Exactly Solvable Models for the First Vlasov Equation
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actly Solvable Models for the First Vlasov Equation E. E. Perepelkina, b, c, d, A. D. Kovalenkoa, B. I. Sadovnikovb, N. G. Inozemtsevac, d, A. A. Tarelkinb, d, R. V. Polyakovaa, *, M. B. Sadovnikovab, N. M. Andronovad, and E. Scherkhanova a
Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia bFaculty of Physics, Moscow State University, Moscow, 119991 Russia c Dubna State University, Dubna, Moscow oblast, 141980 Russia d Moscow Technical University of Communications and Informatics, Moscow, 111024 Russia *e-mail: [email protected] Received March 16, 2020; revised April 14, 2020; accepted April 14, 2020
Abstract—Construction of the method for finding exact solutions of the first equation from the chain of Vlasov equations, formally similar to the continuity equation, is considered. The equation under investigation is written for the scalar function f and the vector field v . Depending on the formulation of the problem, the function f can correspond to the density of probabilities, charge, mass, or the magnetic permeability of a magnetic material. The vector field v can correspond to the probability flow, velocity field of a continuous medium, or magnetic field strength. Mathematically, the same equation is applicable for describing statistical, quantum, and classical systems. The exact solution obtained for one physical system can be mapped onto the exact solution for another system. Availability of exact solutions of model nonlinear systems is important for designing complex physical facilities, such as the SPD detector for the NICA project. These solutions are used as tests for writing a program code and can be encapsulated into finite-difference schemes to numerically solve boundary-value problems for nonlinear differential equations. DOI: 10.1134/S1063779620050068
INTRODUCTION This review embraces several methods of constructing exact solutions of nonlinear problems arising in the field of accelerator physics during simulation of dynamics of systems of many interacting particles. Mathematically, simulation of dynamics of many interacting particles boils down to various types of nonlinear initial boundary value formulations of problems in complex three-dimensional geometrical regions that can usually be solved only by numerical methods. For nonlinear systems, convergence, stability, and accuracy of numerical solutions have been investigated not so widely as for linear systems. Therefore, when writing his own program code or obtaining a numerical result using third-party software, one faces a problem of correctness of the result. In these cases, it is important to have the known exact solutions of nonlinear problems, which can be used as tests for numerical algorithms and as initial approximations in searches for the optimal configuration of a nonlinear system. In the hydrodynamic approximation, the behavior of a many-body system can be described by the charge or mass density function that satisfies the corresponding kinematic equations. In this work, we consider a
kinematic chain of V
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