Dynamical behaviour of rogue wave solutions in a multi-component AB system

Abstract

With the aid of a 3 × 3 matrix spectral problem, we propose a multi-component AB system to describe self-induced transparency phenomenon. Based on the gauge transformation between the spectral problems, a one-fold Darboux transformation of this system can be given directly. The compact determinant forms of the n-fold classical Darboux transformation of the multi-component AB system are obtained by iterating the one-fold Darboux transformation and solving a linear algebraic system. Furthermore, an N-fold generalized Darboux transformation of the multi-component AB system is generated by utilizing the limit technique and Taylor expansion. As applications of the generalized Darboux transformation, all kinds of solutions for the multi-component AB system can be obtained, including Ma breather solutions, Akhmediev breather solutions and rogue wave solutions. In addition, the dynamical behaviours of all types of rogue wave solutions, such as four-peaked, four-petalled, triangular and ring, are discussed in detail by using the generalized Darboux transformation.