Focusing and defocusing coupled modified complex short pulse equations: Darboux transformation and solitonic solutions

In this paper, we develop the Darboux transformation (DT) for the focusing and defocusing coupled modified complex short pulse (cmcSP) equations by employing reciprocal transformation and the associated cmcSP equation. As applications of the DT, various solitonic solutions of the cmcSP equation are also obtained. Specifically, under zero seed solution, we present general soliton solution, double-hump soliton solution, composite breather solution and higher order soliton solution for the focusing cmcSP equation. Under plane wave seed solution, we construct breather solution and rogue wave solution for the focusing cmcSP equation, as well as dark soliton solution, breather solution and rogue wave solution for the defocusing cmcSP equation.