|Date: Tuesday, January 15, 2013
Location: 3866 East Hall (2:10 PM to 3:00 PM)
Title: What is the inverse-scattering transform?
Abstract: The inverse-scattering transform was first discovered in the 1960's by Gardner, Greene, Kruskal, and Miura as a method of solving the initial-value problem for the Korteweg-de Vries equation, a well-known nonlinear partial differential equation modeling (among many other things) the propagation of surface water waves in a channel. It soon became apparent that the method applies more broadly to a wider class of problems of great interest in nonlinear wave theory. I will describe some of the history and then explain how the method can be used to solve the defocusing cubic nonlinear Schr\"odinger equation, as was first discovered by Zakharov and Shabat. As suggested by the name of the method, the key ideas come from the mathematical treatment of the direct and inverse-scattering problems for various linear equations, problems that are of independent interest in applications (see John Schotland's upcoming talk in this seminar).
Although this will be Part II, with particular emphasis on the representation of the inverse scattering problem as a Riemann-Hilbert problem of complex function theory, I will try to make the talk self-contained for those interested people who may have missed Part I last November.
Speaker: Peter Miller
Institution: Univ of Michigan