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

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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About the authors ▼
J. Golenia
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland
Y. A. Prykarpatsky
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland
A. M. Samoilenko
Institute of Mathematics at the NAS, Kiev 01601, Ukraine
A. K. Prykarpatsky
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland
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Received: 2004-03-16.
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J. Golenia, Y. A. Prykarpatsky, A. M. Samoilenko, A. K. Prykarpatsky, The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2, Opuscula Math. 24, no. 1 (2004), 71-83, http://dx.doi.org/10.7494/OpMath.2004.24.1.71.
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