Invertible Darboux Transformations

TL;DR

This paper establishes criteria for when bivariate linear partial differential operators of any order admit invertible Darboux transformations, expanding understanding beyond the classical Laplace transformations.

Contribution

It introduces new criteria for invertible Darboux transformations for arbitrary order bivariate operators, including conditions when Wronskian formulae are applicable.

Findings

Criteria for invertibility of Darboux transformations.

Conditions under which Wronskian formulae fail or succeed.

Extension of invertible Darboux transformations beyond Laplace cases.

Abstract

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the operator kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux transformation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.