Graduate School in Nonlinear Science

Sponsored by the Danish Research Academy

The Dressing Method

by Alexandre Zenchuk
Landau Institute, Kosygina 2
Moscow 117334

Monday, June 15, 1998 15:15 h
at MAT, Building 303, room 026

Abstract:The Fourier method allows one to solve completely a system of linear PDEs with constant coefficients. But this method can in general not be used when nonlinear systems of PDEs are under consideration. One special method that has been developed to solve a large class of nonlinear PDE's is know as the Dressing Method, and the equations that can be treated by it are called completely integrable. The main idea of this method is to relate the nonlinar PDE's to a corresponding set of linear equations, which are integral equations with a kernel of concrete structure, or PDE's with variable coefficients. The problem of solving the the nonlinear equation is then reduced to solving the linear equation.

The Dressing Method has several versions (V.E.Zakharov, A.B.Shabat, S.V.Manakov). One recent version which we will discuss is the so called D-bar problem. It relates solutions of nonlinear PDE with solutions of a linear integral equation having a kernel of a certain type.