by Alexandre Zenchuk
Landau Institute, Kosygina 2
Moscow 117334
Russia
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.
