Proof of the Completeness of Darboux Wronskian Formulae for Order Two

Abstract

Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux Transformations of total order two.

References in corpus (3)

• Two-step Darboux transformations and exceptional Laguerre polynomials
• Factorization in categories of systems of linear partial differential equations
• Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations

Cited by in corpus (7)

• Differential operators on the superline, Berezinians, and Darboux transformations
• Invertible Darboux Transformations
• Darboux transformations for differential operators on the superline
• Classification of Darboux transformations for operators of the form ${\partial }_x{\partial }_y+a{\partial }_x+b{\partial }_y+c$
• Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator
• Factorization of Darboux-Laplace transformations for discrete hyperbolic operatros
• Laplace invariants of differential operators