F. Pedit et Hy. Wu, DISCRETIZING CONSTANT CURVATURE SURFACES VIA LOOP GROUP FACTORIZATIONS - THE DISCRETE SINE-GORDON AND SINH-GORDON EQUATIONS, Journal of geometry and physics, 17(3), 1995, pp. 245-260
The sine- and sinh-Gordon equations are the harmonic map equations for
maps of the (Lorentz) plane into the 2-sphere. Geometrically they cor
respond to the integrability equations for surfaces of constant Gauss
and constant mean curvature. There is a well-known dressing action of
a loop group on the space of harmonic maps. By discretizing the vacuum
solutions we obtain via the dressing action completely integrable dis
cretizations (in both variables) of the sine- and sinh-Gordon equation
s. For the sine-Gordon equation we get Hirota's discretization. Since
we work in a geometric context we also obtain discrete models for harm
onic maps into the 2-sphere and discrete models of constant Gauss and
mean curvature surfaces.