Opuscula Math. 26, no. 1 (2006), 137-150

Opuscula Mathematica

# The Delsarte-Darboux type binary transformations, their differential-geometric and operator structure with applications. Part 1

Yarema A. Prykarpatsky
Anatoliy M. Samoilenko

Abstract. The structure properties of multidimensional Delsarte-Darboux transmutation operators in parametric functional spaces are studied by means of differential-geometric and topological tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutative differential operator expressions related via a Delsarte-Darboux transformation and having a lot of applications in soliton theory.

Keywords: Delsarte transmutation operators, parametric functional spaces, Darboux transformations, inverse spectral transform problem, soliton equations, Zakharov-Shabat equations.

Mathematics Subject Classification: 34A30, 34B05, 34B15.

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