arXiv subjects
Publications and source records attributed to E. I. Ganzha.
We discuss the notion of reduction of a special type of explicit solutions which generalize the solutions appearing in the classical Laplace cascade method of integration of hyperbolic equations of the second order in the plane. We give algorithms of reduction and prove that different natural precise definitions of reduction are equivalent. Keywords: cascade integration method, integrable systems, Euler integrals.
For hyperbolic first-order systems of linear partial differential equations (master equations), appearing in description of kinetic processes in physics, biology and chemistry we propose a new procedure to obtain their complete closed-form non-stationary solutions. The methods used include the classical Laplace cascade method as well as its recent generalizations for systems with more than 2 equations and more than 2 independent variables. As an example we present the complete non-stationary solution (probability distribution) for Verhulst model driven by Markovian coloured dichotomous noise.
In this paper we solve positively the problem of (local) density of the "potentials" M(x,y) of the Moutard equation, u_{xy} = M(x,y) u, u=u(x,y), (used in many papers for construction of exact solutions of (2+1)-dimensional integrable systems) obtainable from a given initial potential with consecutive Moutard transformations.
In this paper we solve positively the problem of (local) density of solutions of the (2+1)-dimentional integrable system describing triply orthogonal curvilinear coordinates in R^3 (a (2+1)-dimensional generalization of the 3-wave system) obtainable from a given initial solution with consecutive Bäcklund transformations (called Ribaucour transformations in classical differential geometry) in the space of all solutions of the system in question.