Birkhoff Normal Form for Some Nonlinear PDEs.

We consider the problem of extending to PDEs Birkhoff normal form the o- rem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation u tt - u xx + g(x, u) = 0, (0.1) with Dirichlet boundary conditions on [0, π]; g is an analytic skewsymmetric function which vanishes for u = 0 and is periodic with period 2π 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 semilinear equations in one space dimension. [ABSTRACT FROM AUTHOR]

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