 02141 Bambusi, D.
 BIRKHOFF NORMAL FORM FOR SOME NONLINEAR PDEs
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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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