Analytical solutions to the Grad–Shafranov equation

Two families of exact analytical solutions of the Grad–Shafranov equation are presented by specifying the highest polynomial dependence of the plasma current density on the flux function Ψ in such a way that the Grad–Shafranov equation becomes a linear inhomogeneous differential equation. Both the pressure profile and the poloidal current profile each have two free parameters. X-points can be represented by superposition of solutions. Examples of the exact equilibrium solution are given for both a D-shaped plasma and a toroidally diverted plasma.