Integrable discretizations of the (2+1)-dimensional sinh-Gordon equation

Abstract

In this paper, we propose two semi-discrete equations and one fully discrete equation and study them by Hirota's bilinear method. These equations have continuum limits into a system which admits the (2+1)-dimensional generalization of the sinh-Gordon equation. As a result, two integrable semi-discrete versions and one fully discrete version for the sinh-Gordon equation are found. Bäcklund transformations, nonlinear superposition formulae, determinant solution and Lax pairs for these discrete versions are presented.