Speaker : 1.Prof. Peter Miller (University of Michigan) 2.Prof. Yuji Kodama (The Ohio State University)
Title : 1.A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrodinger Equation 2.On the rational solutions and the solitons of the KP hierarchy
Time : 2018-07-02 (Mon) 14:00 - 16:30
Place : Seminar Room 617, Institute of Mathematics (NTU Campus)
Abstract: 1.We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrodinger equation (also other equations by natural generalization). The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related "rogue-wave" solutions in an inverse-scattering context for the first time. This allows one to directly study the stability of such solutions. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations, rather than the generalized Darboux transformations in the literature. This latter fact enables the asymptotic analysis of high-order rogue waves by steepest descent techniques, a program that leads to the identification of a new special function solution of the nonlinear Schrodinger equation that we call the rogue wave of infinite order. This project is joint work with Deniz Bilman and Liming Ling.

2.It is well known that the Schur polynomials satisfy the Hirota bilinear equations of the KP hierarchy, and that each Schur polynomial can be parametrized by a unique Young diagram. We also know that the KP solitons (exponential solutions) can be parametrized by certain decomposition of the Grassmannians. In the talk, I will explain the connection between the rational solutions and the KP solitons in terms of the Young diagrams. More explicitly, I will show how one gets a rational solution from a KP soliton. I will also discuss a connection between quasi-periodic solutions (theta or sigma functions) and the KP solitons. The rational solutions then give theta divisors of certain algebraic curves.