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Nonlinear hyperbolic equations in surface theory: integrable discretizations and approximation results, arXiv:math.NA/ 0208042

By Alexander I. Bobenko, Daniel Matthes, Yuri and B. Suris


Abstract. A lattice-discretization of the Goursat problem for a class of nonlinear hyperbolic systems is proposed. Local C ∞-convergence of the discrete solutions is proven, and the approximation error for the smooth limit is estimated. The results hold in arbitrary dimensions, and for an arbitrary number of dependent variables. The abstract approximation theory is matched by a guiding example, which is the sine-Gordon-equation. 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 proven. The result extends to the simultaneous approximation of Bäcklund transformations. This puts on a firm basis on the generally accepted belief that the theory of integrable surfaces and their transformations may be obtained as the continuum limit of a unifying multi–dimensional discrete theory. 1

Year: 2002
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