Classification of Darboux transformations for operators of the form ∂x∂y+a∂x+b∂y+c

Abstract

Darboux transformations are nongroup-type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_{1}M$ or the intertwining relation $ML=L_{1}N$ in the cases when the former is too restrictive.

Here we show that Darboux transformations for operators of the form $L={\partial }_x{\partial }_y+a{\partial }_x+b{\partial }_y+c$ (sometimes referred to as 2D Schrödinger operators or Laplace operators) are always compositions of atomic Darboux transformations of two different well-studied types of Darboux transformations, provided that the chain of Laplace transformations for the original operator is long enough.