Graduate School in Nonlinear Science

Sponsored by the Danish Research Academy

The D-bar problem and Nonlinear Integrable PDEs

by Alexandre Zenchuk
Landau Institute, Kosygina 2
Moscow 117334

Tuesday, June 16, 1998 15:00 h
at MIDIT, Building 305 room 027

MIDIT-seminar No. 406

Abstract:The D-bar problem is a linear integral-differential equation whose solutions are used to construct solutions to a wide class of completely integrable nonlinear Partial Differential Eqations by means of a special procedure -- the so-called dressing method.
The nonlinear Schrödinger Equation and the Kadomtsev-Petviashvili Equation are important examples of this class of equations. Several modifications of this method have been developed recently, and some of them will be discussed here. One is the Dual D-bar problem (S.V.Manakov, A.I. Zenchuk, TMF (Russia), which gives a simple way of performing symmetry reductions; the second one is the dressing procedure with arbitrary functions of independent variables. This leads to new integrable equations related to those already known (A.Degasperis, S.V. Manakov, A.I.Zenchuk, to appear). The modification permits the construction of a class of small deformations of integrable equations, which does not disturb the integrability (A.I.Zenchuk, JETP Letters).
Another generalization of the D-bar problem is related to the generalized normalization function in the D-bar equation, permitting construction of Miura transformations for integrable equations.