We derive in 3 + 0 dimensions exact solutions of Liouville's equation V a = exp a, by applying the Biicklund transformation technique in conjunction with the principle of nonlinear superposition. The procedure, which is later extended to 3 + 1 dimensions, yield, as a byproduct, particular solutions of V2/3 = 0 and V2a = exp (a + a/3).
The desire of mathematical physicists to find exact solutions of second-order nonlinear partial differential equations is probably matched only by their ambition to derive new ma}hematical tools that will aid in the analysis of these and related systems. Two of the principal tools available today are the method of B~icklund transformations [ 1 ] and the inverse scattering technique [2] which have both been applied with remarkable success to a large variety of nonlinear problems [3].
The purpose of this note is to apply the technique of B/icklund transformations (BT) to yet another nonlinear system, namely to Liouville's equation [4] in three spatial dimensions:
with suitable boundary conditions on X, where X is a scalar field and k, a are real constants appropriate to a certain physical situation. Equation (1 a) has an astonishingly large number of applications. It occurs, for example, in hydrodynamics in the study of vortices [5,6], in cosmology in the context of the nebular theory [7], as well as in the discussion of isothermal gas spheres [8]. It was also used in electrostatics by Richardson [9] and v. Laue [10] to describe, for example, the distribution of electric charge around a glowing wire. Equation (la) was first solved in 2 + 0 dimensions by J. Liouville [4] in 1853, and has subsequently been studied -still in two dimensions -by such renowned mathematicians as Poincard, Picard and Bieberbach [11]. Unfortunately, little progress has been made in higher dimensions, even though v. Laue [10], Bateman [5] and others had stressed its physical relevance also in three dimensions. The aim of this note is to report exact solutions of eqn. (1 a) in three spatial dimensions (with a = k = +1, for convenience),