Abstract:
On the basis of a three-dimensional non stationary model of a convective cloud with detailed description of dynamic, thermodynamic and microphysical processes, numerical experiments were conducted to study the formation of parameters of convective clouds under unstable stratification of the atmosphere. Numerical experiments have been carried out to study the formation of convective processes in the atmosphere. The thermo hydrodynamic parameters in the zone of a thunderstorm cloud are determined, and regions with a vortex motion of air are identified. The main flows feeding the convective cloud in the mature stage are determined. Due to the means of visualization, the areas of formation and growth of precipitation particles are identified. In a three-dimensional form, the interaction of dynamic and thermodynamic processes is analyzed. The interaction of fields is manifested in the form of deformation of fields of thermodynamic parameters under the influence of dynamic processes. Trajectories of air streams around a cloud and the trajectories of drops in a cloud are determined. The results of numerical experiments confirm that dynamic processes significantly influence the formation of fields of thermodynamic parameters in the cloud, which also determine the course of microphysical processes and the nature of the growth of precipitation particles.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
The state of the physics of convective clouds and cloud seeding is discussed briefly. It is noted that at the present time there is a transition from the stage of investigation of “elementary” processes in the clouds to the stage of studying the formation of macro- and microstructural characteristics of clouds as a whole, taking into account their system properties. The main directions of the development of cloud physics at the upcoming stage of its development are discussed. The paper points out that one of these areas is the determination of the structure-forming factors for the clouds and the study of their influence on their formation and evolution. It is noted that one of such factors is the interaction of clouds with their surrounding atmosphere, and the main method of studying its role in the processes of cloud formation is mathematical modeling. A three-dimensional nonstationary model of convective clouds is presented with a detailed account of the processes of thermohydrodynamics and microphysics, which is used for research. The results of modeling the influence of the wind field structure in the atmosphere on the formation and evolution of clouds are presented. It is shown that the dynamic characteristics of the atmosphere have a significant effect on the formation of macro- and microstructural characteristics of convective clouds: the more complex the structure of the wind field in the atmosphere (i.e., the more intense the interaction of the atmosphere and the cloud), the less powerful the clouds are formed.

Abstract:
Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator

Abstract:
Two analytical methods have been developed for constructing approximate solutions to a nonlocal generalization of the 1D Fisher-Kolmogorov-Petrovskii-Piskunov equation. This equation is of special interest in studying the pattern formation in microbiological populations. In the greater part of the paper, we consider in detail a semiclassical approximation method based on the WKB-Maslov theory under the supposition of weak diffusion. The semiclassical asymptotics are sought in a class of trajectory concentrated functions. Such functions are localized in a neighborhood of a point moving in space. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval which can be small in the sense that a pattern has no time to form in this interval. In the final part of the paper, we have constructed asymptotics which are different from the semiclassical ones and can describe the evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. These asymptotics represent small perturbations on the background of an exact quasi-stationary solution. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as pattern formation.

Abstract:
The general construction of semiclassically 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 ℏ (ℏ→0), are constructed with a power accuracy of O(ℏ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of semiclassically concentrated solutions of Hartree type equations. The results obtained are exemplified by a one-dimensional Hartree type equation with a Gaussian potential.

Abstract:
For the nonlocal $T$-periodic Gross-Pitaevsky operator, formal solutions of the Floquet problem asymptotic in small parameter $\hbar$, $\hbar\to0$, up to $O(\hbar^{3/2})$ have been constructed. The quasi-energy spectral series found correspond to the closed phase trajectories of the Hamilton-Ehrenfest system which are stable in the linear approximation. The monodromy operator of this equation has been constructed to within $\hat O(\hbar^{3/2})$ in the class of trajectory-concentrated functions. The Aharonov-Anandan phases have been calculated for the quasi-energy states.

Abstract:
The Cauchy problem for the Gross--Pitaevsky equation with quadratic nonlocal nonlinearity is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Gross--Pitaevsky equations is considered.