Loop, cuspon, and soliton solutions of a multicomponent discrete complex short-pulse equation

We present an integrable discretization of a multicomponent discrete complex short-pulse (dCSP) equation in terms of a Lax pair representation and a Darboux transformation (DT). The Lax pair representation is explored using block matrices by extending the $2\times2$ Lax matrices to $2^L\times2^L$ Lax matrices. The DT on the matrix solutions is studied and is used to generate solutions of the multicomponent dCSP equation by using the properties of quasideterminants. By expanding the quasideterminants, we then show the soliton solutions to be expressed as ratios of ordinary determinants. Further, an appropriate continuum limit is applied to obtain multisoliton solutions of the continuous complex short-pulse equation.