Opuscula Math. 24, no. 1 (2004), 71-83

Opuscula Mathematica

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

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

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.

Mathematics Subject Classification: 34A30, 34B05, 34B15.

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