Soliton and breather interactions for a coupled system

Under investigation in this paper is a higher-order nonlinear Schrodinger-Maxwell-Bloch system with the sextic term which might describe the ultrashort optical pulses, up to the attosecond duration, in an erbium-doped fiber. We derive the Lax pair and Darboux transformation, which are both related to and , the coefficient for the higher-order terms and the detuning of the atomic transition frequency from the incoming radiation frequency, respectively. Bright and dark solitons and breathers are constructed by virtue of the Darboux transformation. Besides, based on the breather solutions, we get the first-order rogue wave solutions via the limiting procedure. We see that both and affect velocities and widths of the solitons, also decide whether the solution is a bright or dark soliton while have no effect on the periods of the two types of breathers. The interaction between the solitons is elastic, and the periodic structure comes into being with the interaction between the two solitons when the velocities of the two interacted solitons are equal to each other. With different eigenvalues, different interaction forms between the two types of the breathers are obtained. Interaction between the two breathers with the adjacent frequencies forms the bound breather with a periodic attractive-repulsive structure. Especially, two near-degenerate cases of the interactions are also exhibited.