Separation of variables for the 1-dimensional non-linear diffusion equation

The class of separable solutions of a 1-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group. It includes all solutions invariant under a subgroup of the generic group. An equation which admits separation of variables in some field coordinate has separable solutions not invariant under any subgroup, as in the linear case. The class of separable equations significantly extends the class of equations having non-generic symmetry, i.e. those with exponential or power law diffusivities, for which separation of variables is a trivial process. We derive a complete list of canonical forms for diffusion equations which admit separation of variables in some coordinate, and we describe the separation mechanism for these equations. It involves the integration of a fixed third order ordinary differential equation, generally non-linear, and the subsequent integration of a first order ordinary differential equation which depends on the particular solution of the third order equation. The procedure yields a 3-parameter family of separable solutions of the given diffusion equation. Several non-symmetric examples are analyzed in detail, leading to explicit non-invariant solutions. (C) 1997 Elsevier Science Ltd.