Invertible Darboux Transformations

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.

References in corpus (3)

• Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation
• Proof of the Completeness of Darboux Wronskian Formulae for Order Two
• Discrete SL2 Connections and Self-Adjoint Difference Operators on the Triangulated 2-manifold

Cited by in corpus (6)

• Classification of Multidimensional Darboux Transformations: First Order and Continued Type
• Darboux transformations for differential operators on the superline
• Classification of Darboux transformations for operators of the form ${\partial }_x{\partial }_y+a{\partial }_x+b{\partial }_y+c$
• Laplace maps and constraints for a class of third order partial differential operators
• Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator
• Laplace invariants of differential operators