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Some analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations. Part. 1

100%

Pytel-Kudela M. , Prykarpatsky A. K.

tom Vol. 26, no. 1

131-136

The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bilinear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.

A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation

80%

Prykarpatsky Y. A. , Blackmore D. , Golenia J. , Prykarpatsky A. K.

Opuscula Mathematica

2013

tom Vol. 33, no. 1

139-149

An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type hydrodynamical hierarchy is devised. A functional representation generating an infinite hierarchy of dispersive Lax type integrable flows is obtained.

Isospectral integrability analysis of dynamical systems on discrete manifolds

80%

Blackmore D. , Prykarpatsky A. K. , Prykarpatsky Y. A.

Opuscula Mathematica

2012

tom Vol. 32, no. 1

41-66

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Riemann-Burgers dynamical systems.

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

80%

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

Opuscula Mathematica

2004

tom 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

80%

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

Opuscula Mathematica

2003

tom 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.