BIRKHOFF NORMAL FORM FOR PDEs WITH TAME MODULUS

Birkhoff Normal Form and Hamiltonian PDEs

These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centere...

Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization

We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite dimensional Birkhoff normal form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over...

BIRKHOFF NORMAL FORM FOR SOME NONLINEAR PDEs

ar X iv : m at h / 06 04 13 2 v 2 [ m at h . A P ] 8 N ov 2 00 6 BIRKHOFF NORMAL FORM AND HAMILTONIAN

— These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some normal form technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centere...

A Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs

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...