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# Proof of the Completeness of Darboux Wronskian Formulae for Order Two

Published:2012-07-27
Printed: Jun 2013
• E. Shemyakova,
Mathematics Department, University of Western Ontario, London, ON
 Format: LaTeX MathJax PDF

## 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.
 Keywords: completeness of Darboux Wronskian formulas, completeness of Darboux determinants, Darboux transformations, invariants for solution of PDEs
 MSC Classifications: 53Z05 - Applications to physics 35Q99 - None of the above, but in this section