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

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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 Backlund equations are also recorded.
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Document Type: Research Article

Affiliations: 1: University of Glasgow 2: University of New South Wales

Publication date: April 1, 1998

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