Generalized (m, N − m)-fold Darboux transformation and localized waves of the complex Kraenkel-Manna-Merle system in a ferrite

Abstract

In this paper, we focus on a complex Kraenkel-Manna-Merle system, which could characterize certain short waves in a ferrite. On the basis of the existing N-fold Darboux transformation, we have determined a generalized (m, N − m)-fold Darboux transformation which admits m spectral parameters, with N and m being the positive integers. Rogue-wave, breather and mixed wave solutions of that system are derived utilizing the generalized (m, N − m)-fold Darboux transformation. We show the first-order rogue wave with one hump and two valleys, as well as the first-order rogue wave with one hump and one valley. We obtain the second-order rogue wave and also show that the second-order rogue wave divides into three first-order rogue waves which are arranged in the triangle structure. We present the third-order rogue wave and observe that the third-order rogue wave divides into six first-order rogue waves which are arranged in the triangle and pentagonal structures. The first-order breather and interaction between the two first-order breathers are illustrated. In addition, we present the interactions between the first-order/second-order rogue wave and first-order breather.