Birkhoff Normal Form for Some Nonlinear PDEs |
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Authors: | Dario Bambusi |
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Institution: | (1) Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133 Milano, Italy, IT |
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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 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.
Received: 15 May 2002 / Accepted: 13 September 2002 Published online: 24 January 2003 |
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