skip to content
 

Solving Nonlinear Dispersive Equations in Dimension Two by the Method of Inverse Scattering

Presented by: 
P Perry University of Kentucky
Date: 
Tuesday 8th November 2011 - 14:00 to 15:00
Venue: 
INI Seminar Room 2
Abstract: 
The Davey-Stewartson II equation and the Novikov-Veselov equations are nonlinear dispersive equations in two dimensions, respectively describing the motion of surface waves in shallow water and geometrical optics in nonlinear media. Both are integrable by the $\overline{\partial}$-method of inverse scattering, and may be considered respective analogues of the cubic nonlinear Schrodinger equation and the KdV equation in one dimension. We will prove global well-posedness for the defocussing DS II equation in the space $H^{1,1}(R^2)$ consisting of $L^2$ functions with $\nabla u$ and $(1+|\, \cdot \,|) u(\, \cdot \, )$ square-integrable. Using the same scattering and inverse scattering maps, we will also show that the inverse scattering method yields global, smooth solutions of the Novikov-Veselov equation for initial data of conductivity type, solving an open problem posed recently by Lassas, Mueller, Siltanen, and Stahel.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons