Classification of Multidimensional Darboux Transformations: First Order and Continued Type

Abstract

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.

Factorization of Darboux—Laplace Transformations for Discrete Hyperbolic Operators

“…Therefore it is sufficient to consider only operators of this kind. In this case if c n = 0, then equations (20) yieldγ n = − ln(ϕ n+1,x ) x (otherwise one of the Laplace invariants vanishes, and this contradicts our assumption). Introduce operatorD 1 := ∂ x − ln(ϕ n+1,x ) x and use relation (1):…”

“…It is not difficult to prove that any solution of this equation has the form γ n = −(ln ϕ n ) x , where ϕ ∈ Ker L. Conversly, if a n = 0, then for arbitrary function ϕ ∈ Ker L define u n := b n−1 + (ln ϕ n ) x and coefficients of operatorsL, D andD using formulas (20). It is easy to check that equations (7)- (14) are satisfied in this case.…”

On soliton solutions of multi-component semi-discrete short pulse equation

“…DT is one of solution generating technique in soliton theory that allow us to express the solutions for a given integrable equation in simple explicit form [28][29][30][31][32][33]. In this section, we construct the DT of the multicomponent .…”

Laplace maps and constraints for a class of third-order partial differential operators

“…Further classes of generalization include Darboux transformations of type I [17,18,19] and of continued type [9]. A looser and more general notion of intertwining Laplace transformation than that presented in this paper can be found in [7].…”