Discrete Differential Geometry. Integrability as Consistency

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• 作者：Alexander I. Bobenko
• 刊名：Lecture Notes in Physics
• 出版年：2004
• 出版时间：2004
• 年：2004
• 卷：644
• 期：1
• 页码：pp.85-110
• 全文大小：386 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical Methods in Physics
  Mathematical and Computational Physics
  Astronomy, Astrophysics and Cosmology
  Atoms, Molecules, Clusters and Plasmas
  Relativity and Cosmology
  Extraterrestrial Physics and Space Sciences
  Condensed Matter
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1616-6361

文摘

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.