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Proof of the Completeness of Darboux Wronskian Formulae for Order Two

Published online by Cambridge University Press:  20 November 2018

E. Shemyakova
E. Shemyakova*
Affiliation:
Mathematics Department, University of Western Ontario, LondonON
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Abstract

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Darboux Wronskian formulas allow us 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 to discover what other exceptions exist. 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

53Z0535Q99completeness of Darboux Wronskian formulascompleteness of Darboux determinantsDarboux transformationsinvariants for solution of PDEs
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 3 , 01 June 2013 , pp. 655 - 674
Copyright
Copyright © Canadian Mathematical Society 2013

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