Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)–dimensional sine–Gordon system

Abstract

Superposition principles, both linear and nonlinear, associated with the Moutard transformation are found. On suitable reinterpretation, these constitute an integrable discrete nonlinear system and its associated linear system. Further, it is shown that, in a particular form, this system is an integrable discretization of a (2+1)–dimensional sine–Gordon system. Solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the Moutard transformation. Included in these solutions are discrete analogues of the kink solutions of the continuous system.