Nonlinear d'Alembert formula for discrete pseudospherical surfaces

TL;DR

This paper introduces a discrete nonlinear d'Alembert formula for constructing pseudospherical surfaces with constant negative curvature in Euclidean space, based on loop group decompositions, and demonstrates its application through detailed examples.

Contribution

It develops a novel discrete nonlinear d'Alembert formula using loop group decompositions for pseudospherical surfaces, extending the separation of variables method to discrete differential geometry.

Findings

Derived a discrete nonlinear d'Alembert formula for pseudospherical surfaces

Provided detailed examples illustrating the formula's application

Extended continuous methods to discrete geometric surfaces

Abstract

On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature (pseudospherical) surfaces in Euclidean three space. We also compute two examples by this formula in detail.