Nonlinear periodic solutions for 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 known as the Grad–Shafranov equation. Specifying the arbitrary function in the latter equation, yields a nonlinear elliptic equation. Analytical nonlinear periodic solutions of this elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field: e.g. a model for the solar atmosphere. We obtained several classes of exact solutions of five nonlinear evolution equations (Liouville, sinh–Poisson, double sinh–Poisson, sine–Poisson and double sine–Poisson) using the generalized tanh method. Moreover, the Bäcklund transformations are used to generate further new classes of solutions. The final results may be used to investigate some models in solar physics.