Abundant exact solutions to the discrete complex mKdV equation by Darboux transformation

In this paper, an N-fold Darboux transformation is constructed for the discrete complex modified Korteweg-de Vries equation of focusing type, in terms of determinants. Through the obtained one-fold and two-fold Darboux transformations, a variety of new exact solutions, including an anti-dark soliton solution, a breather solution, a periodic solution, and a two-soliton solution, are derived from a nonzero constant and plane-wave seed solution. Via numerical simulation, a new kind of dynamical behavior of the two-soliton solution is exhibited, which tells that the two-soliton solution includes an anti-dark solitary wave and a w-shaped solitary wave.