Solitons, breathers and rogue waves of the coupled Hirota system with 4 × 4 Lax pair

Solitons, breathers and rogue waves of the coupled Hirota system are investigated. To obtain these solutions, we construct the 4 × 4 Lax pair and Darboux transformation (DT). With zero backgrounds, we derive the one- and two-peak solitons. With non-zero and mixed backgrounds, we produce a family of the breather solutions and rational solutions for the purpose of describing the rogue waves. The second-order rogue-wave solutions on the mixed backgrounds are obtained. The existence and properties of bright-dark breathers and rogue waves are discussed. By adjusting the parameters in DT, we show the conversions between the bright breathers/rogue waves and the dark breathers/rogue waves in the course of evolution. We note that for such system if there is the modulation instability, it is of baseband type only.