Персона: Аксенов, Александр Васильевич
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Институт лазерных и плазменных технологий
Стратегическая цель Института ЛаПлаз – стать ведущей научной школой и ядром развития инноваций по лазерным, плазменным, радиационным и ускорительным технологиям, с уникальными образовательными программами, востребованными на российском и мировом рынке образовательных услуг.
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- ПубликацияТолько метаданныеApplication of invariants of characteristics to construction of solutions without gradient catastrophe(2022) Aksenov, A. V.; Druzhkov, K. P.; Kaptsov, O. V.; Аксенов, Александр ВасильевичThe one-dimensional system of equations of isentropic gas dynamics is considered. First-order invariants of characteristics of this system are classified. Second-order invariants of characteristics are classified for polytropic processes. The infinite sequence of Darboux integrable systems is described. The approach to construction of smooth solutions without gradient catastrophe is proposed. Examples of solutions without gradient catastrophe are presented.
- ПубликацияТолько метаданныеOn the Correspondence between the Variational Principles in the Eulerian and Lagrangian Descriptions(2021) Druzhkov, K. P.; Aksenov, A. V.; Аксенов, Александр Васильевич© 2021, Pleiades Publishing, Ltd.Abstract: The relationship between the variational principles for equations of continuum mechanics in Eulerian and Lagrangian descriptions is considered. It is shown that, for a system of differential equations in Eulerian variables, the corresponding Lagrangian description is related to introducing nonlocal variables. The connection between the descriptions is obtained in terms of differential coverings. The relation between the variational principles of a system of equations and its symplectic structures is discussed. It is shown that, if a system of equations in Lagrangian variables can be derived from a variational principle, then there is no corresponding variational principle in the Eulerian variables.
- ПубликацияТолько метаданныеGroup Classification of the System of Equations of Two-Dimensional Shallow Water over Uneven Bottom(2020) Druzhkov, K. P.; Aksenov, A. V.; Аксенов, Александр Васильевич© 2020, Pleiades Publishing, Ltd.Abstract: A system of equations of two-dimensional shallow water over an uneven bottom is considered. An overdetermined system of equations for finding the corresponding symmetries is obtained. The compatibility of this overdetermined system of equations is investigated. A general form of the solution of the overdetermined system is found. The kernel of the symmetry operators is found. The cases of kernel extensions of symmetry operators are presented. The results of group classification indicate that the system of equations of two-dimensional shallow water over an uneven bottom cannot be linearized by point transformation, in contrast to the system of equations of one-dimensional shallow water in the cases of horizontal and inclined bottom profiles.
- ПубликацияТолько метаданныеConservation laws of the equation of one-dimensional shallow water over uneven bottom in Lagrange's variables(2020) Druzhkov, K. P.; Aksenov, A. V.; Аксенов, Александр Васильевич© 2019 Elsevier LtdThe systems of equations of one-dimensional shallow water over uneven bottom in Euler's and Lagrange's variables are considered. The intermediate system of equations is introduced. Hydrodynamic conservation laws of intermediate system of equations are used to find all first order conservation laws of shallow water equations in Lagrange's variables for all bottom profiles. The obtained conservation laws are compared with the hydrodynamic conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Euler's variables. Bottom profiles, providing additional conservation laws, are given. The problem of group classification of contact transformations of the shallow water equation in Lagrange's variables is solved. First order conservation laws of the shallow water equation in Lagrange's variables are obtained using Noether's theorem. In the considered cases, the correspondence between non-divergence symmetries and the original Lagrangian is shown. A similar correspondence is valid for an arbitrary ordinary differential equation of the second order. It is shown that the application of Lagrange's identity did not find all first order conservation laws of the shallow water equation in Lagrange's variables.
- ПубликацияОткрытый доступArticle methods for constructing complex solutions of nonlinear pdes using simpler solutions(2021) Polyanin, A. D.; Aksenov, A. V.; Аксенов, Александр Васильевич© 2021 by the authors. Licensee MDPI, Basel, Switzerland.This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u = u(x, t), these equations contain the same function at a past time, w = u(x, t − τ), where τ > 0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u = u(x, t), also contain the same functions with dilated or contracted arguments, w = u(px, qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.