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The study of Delsarte-Lions type binary transformations, their differential-geometric and operator structure with applications. Pt. 2

Prykarpatsky Y. A., Samoilenko A. M.

Opuscula Mathematica

2007

Vol. 27, no. 1

113-130

The Gelfand-Levitan integral equations for Delsarte-Lions type transformations in multidimension are studied. The corresponding spectral and analytical properties of Delsarte-Lions transformed operators are analyzed by means of the differential-geometric and topological tools. An approach for constructing Delsarte-Lions type transmutation operators for multidimensional differential expressions is devised.

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.

The multidimensional Delsarte transmutation operators, their differential-geometric structure and applications. Pt. 1

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

Opuscula Mathematica

2003

Vol. 23

71-80

A differential-geometric structure of Delsarte transmutation operators in multidimension is desribed, application to the inverse spectral transform problem is discussed.