On singularity formation in Hamiltonian evolution equations

Hamiltonian evolution equations arise in the description of nonlinear phenomenons in various instances from nonlinear optics to astrophysics or fluid mechanics, but the description of most even simplified models still remains a mathematical challenge. Substantial progress have been made since the 1980's for the qualitative description of solutions through the importation and mixing of various ideas from dynamical systems, functional analysis, harmonic analysis and the calculus of variations. I will report in this survey on recent progress on the study of one specific scenario: singularity formation, that is the ability for non linear waves to concentrate their energy while propagating in some nonlinear medium. A new methodology has emerged in the last two decades on canonical models like the non linear Schrodinger or wave equations both for the construction and the classification of singular regimes, with applications also to parabolic models. A special class of solutions plays a distinguished role in the structure of the corresponding blow up bubbles: the solitary wave.