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.

中文翻译：

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第一个Vlasov方程的精确可解模型

摘要

考虑构造从形式上类似于连续性方程的Vlasov方程链中找到第一个方程的精确解的方法。正在研究的方程式是针对标量函数\（f \）和向量场\（\ left \ langle {\ vec {v}} \ right \ rangle \）编写的。根据问题的表达，函数\（f \）可以对应于磁性材料的概率密度，电荷，质量或磁导率。向量字段\（\ left \ langle {\ vec {v}} \ right \ rangle \）可以对应于概率流，连续介质的速度场或磁场强度。在数学上，相同的等式适用于描述统计，量子和经典系统。可以将一个物理系统的精确解映射到另一系统的精确解。模型非线性系统精确解的可用性对于设计复杂的物理设施（例如NICA项目的SPD检测器）非常重要。这些解决方案用作编写程序代码的测试，并且可以封装为有限差分方案，以数值方式求解非线性微分方程的边值问题。