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Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations. (English) Zbl 0856.58020

The authors apply loop factorizations and the corresponding dressing actions to the sine-Gordon equation, which describes negative constant Gauss curvature surfaces (K-surfaces), to obtain integrable discrete K-surfaces. The same technique is used to rise discrete constant mean curvature surfaces starting from the corresponding sinh-Gordon equation.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53A05 Surfaces in Euclidean and related spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E20 Harmonic maps, etc.
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