Moutard transforms for the conductivity equation

Abstract

We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrödinger equation at zero energy are also shown.

The Moutard transformation for the Davey-Stewartson II equation and its geometrical meaning

“…Let us consider the simplest examples of solutions of type (19). They correspond to cases when ϕ 1 is a polynomial in z of degree n ≤ 3.…”

“…For similar purposes, Moutard-type transformations of two-dimensional Dirac operators, except for cite T152 and the recent article, have been successfully applied in [18,19,20].…”

Darboux Moutard Transformations and Poincaré—Steklov Operators

“…In [19,20], it was established a relation of the generalized Moutard transformation for two-dimensional Dirac operators [21] to the conformal geometry of surfaces in three-and four-dimensional spaces, and with this were constructed blowing-up solutions of the modified Novikov-Veselov equation with regular initial data [22,23]. A generalization of the Moutard transformation to the case of generalized analytic functions, in particular, gave an approach to constructing the theory of generalized analytic functions with contour poles [24,25,26,27] and also allowed to construct a Moutard-type transformation for the conductivity equation [28].…”