The sine-Gordon equation: its geometry and applications of current interest

Abstract

This chapter is devoted to geometrical aspects in the study of the sine-Gordon equation as a canonical (from the point of view of non-Euclidean hyperbolic geometry) nonlinear equation that has wide applications in contemporary mathematical physics. A far-reaching fact that enables the realization of diverse approaches to the investigation of problems connected with the sine-Gordon equation is the intimate association of this equation with surfaces of constant negative curvature, i.e., with pseudospherical surfaces.