Small-amplitude solutions for space-multidimensional hamiltonian PDEs under periodic boundary conditions

I will discuss the problem of studying the long-time behaviour of small solutions for nonlinear Hamiltonian PDEs on $T^d$, and will explain that in certain sense the behaviour for solutions of space-multidimensional equations ($d>1$) significantly differs from that for the $1d$ systems.

^* The talk is based on my recent joint work with H. Eliasson and B.Grebert, arXiv 1604.01657