  The Darboux transformation (DT) for the coupled complex short pulse (CCSP)
equation is constructed through the loop group method. The DT is then utilized
to construct various exact solutions including bright soliton, dark-soliton,
breather and rogue wave solutions to the CCSP equation. In case of vanishing
boundary condition (VBC), we perform the inverse scattering analysis to
understand the soliton solution better. Breather and rogue wave solutions are
constructed in case of non-vanishing boundary condition (NVBC). Moreover, we
conduct a modulational instability (MI) analysis based on the method of squared
eigenfunctions, whose result confirms the condition for the existence of rogue
wave solution.
