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Nonlinear Schrödinger equation, short pulse equation, and complex short pulse equation have important applications in nonlinear optics. They can be derived from the Maxwell equations. In this paper, we 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 constructed. Furthermore, the breather solutions 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 a fact that the bright-bright soliton collision can be either elastic or inelastic, but the bright-dark and dark-dark soliton collision can only be elastic.

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