Bäcklund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere. (English) Zbl 0959.35056

Khater, A. H.; Callebaut, D. K.; El-Kalaawy, O. H.

IMA J. Appl. Math. 65, No. 1, 97-108 (2000).

Summary: The equations of magnetostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with an ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential \(u\) known as the Grad-Shafranov equation. By specifying the arbitrary functions in this equation, a Liouville equation is obtained. Bäcklund transformations are described and applied to obtain exact solutions for the Liouville equation modelling an isothermal magnetostatic atmosphere, in which the current density \(J\) is proportional to the exponential of the magnetic potential and moreover falls off exponentially with distance vertical to the base with an \(e\)-folding distance equal to the gravitational scale height.

MSC:

35J60	Nonlinear elliptic equations
86A25	Geo-electricity and geomagnetism
35Q60	PDEs in connection with optics and electromagnetic theory
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
76W05	Magnetohydrodynamics and electrohydrodynamics
35A30	Geometric theory, characteristics, transformations in context of PDEs

Keywords:

magnetostatic equilibria; gravitational field; Liouville equation