Solutions of local and nonlocal discrete complex modified Korteweg-de Vries equations and continuum limits

Abstract: Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two proper' discrete Ablowitz-Kaup-Newell-Segur equations and twounproper' discrete Ablowitz-Kaup-Newell-Segur equations are derived. The proper' equations admit local reduction, while theunproper' equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.