Bäcklund transformations and Painlevé analysis: Exact solutions for the nonlinear isothermal magnetostatic atmospheres

The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential μ̃, known as the Grad–Shafranov equation. Specifying the arbitrary functions in the latter equation, one gets the nonlinear elliptic equation. Analytical solutions of the elliptic equation are obtained for the case of a nonlinear isothermal atmosphere in a uniform gravitational field. The solutions are obtained by using the Bäcklund transformations technique and Painlevé analysis, which are adequate for describing parallel filaments of diffuse, magnetized plasma suspended horizontally in equilibrium in a uniform gravitational field.