ABSTRACT
Attempts to understand the origin of the inverse scattering Lax representation of completely integrable nonlinear evolution equations have resulted in an interesting differential geometric and group theoretical interpretation of solitons. This chapter discusses differential-geometric character in the context of a specific example, namely, the discrete sine-Gordon equation, which is perhaps the simplest example in which all the properties of the continuous case carry over to the discrete case. The gauge property of Bäcklund transformations may be used to derive an algebraic combination formula among four solutions of the sine-Gordon equation. The chapter also discusses discrete nonlinear sigma model.
