Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 010, 20 pages      arXiv:1605.04362      https://doi.org/10.3842/SIGMA.2017.010

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

David Hobby and Ekaterina Shemyakova
1 Hawk dr., Department of Mathematics, State University of New York at New Paltz, USA

Received October 10, 2016, in final form February 14, 2017; Published online February 24, 2017

Abstract
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, which was previously introduced. There is also a modification of this type of Darboux transformations, continued Wronskian type, which generalize Wronskian type Darboux transformations.

Key words: Darboux transformations; Laplace transformations; linear partial differential operators; continued Darboux transformations.

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