Darboux transformation and solitonic solution to the coupled complex short pulse equation

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