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