Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits

We study local and nonlocal complex reductions of a discrete negative-order Ablowitz–Kaup–Newell–Segur equation. For the resulting local and nonlocal complex discrete sine-Gordon equations, we construct solutions of the Cauchy matrix type, including soliton solutions and Jordan-block solutions. The dynamics of $1$-soliton solutions are analyzed and illustrated. Continuum limits of the resulting local and nonlocal complex discrete sine-Gordon equations are discussed.