1. CJM 2012 (vol 65 pp. 655)
|Proof of the Completeness of Darboux Wronskian Formulae for Order Two|
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
2. CJM 2001 (vol 53 pp. 278)
|Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting |
In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\KP$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau$-functions is given.
Keywords:KP hierarchy, Darboux transformation, Grassmann manifold
Categories:22E65, 22E70, 35Q53, 35Q58, 58B25