ,\autoeqno{1} $$ with Dirichlet boundary conditions on $[0,\pi]$; $\pert$ is an analytic skewsymmetric function which vanishes for $u=0$ and is periodic with period $2\pi$ in the $x$ variable. We prove, under a nonresonance condition which is fulfilled for most $g$'s, that for any integer $M$ there exists a canonical transformation that puts the Hamiltonian in Birkhoff normal form up to a reminder of order $M$. The canonical transformation is well defined in a neighbourhood of the origin of a Sobolev type phase space of sufficiently high order. Some dynamical consequences are obtained. The technique of proof is applicable to quite general equations in one space dimension.

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