Solitons, Breathers and Modulation Instability for a Higher-Order Coupled Nonlinear Schrodinger System for the Ultrashort Optical Pulses in a Nonlinear Medium

In this paper, we investigate some soliton and breather solutions for a higher-order coupled nonlinear Schrodinger system which may describe the ultrashort optical pulses in a nonlinear medium. With the help of the existing Darboux transformation, we construct the first-and second-order soliton solutions, as well as the first-and second-order breather solutions. We present two types of the solitons, i.e., two-hump solitons and one-hump solitons. Interaction between the two one-hump solitons, and interaction between a two-hump soliton and a one-hump soliton are presented. When the velocities of two solitons are equal, we obtain the bound state of the two solitons. Real constant epsilon in the system affects the velocities of the solitons. We show the one breather and interaction between the two breathers. Velocities and shapes of the breathers are also affected by epsilon. We discuss the modulation instability of that system through the linear stability analysis. That system may describe the ultrashort optical pulses in a nonlinear medium, therefore the wave phenomena proposed in our paper may provide certain theoretical references for the related experiments in the future.