Separation of variables for scalar evolution equations in one space dimension

Abstract

The flow of an autonomous kth-order evolution equation in one space dimension is generated by a kth-order ordinary differential operator on the space of fields. The characteristic fields are the solutions of the characteristic equation of the operator. A solution of the evolution equation is separable if and only if it is a curve in the .k C 1/-parameter space of characteristic fields. The non-stationary characteristic fields with separable evolution are those which remain characteristic under small dilation. Every characteristic field has local separable evolution if and only if the characteristic equation is infinitesimally dilation invariant. This is the case when the evolutionary generator is an infinitesimal symmetry of the characteristic equation, so that its flow stabilizes the space of characteristic fields. The evolution equation is then locally homogeneous if and only if the characteristic value is an invariant of the restricted flow, which is a non- generic property. The separable kth-order evolution equations are parametrized by the kth-order invariants of the locally homogeneous .k C 1/th-order ordinary differential equations.