Exact solutions of a class of nonlinear elliptic systems of a special type

In paper the problem of construction of exact solutions for nonlinear system of two equations of elliptic type is studied. Nonlinear systems of the equations of elliptic type are applied as mathematical models in the theory warm and a mass transfer of the reacting systems, in the theory of chemical reactors, the theory of burning and mathematical biology. In a one-dimensional case can carry the model of magnetic isolation of the vacuum diode described by the ordinary differential equations to the same class of the equations. Finding of exact solutions for nonlinear elliptic systems plays an important role as for development of the theory and establishment of properties of all set of solutions, and for applications. Exact solutions can be used for testing and verification of numerical methods of the solution of boundary value problems. In this paper the system of two equations of elliptic type with one nonlinearity depending on a difference of squares of required functions is considered. Conditions on nonlinearity under which the system is reduced to one equation are found. It is shown that in this case the system is reduced to the semi-linear elliptic equation of a special kind, only one component different from Helmholtz's equation. The case of the system which isn't reduced is separately studied at any nonlinearity to one equation. For this case the integro-differential equation to which have to satisfy radially symmetric solutions is obtained. Cases when this equation is reduced to the ordinary differential equation are specified and is integrated in an explicit form. A number of examples of construction of the exact solutions set by elementary functions for systems with the two-dimensional and three-dimensional operator of Laplace is given.