Bambusi, D.
BIRKHOFF NORMAL FORM FOR SOME NONLINEAR PDEs
(438K, Postscript)

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.