FUNCTIONAL SEPARATION OF VARIABLES FOR LAPLACE EQUATIONS IN 2 DIMENSIONS

We say that a solution PSI of a partial differential equation in two real variables x1, x2 is functionally separable in these variables if PSI(x1, x2) = phi(A(x1) + B(x2)) for single variable functions phi, A, B such that phi'A'B' not-equal 0. In this paper we classify all possibilities for regular functional separation in local coordinates for equations of the form DELTA2PSI = f(PSI, x1, x2) where DELTA2 is the Laplace-Beltrami operator on a two-dimensional Riemannian or pseudo-Riemannian space. If the dependence of f on x1, x2 is non-trivial then separation can occur for conformal Cartesian coordinates on any space. If f = G(PSI) then for orthogonal coordinates we find that true functional separation, i.e. separation other than additive or multiplicative, occurs precisely for Cartesian coordinates in the Euclidean and pseudo-Euclidean planes. (The sine-Gordon equation provides an example of this separation.) For many of these cases the separated solutions A, B can be expressed in terms of elliptic functions. For non-orthogonal coordinates and f = G(PSI) true functional separation occurs precisely for Cartesian coordinates in the pseudo-Euclidean plane and for a coordinate system on the hyperboloid of one sheet, a pseudo-Riemannian space of constant curvature.