An Integrable Discretization of a 2+1‐dimensional Sine‐Gordon Equation

Abstract

We show that the complex discrete BKP equation that has been recently identified as an integrable discretization of the 2+1‐dimensional sine‐Gordon system introduced by Konopelchenko and Rogers admits a natural reduction to a discrete 2+1‐dimensional sine‐Gordon equation. We discuss three important properties of this equation. First, it may be interpreted as a superposition principle associated with a constrained Moutard transformation. Second, the complexified discrete sine‐Gordon equation constitutes an eigenfunction equation for the discrete sine‐Gordon system. Third, we derive a form of the equation in terms of trigonometric functions that has been studied by Konopelchenko and Schief in a discrete geometric context. A discrete Moutard transformation for the discrete sine‐Gordon equation and the corresponding Bäcklund equations are also recorded.