Multidimensional Exact Solutions of a Class of Elliptic Systems

In plasma modeling, partial differential equations and equation systems are usually applied, such as Boltzmann or Vlasov equations. Their solutions must meet initial and boundary conditions which presents a stubborn problem. Thus, the task is commonly reduced to a simpler one, e.g., to solving ordinary differential equations. This is the basis for model of magnetic electron isolation in vacuum diode proposed by a group of French mathematicians. The model is described by a system of two nonlinear ordinary second-order differential equations, and the problem of finding all exact solutions, i.e. full integration is concerned. In this paper, the whole concept is further developed into a class of elliptic equation systems with multidimensional Laplace operator, including both generalization of the above vacuum diode model and other systems applied in chemical technology, mathematical biology, etc. It is established that only solutions of Helmholtz linear equation can be solutions of the elliptic systems considered, and the properties of the former solutions can be inherited by the latter ones. Method of finding radially symmetric exact solutions is offered. A series of example control systems are observed, for which parametrical families of exact solutions (including those anisotropic by spatial variables) described by elementary or harmonious functions are found. Examples of global solutions defined on entire space are specified. The explicit expressions of exact solutions obtained have both theoretical and applied value as they can be used for testing, development and adaptation of numerical methods and algorithms of finding approximate solutions for boundary problems within the generalized model of magnetic isolation.