Dynamics of collapses in the focusing Nonlinear Schrodinger Equation
I will consider the focusing nonlinear Schrodinger equation in one, two and three space dimensions with different powers of nonlinearities (including cubic and quintic powers) and their global solutions with finite energy initial data. My discussion will focus on collapsing (blow up) solutions and their dynamics. In particular, I will show different regimes and rates as well as the geometric sets on which collapses occur. For example, solutions can blow up not only on a single point set but on circles, spheres, cylinders, not necessarily fixed in time, while remaining regular away from the blow-up core.