Discrete Differential Geometry. Integrability as Consistency
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We discuss a new geometric approach to discrete integrability coming from discrete differential geometry. A d–dimensional equation is called consistent if it is valid for all d–dimensional sublattices of a (d+1)–dimensional lattice. This algorithmically verifiable property implies analytical structures characteristic of integrability, such as the zero-curvature representation, and allows one to classify discrete integrable equations within certain natural classes. These ideas also apply to the noncommutative case. Theorems about the smooth limit of the theory are also presented.
KeywordsGordon Equation Elementary Quadrilateral Loop Group Dimensional Lattice Discrete Surface
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