A Birkhoff Normal Form Theorem for Partial Differential Equations on torus

Abstract: We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations on torus. The normal form is complete up to arbitrary finite order. The proof is based on a valid non-resonant condition and a suitable norm of Hamiltonian function. Then as two examples, we apply this theorem to nonlinear wave equation in one dimension and nonlinear Schr\"{o}dinger equation in high dimension. Consequently, the polynomially long time stability is proved in Sobolev spaces $H^s$ with the index $s$ being much smaller than before. Further, by taking the iterative steps depending on the size of initial datum, we prove sub-exponentially long time stability for these two equations.