Discreditizing constant curvature surfaces via loop group factorizations - the discrete sine-Gordon and Sinh-Gordon equations
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The sine- and sinh-Gordon equations are the harmonic map equations for maps of the (Lorentz) plane into the 2-sphere. Geometrically they correspond 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 discretizations (in both variables) of the sine- and sinh-Gordon equations. For the sine-Gordon equation we get Hirota's discretization. Since we work in a geometric context we also obtain discrete models for harmonic maps into the 2-sphere and discrete models of constant Gauss and mean curvature surfaces.
F Pedit and HY Wu. "Discreditizing constant curvature surfaces via loop group factorizations - the discrete sine-Gordon and Sinh-Gordon equations" Journal of Geometry and Physics 17.3 (1995): 245-260.
Available at: http://works.bepress.com/franz_pedit/2
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