On the separability of the sine-Gordon equation and similar quasilinear partial differential equations

The separability of the sine-Gordon equation (SGE) is defined and studied in detail. We find a general class of dependent-variable transformations under which the SGE is separable. This class may be reduced to a two-parameter generalization of the usual transformation adopted, by requiring the transformations to reduce to the identity in the linear limit of the SGE (i.e., the Klein–Gordon equation). The method developed for studying the separability of the SGE is then applied to more general quasilinear equations and a discussion of the limitations of the method, and of separable solutions in general, is also given.