Pierre Raphael

Abstract:   Singularity formation in nonlinear evolution problems has attracted a considerable attention for the past thirty years. Some recent progress have been made on the question of existence of such singular dynamics in Hamiltonian wave propagation dynamics. The important feature of these works is to provide a qualitative description of the singularity formation and the bubble of energy concentration through a robust approach which goes far beyond the usual obstructive argument on global existence. I will illustrate these progress on some examples, in particular nonlinear Schrodinger equation dynamics and geometrical equations like wave or Schrodinger maps, which reveal a critical structure. I will show how these questions are deeply related to the study of the flow near exceptional nonlinear waves, the ground solitary wave, and will present a new type of theorems on the classification of the flow near the solitary wave in the presence of singular dynamics.