Functional separation of variables for Laplace equations in two dimensions

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Abstract
The authors 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=0. They classify all possibilities for regular functional separation in local coordinates for equations of the form Delta 2 Psi =f( Psi ,x1,x2) where Delta 2 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 they 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.