Classification of Multidimensional Darboux Transformations: First Order and Continued Type

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.

References in corpus (8)

• Supersymmetry and Quantum Mechanics
• Darboux Transformations for Supersymmetric Korteweg - de Vries Equations
• Darboux Transformation for the Manin-Radul Supersymmetric KdV equation
• Intertwining relations of non-stationary Schrödinger operators
• Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation
• Crum Transformations and Wronskian Type Solutions for Supersymmetric KdV equation
• Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics
• Invertible Darboux Transformations

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• Factorization of Darboux-Laplace transformations for discrete hyperbolic operatros
• Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator
• Laplace invariants of differential operators