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

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.