arXiv subjects

A. V. Shapovalov

Publications and source records attributed to A. V. Shapovalov.

22 records · Page 2Linked to original sources

arXiv2000

The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation

The general construction of quasi-classically concentrated solutions to the Hartree-type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter \h (\h\to0), are constructed with a power accuracy of O(\h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator Physics

math-ph↗

arXiv1998

The geometry of the Fisher selection dynamics

We study the Fisher model describing natural selection in a population with a diploid structure of a genome by differential- geometric methods. For the selection dynamics we introduce an affine connection which is shown to be the projectively Euclidean and the equiaffine one. The selection dynamics is reformulated similar to the motion of an effective particle moving along the geodesic lines in an 'effective external field' of a tensor type. An exact solution is found to the Fisher equations for the special case of fitness matrix associated to the effect of chromosomal imprinting of mammals. Biological sense of the differential- geometric constructions is discussed. The affine curvature is considered as a direct consequence of an allele coupling in the system. This curving of the selection dynamics geometry is related to an inhomogenity of the time flow in the course of the selection.

physics.bio-ph↗

arXiv1997

Separation of variables via integral transformations

For a system of linear partial differential equations (LPDEs) we introduce an operator equation for auxiliary operators. These operators are used to construct a kernel of an integral transformation leading the LPDE to the separation of variables (SoV). The auxiliary operators are found for various types of the SoV including conventional SoV in the scalar second order LPDE and the SoV by the functional Bethe anzatz. The operators are shown to relate to separable variables. This approach is similar to the position-momentum transformation to action angle coordinates in the classical mechanics. General statements are illustrated by some examples.

solv-int↗

arXiv1997

Hamiltonian Dynamics of Darwin Systems

We present a Hamiltonian approach for the wellknown Eigen model of the Darwin selection dynamics. Hamiltonization is carried out by means of the embedding of the population variable space, describing behavior of the system, into the space of doubled dimension by introducing additional dynamic variables. Besides the study of the formalism, we try to interpret its basic elements (phase space, Hamiltonian, geometry of solutions) in terms of the theoretical biology. A geometric treatment is given for the considered system dynamics in terms of the geodesic flows in the Euclidean space where the population variables serve as curvilinear coordinates. The evolution of the distribution function is found for arbitrary distributed initial values of the population variables.

physics.bio-ph↗