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Bäcklund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere. (English) Zbl 0959.35056
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.
35J60Nonlinear elliptic equations
86A25Geo-electricity and geomagnetism
35Q60PDEs in connection with optics and electromagnetic theory
35A05General existence and uniqueness theorems (PDE) (MSC2000)
76W05Magnetohydrodynamics and electrohydrodynamics
35A30Geometric theory for PDE, characteristics, transformations