A coupled focusing-defocusing complex short pulse equation: multisoliton, breather, and rogue wave

Nonlinear Schr\"odinger equation, short pulse equation and complex short pulse equation have important application in nonlinear optics. They can be derived from the Maxwell equation. In this paper, we introduce and investigate a coupled focusing-defocusing complex short pulse equation. The bright-bright, bright-dark and dark-dark soliton solutions of the coupled focusing-defocusing complex short pulse equation are given. Then the breathers are derived from the dark soliton solution. The rogue wave solutions are also constructed. The dynamics and the asymptotic behavior of the soliton solutions are analyzed, which reveals that there exist the elastic or inelastic collision in bright-bright soliton solution. But the interactions of bright-dark and dark-dark soliton solutions are both elastic.