Exactly Solvable Models for the First Vlasov Equation

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 $$\left\langle {\vec {v}} \right\rangle $$ . 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 $$\left\langle {\vec {v}} \right\rangle $$ 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.