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Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis

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Symmetries and Integrability of Difference Equations

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

The field of discrete differential geometry lies on the border of classical differential geometry and discrete geometry. Its aim is to develop discrete geometric theories which respect fundamental aspects of the corresponding smooth ones. Also, these discretizations often clarify structures of the smooth theory.

In our presentation, we focus on the area of discrete complex analysis. In particular, we introduce several concepts of discrete holomorphic functions based on a linear approach and on nonlinear theories concerning cross-ratio systems, circle patterns and discrete conformal equivalence. These examples are used to illustrate some characteristic features in discrete differential geometry like integrability as consistency and Bäcklund–Darboux transformations.

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Acknowledgements

The author is supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

Furthermore, I am grateful to Nikolay Dimitrov for discussions on integrability in the context of discrete differential geometry, in particular concerning cross-ratio systems and conformally equivalent triangulations. Many thanks also to the referee for his careful reading and advice.

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  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Ulrike Bücking

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  1. Ulrike Bücking
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Correspondence to Ulrike Bücking .

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  1. Mathematics and Physics Departments, and INFN, Sezione Roma Tre, Roma Tre University, Roma, Italy

    Decio Levi

  2. Collège Ahuntsic, Montréal, Québec, Canada

    Raphaël Rebelo

  3. Centre de Recherches Mathématiques and Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada

    Pavel Winternitz

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Bücking, U. (2017). Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis. In: Levi, D., Rebelo, R., Winternitz, P. (eds) Symmetries and Integrability of Difference Equations. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-56666-5_4

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