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Algebra i Analiz, 2005, Volume 17, Issue 1, Pages 53–83 (Mi aa646)  

This article is cited in 3 scientific papers (total in 3 papers)

Research Papers

Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results

A. I. Bobenkoa, D. Matthesb, Yu. B. Surisa

a Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
b Institut für Mathematik, Universität Mainz, Mainz, Germany

Abstract: A discretization of the Goursat problem for a class of nonlinear hyperbolic systems is proposed. Local $C^\infty$-convergence of the discrete solutions is proved, and the approximation error is estimated. The results hold in arbitrary dimensions, and for an arbitrary number of dependent variables. The sine-Gordon equation serves as a guiding example for application of the approximation theory. As the main application, a geometric Goursat problem for surfaces of constant negative Gaussian curvature ($K$-surfaces) is formulated, and approximation by discrete $K$-surfaces is proved. The result extends to the simultaneous approximation of Bäcklund transformations. This rigorously justifies the generally accepted belief that the theory of integrable surfaces and their transformations may be obtained as the continuum limit of a unifying multidimensional discrete theory.

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English version:
St. Petersburg Mathematical Journal, 2006, 17:1, 39–61

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Received: 01.09.2004
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Citation: A. I. Bobenko, D. Matthes, Yu. B. Suris, “Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results”, Algebra i Analiz, 17:1 (2005), 53–83; St. Petersburg Math. J., 17:1 (2006), 39–61

Citation in format AMSBIB
\Bibitem{BobMatSur05}
\by A. I. Bobenko, D.~Matthes, Yu. B. Suris
\paper Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results
\jour Algebra i Analiz
\yr 2005
\vol 17
\issue 1
\pages 53--83
\mathnet{http://mi.mathnet.ru/aa646}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2140674}
\zmath{https://zbmath.org/?q=an:1098.53005}
\transl
\jour St. Petersburg Math. J.
\yr 2006
\vol 17
\issue 1
\pages 39--61
\crossref{https://doi.org/10.1090/S1061-0022-06-00892-2}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Adler V. E., “Some incidence theorems and integrable discrete equations”, Discrete Comput. Geom., 36:3 (2006), 489–498  crossref  mathscinet  zmath  isi  elib  scopus
    2. Xia Qiaoling, “Generalized Weierstrass representations of surfaces with the constant Gauss curvature in pseudo-Riemannian three-dimensional space forms”, J. Math. Phys., 48:4 (2007), 042301, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Burstall F.E., Donaldson N.M., Pedit F., Pinkall U., “Isothermic submanifolds of symmetric R-spaces”, J Reine Angew Math, 660 (2011), 191–243  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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