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

A. I. Bobenko, D. Matthes, Yu. B. Suris

TL;DR

This work develops a rigorous, multidimensional discretization framework for nonlinear hyperbolic systems arising in surface theory, anchored by the Goursat problem and discrete differential geometry. It proves $C^1$- and higher-order convergence of discrete schemes to their continuous counterparts, and applies the results to sine-Gordon dynamics for $K$-surfaces, including the convergence of discrete surfaces, their associated families, and Backlund transformations. By establishing a priori bounds, regularity transfer, and multidimensional compatibility, the paper demonstrates that classical differential geometry of integrable surface classes can be recovered from a refined multidimensional discrete theory. The methods extend to general hyperbolic systems in arbitrary dimensions, providing a broad numerical-analytic foundation for discretizations that preserve integrability and geometric structure.

Abstract

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

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

TL;DR

This work develops a rigorous, multidimensional discretization framework for nonlinear hyperbolic systems arising in surface theory, anchored by the Goursat problem and discrete differential geometry. It proves - and higher-order convergence of discrete schemes to their continuous counterparts, and applies the results to sine-Gordon dynamics for -surfaces, including the convergence of discrete surfaces, their associated families, and Backlund transformations. By establishing a priori bounds, regularity transfer, and multidimensional compatibility, the paper demonstrates that classical differential geometry of integrable surface classes can be recovered from a refined multidimensional discrete theory. The methods extend to general hyperbolic systems in arbitrary dimensions, providing a broad numerical-analytic foundation for discretizations that preserve integrability and geometric structure.

Abstract

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

Paper Structure

This paper contains 7 sections, 18 theorems, 178 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Two-dimensional theory
  3. Additional Smoothness
  4. Three--dimensional theory: approximating Bäcklund transformations
  5. Approximation theorems for K-Surfaces
  6. General hyperbolic systems
  7. Appendix

Key Result

Proposition 1

The Goursat problem for a discrete 2D hyperbolic system (D2DH) has a unique solution $(a^\epsilon,b^\epsilon)$ on $\Omega^\epsilon(r)$.

Figures (5)

  • Figure 1: Surfaces and their transformations as a limit of multidimensional lattices
  • Figure 2: A countinous and a discrete Amsler surfaces
  • Figure 3: Maximum norm of the error $F^\epsilon -F$ vs. mesh size $\epsilon$. $F:[0,r]\times[0,r]\rightarrow{\mathbb R}^3$ is a smooth K-surface, $F^\epsilon:[0,r]^{\epsilon}\times[0,r]^\epsilon\rightarrow{\mathbb R}^3$ are discrete K-surfaces. (See section \ref{['Sect 2d']} for notations.) The lower line corresponds to $r=1.0$, the upper one to $r=4.0$.
  • Figure 4: An elementary quadrilateral
  • Figure 5: Three-dimensional consistency

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 1
  • Lemma 4
  • ...and 14 more