Tanh-travelling wave solutions, truncated Painlevé expansion and reduction of Bullough–Dodd equation to a quadrature in magnetohydrodynamic equilibrium

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation for the magnetic potential ũ, known as the Grad–Shafranov equation. Specifying the arbitrary functions in this equation, the Bullough–Dodd equation can be obtained. The truncated Painlevé expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the travelling wave solutions of the Bullough–Dodd equation for the case of isothermal magnetostatic atmosphere, in which the current density J is proportional to the exponentially of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height.