Penn Arts & Sciences Logo

Analysis Seminar

Wednesday, April 25, 2001 - 3:00pm

Peter Miller

University of Michigan

Location

University of Pennsylvania

4C2 DRL

We will present joint work with Spyros Kamvissis, Ken McLaughlin, and Peter Miller, on semiclassical asymptotics for solutions of the focusing nonlinear Schrödinger equation. Here is a brief outline of the presentations we hope to give. 1. Introduction and setup     - soliton ensembles with no reflection coefficient     - WKB spectral data     - formulation of the inverse problem as a Riemann--Hilbert problem.     -outline of asymptotic calculation       A. seeking a g function: conditions for asymptotically viable Riemann--Hilbert problem.       B. Riemann--Hilbert problem for the error.       C. New variational problem     - seeking a g function: genus 0 calculations relate to elliptic problem.     - asymptotic results 2. Local problem at z=0,variational problem, and construction at t=0. -- analysis at z=0.      A. asymptotics for jump matrix at z=0.      B. local model problem      C. existence of solution to local model problem. -- Details about variational problem -- construction at t=0 3. Existence of contour and analysis of g function     - existence of the contour for small time.     - computation of the contours     - conditions for first failure of genus 0 problem     - onset of higher genus behavior       A. equivalence with degenerate genus 2.       B. perturbation theory for gap opening.