Breathers and rogue waves on the periodic background for the Gerdjikov-Ivanov equation for the Alfvén waves in an astrophysical plasma

Alfvén waves in the magnetized astrophysical plasmas are invoked to explain the heating of the stellar coronae, acceleration of the stellar winds, as well as the origin and formation of the galactic and extragalactic jets. Under investigation in this paper is the Gerdjikov-Ivanov equation which is considered as a plasma-physics model for the Alfvén waves propagating parallel to the ambient magnetic field. Based on the existing N-th order analytic solutions for the Gerdjikov-Ivanov equation with N being a positive integer, we construct two types of the breathers on the periodic background, and the k-th order rogue waves on the periodic background in terms of the determinant expression for the transverse magnetic field perturbation are derived via the Taylor expansion when N=2k+1 with k being a non-negative integer. We show that the breathers and rogue waves are located in the periodic background rather than in the constant background. We obtain four different structures, which are the so-called fundamental, triangular, ring and ring-triangular structures, for the higher-order rogue waves on the periodic background. We graphically demonstrate some solutions for the transverse magnetic field perturbation to analyse the characteristics of the four different structures.