Riemann–Hilbert approach for discrete sine‐Gordon equation with simple and double poles

Abstract In this paper, we present the inverse scattering transformation (IST) for discrete sine‐Gordon equation via the Riemann–Hilbert approach. In the direct scattering part, we establish analyticity, symmetries, and asymptotic properties of Jost solutions and reflection coefficients. In the inverse part, we solve the discrete sine‐Gordon equation by establishing a Riemann–Hilbert problem with simple and double poles, respectively. N ‐soliton solutions are obtained via the reconstruction formula correspondent to the Riemann–Hilbert problem. As examples of N ‐solitons, we present some explicit solutions such as breathers, degenerate solitons for discrete sine‐Gordon equation, and the dynamical properties of these solutions are further analyzed.