A Birkhoff--Lewis-Type Theorem for Some Hamiltonian PDEs
Abstract
In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schrödinger equation with smoothing nonlinearity.
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Published In
SIAM Journal on Mathematical Analysis
Pages: 83 - 102
ISSN (online): 1095-7154
Copyright
Copyright © 2005 Society for Industrial and Applied Mathematics.
History
Published online: 1 August 2006
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