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

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

2002-08-06

arxiv.org/abs/math/0208042v1

Algebra Anal., 2005, vol. 17, Nr. 1, p.53-83, English translation: St. Petersburg Math. J., 2006, vol. 17, Nr. 1, p.39-61

Mathematics

Numerical Analysis

42 pages, 3 figures

Scientific paper

A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are applied to hyperbolic systems of differential-geometric origin, like the sine-Gordon equation describing the surfaces of the constant negative Gaussian curvature (K-surfaces). In particular, we prove the convergence of discrete K--surfaces and their Backlund transformations to their continuous counterparts. This puts on a firm basis the generally accepted belief (which however remained unproved untill this work) that the classical differential geometry of integrable classes of surfaces and the classical theory of transformations of such surfaces may be obtained from a unifying multi-dimensional discrete theory by a refinement of the coordinate mesh-size in some of the directions

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