Proof of the Completeness of Darboux Wronskian Formulae for Order Two

TL;DR

This paper proves that all Darboux transformations of total order two can be represented by Darboux Wronskian formulas, completing the understanding of their completeness and providing explicit invariant descriptions.

Contribution

It establishes the completeness of Darboux Wronskian formulas for order two transformations, extending previous results and offering explicit invariant characterizations.

Findings

No exceptions for Darboux transformations of total order two.

Complete characterization of Darboux transformations of order two.

Explicit invariant description of all such transformations.

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.