Speaker: Svetlana Roudenko, George Washington University

Title: Collapse phenomenon in dispersive equations: theories and predictions.

Abstract: Evolution processes can be described by partial differential equations, and one type of such equations is dispersive, wave-like, equations, where different families of frequencies propagate with different velocities (thus, disperse). It is often needed to understand the global in time behavior of solutions to such equations: whether a solution behaves like a linear solution as time goes to infinity, or it forms a soliton (solitary wave), or it forms a singularity in some finite time such as shock, wave breaking, blow up, collapse. The last possibility is especially intriguing, and on an example of one of the simplest dispersive equations, the Schroedinger equation, I will discuss current understanding of singularity formation in its solutions.

In particular, I will consider the focusing nonlinear Schroedinger 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. I will show that the collapse solutions can have different rates, profiles, concentrations, moreover, can blow up not only on a single point set but on various geometries.

Place: Science and Technology Building I, Room 242

Refreshments will be served before the talk at 3:00 p.m. in Room 222.

Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491