Geometric deformations of the evolution equations and bäcklund transformations

Abstract

The sine-Gordon equation (2ω)xi + sin(2ω) = 0 describes a surface with constant negative curvature. Sasaki [3], demonstrated that pseudospherical surfaces are associated with many evolution equations. The surfaces of negative curvature (not necessarily constant), which can be thought of as deformations of the pseudospherical surfaces, are defined by solutions of a system of equations. We call such a system a deformed sine-Gordon system. The geometry of such systems has been extensively studied in the theory of line congruences [2]. It turns out that the more appropriate setting for the study of the geometry in full generality is not the group of euclidean motions but the projective group. In that case we talk about W-systems. It is clear that such a theory is an important special case of the study of the σ-models. The significant feature of the deformed sine-Gordon and W-systems is’ the existence of Backlund transformations and the validity of the permutability theorem.