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 utt-uxx+g(x,u)=0, 0.1 with Dirichlet boundary conditions on [0,?]; g is an analytic skewsymmetric function which vanishes for u=0 and is periodic with period 2? 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.
Birkhoff normal form for some nonlinear PDEs / Dario Bambusi. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 234:2(2003), pp. 253-285..
|Titolo:||Birkhoff normal form for some nonlinear PDEs|
|Settore Scientifico Disciplinare:||Settore MAT/07 - Fisica Matematica|
|Data di pubblicazione:||2003|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1007/s00220-002-0774-4|
|Appare nelle tipologie:||01 - Articolo su periodico|