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

The structure properties of multidimensional Delsarte transmutation operators in parametirc 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 solition theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators.Comment: 10 page

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

An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type Gurevicz-Zybin hydrodynamical hierarchy is devised. A functional representation generating an infinite hirerachy of dispersive Lax type integrable flows is obtaned.Comment: 6 page