Modulation instability, state transitions and dynamics of multi-peak rogue wave in a higher-order coupled nonlinear Schrodinger equation

The coupled nonlinear Schrodinger equation provides an effective description of the propagation of four optical fields in the fibers. The present work explores the dynamics of rogue waves that arise in a coupled nonlinear Schrodinger equation of higher-order that is acquired using the Ablowitz-Kaup-Newell-Segur method. The generalized (m,N - m)-fold Darboux transformation is used to determine the exact rogue wave solutions, which show multiple peaks and depressions. In addition, a detailed analysis is conducted of the modulation instability of three different kinds of plane-wave solutions. The results indicate that the coefficients of the second and third derivative terms, rather than just the coefficients of the third derivative term, have an impact on modulation instability. Lastly, the transition between rogue waves and solitons under the influence of higher-order dispersion effects is elucidated, with explicit soliton solutions provided within the modulation stability region.