Dynamical behaviors of vector localized wave solutions for the coupled modified Korteweg-de Vries equations

Under investigation in this paper is the coupled modified Korteweg-de Vries equations. Based on Lax pair and Darboux transformation, various vector localized wave solutions are derived and analyzed. We notice that the localized waves have different dynamical behaviors in several components, for example, the bell-shaped soliton and flat-top soliton, or the bell-shaped soliton and two-peak soliton, are obtained in two components, respectively. Furthermore, the interactions between two solitons are different in distinct components. In addition, the vector breather and rogue wave solutions are constructed. Especially, the various structures of first-order rogue waves are observed, such as the two-peak rogue waves and flat-top rogue wave, or the dark rogue waves and bright rogue waves. Finally, the vector second-order rogue wave solutions are derived and the strong or weak interaction patterns of second-order rogue waves are achieved.