## Abstract

The authors say that a solution Psi of a partial differential equation in two real variables x_{1},x_{2} is functionally separable in these variables if Psi (x_{1},x_{2})= phi (A(x _{1})+B(x_{2})) 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 ,x_{1},x_{2}) where Delta _{2} is the Laplace-Beltrami operator on a two-dimensional Riemannian or pseudo-Riemannian space. If the dependence of f on x_{1},x_{2} 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.

Original language | English (US) |
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Article number | 017 |

Pages (from-to) | 1901-1913 |

Number of pages | 13 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 26 |

Issue number | 8 |

DOIs | |

State | Published - 1993 |

Externally published | Yes |