The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2

Abstract

The structure properties of multidimensional Delsarte transmutation operators in parametric functional spaces are studied by means of differential-geometric 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 commutive differential operator expressions related via a Darboux-Backlund transformation having a lot of applications in soliton theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators.

Keywords

• Delsarte transmutation operators
• parametric functional spaces
• Darboux transformations
• inverse spectral transform problem
• soliton equations
• Zakharov-Shabat equations
• polynomial operator pencils