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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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

Authors: A. I. Bobenko, D. Matthes and Yu. B. Suris
Original publication: Algebra i Analiz, tom 17 (2005), nomer 1.
Journal: St. Petersburg Math. J. 17 (2006), 39-61
MSC (2000): Primary 65N22, 35L45
Published electronically: January 19, 2006
MathSciNet review: 2140674
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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 applications 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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Additional Information

A. I. Bobenko
Affiliation: Institut für Mathematik, Technische Universität Berlin, Str. des 17. June 136, Berlin 10623, Germany

D. Matthes
Affiliation: Institut für Mathematik, Universität Mainz, Staudingerweg 9, Mainz 55128, Germany

Yu. B. Suris
Affiliation: Institut für Mathematik, Technische Universität Berlin, Str. des 17. June 136, Berlin 10623, Germany

Keywords: Hyperbolic system, Goursat problem, integrability, discretization, $K$-surface, B\"acklund transformations
Received by editor(s): September 1, 2004
Published electronically: January 19, 2006
Additional Notes: The first and the third authors were partially supported by the DFG Research Center Matheon “Mathematics for key technologies”.
The second author was partially supported by the DFG Sonderforschungsbereich 288 “Differential Geometry and Quantum Physics”.
Dedicated: Dedicated to L. D. Faddeev on the occasion of his 70th birthday
Article copyright: © Copyright 2006 American Mathematical Society

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