A classification of the separable solutions of the two-dimensional sine-Gordon equation and of its Laplacian variant

Abstract

In this paper we define, develop and classify a set of separable solutions of the sine-Gordon equation in one space and one time dimension and of its Laplacian or elliptic variant. The solutions we obtain depend, in general, on two real parameters and are classified under the following boundary conditions: i) degeneracy with respect to the parameters, ii) position in the complex-solution space and iii) periodicity with respect to the independent variables. We find three structural groups:a) a one-soliton sector containing the single soliton and antisoliton,b) a two-soliton sector which includes the doublet solutions andc) a general sector in which the solutions are products of Jacobian elliptic functions with coupled periods. By extending the solutions from each of these sectors into a complex domain, we find a generic set of complex solitons whose limiting forms include the real solitons. Furthermore, the existence of the complex doublets enables us to establish a continuous connection between the general sector and the real twosoliton solutions, so that the latter now form part of the boundary of the general sector. Since the vacuum state also lies on this boundary, this has implications for the stability of the two-soliton sector. Additionally, the real solutions of the sine-Gordon equation in this general sector are expressed in standard computable forms. Finally, we discuss the usefulness of the solutions together with questions relating to their stability in the case of the Laplacian variant.