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

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

Opuscula Mathematica

2013

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.

The Cartan-Monge geometric approach to the generalized characteristics method and its application to the heat equation Ut - Uxx = 0

Golenia J., Prykarpatsky Y. A., Wachnicki E.

Opuscula Mathematica

2009

Vol. 29, no. 1

27-39

The generalized Cartan-Monge type approach to the characteristics method is discussed from the geometric point of view. Its application to the classical one-dimensional linear heat equation Ut - uxx = 0 is presented. It is shown that the corresponding exact solution of the Cauchy problem can be represented in a classical functional-analytic Gauss type form.

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.