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

Prykarpatsky Y.A., Samoilenko A.M.

Opuscula Mathematica

2006

Vol. 26, no. 1

137-150

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.

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

Golenia J., Prykarpatsky Y.A., Samoilenko A.M., Prykarpatsky A.K.

Opuscula Mathematica

2004

Vol. 24, no. 1

71-83

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.