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A Birkhoff-Lewis-type theorem for some Hamiltonian PDEs. (English) Zbl 1105.37045

The authors prove the existence of infinitely many periodic orbits of infinite-dimensional Hamiltonian systems corresponding to semilinear, nonintegrable Hamiltonian partial differential equations. The periodic orbits accumulate at the origin, have large periods, and bifurcate from resonant finite-dimensional invariant tori of the fourth-order normal form of the system. The paper contains a general bifurcation theorem which is applied to a nonlinear beam equation and a nonlinear Schrödinger equation. In addition to standard nonresonance and nondegeneracy conditions, a regularizing property of the nonlinearity is required. The proof of the general result consists of first transforming the Hamiltonian via an analytic, symplectic change of variables into the seminormal form \(H_0+G+K\) where \(H_0\) is the quadratic part, \(G\) consists of terms of fourth order, and \(K\) has a zero of order \(6\) at the origin. This Hamiltonian is treated as a perturbation of \(H_0+G\) and a variational Lyapunov-Schmidt reduction is applied.

MSC:

37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
35B10 Periodic solutions to PDEs
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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