Mathematical model of magnetic insulation vacuum diode and its exact solutions

In this paper we obtain new exact solutions of the stationary model of magnetic insulation for the vacuum diode. This mathematical model described by the system of nonlinear singular elliptic equations in two-dimensional coordinate space. Two physically important cases are considered, both with constant and variable current density. The specified class of infinite-dimensional systems of equations of magnetic insulation with variable current density is determined. They are reduced to equations of similar species with a constant current density via nontrivial transformations. The corresponding theorems are proved and illustrative examples are given.