DISCRETIZING CONSTANT CURVATURE SURFACES VIA LOOP GROUP FACTORIZATIONS - THE DISCRETE SINE-GORDON AND SINH-GORDON EQUATIONS

Publication Date

1995

Journal or Book Title

JOURNAL OF GEOMETRY AND PHYSICS

Abstract

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.

Comments

The published version is located at http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ8-3Y6PFH9-8&_user=1516330&_coverDate=11%2F30%2F1995&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1663833072&_rerunOrigin=google&_acct=C000053443&_version=1&_urlVersion=0&_userid=1516330&md5=92a8dd65f7423e8eda91a9898d53e413&searchtype=a

Pages

245-260

Volume

17

Issue

3

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