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

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