Backlund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere

The equations of magnetostatic equilibria for a plasma in a gravitational f ield are investigated analytically. For equilibria with an ignorable spatia l coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential u known as the Grad-Shafranov equation. By speci fying the arbitrary functions in this equation, a Liouville equation is obt ained. Backlund transformations are described and applied to obtain exact s olutions for the Liouville equation modelling an isothermal magnetostatic a tmosphere, in which the current density J is proportional to the exponentia l of the magnetic potential and moreover falls off exponentially with dista nce vertical to the base with an e-folding distance equal to the gravitatio nal scale height.