The spherical Liouville and associated differential equations

Abstract

This paper is concerned with an ordinary non-linear differential equation that occurs in the theory of thermal explosion and, for the spherically symmetric case, in the theory of stellar structure. For plane and axial symmetry, closed form solutions are well known, but the spherically symmetric case can so far only be obtained numerically. By examining the problem in the phase plane, Enig (1967, Critical parameters in the Poisson-Boltzmann equation of steady-state thermal explosion theory. Combust. Flame, 10, 197–199enig1967 was able to obtain an equation from which the critical parameters of the equations can be determined. The equation is of Abel type and there is some difficulty in determining an integrating factor. We note that the partial differential equation for the integrating factor is more difficult to solve then the original equation. A general method is presented that allows the solution to be found in parametric form. Methods of solution for the spherically symmetric case have been presented by various authors. It is shown that these may all be reduced to the equation of Enig. The spherically symmetric case has been examined in some detail near its singular point. In a brilliant but largely ignored paper, it was Jules Enig who first discovered the existence of a vortex in the phase plane. It is hoped that my contribution has solved some of the mysteries of the vortex.