The modulational instability, classifications and dynamical properties of the localized wave solutions in the Hirota equation

In this paper, we introduce the generalized perturbation ($n,N$-$n$)-fold Darboux transformation (gDT) to investigate the Hirota equation, which has been viewed as a better physical model to accurately describe wave propagations in the ocean and optical fibers than the NLS equation. Firstly, we analyze the modulational instability and build the mechanism of generating different localized waves for the Hirota equation. Then, by using ($n,N$-$n$)-fold gDT, we obtain abundant localized wave solutions, including higher-order rogue waves (RWs), higher-order periodic waves and their interactions. Dynamical properties of these solutions are investigated numerically and theoretically. Based on the detailed numerical and theoretical analysis, we make a classification of the higher-order RWs based on their structures. In particular, for the first-order RWs, we give the analytical formulas of their length and width via the contour line method. And for the second-order RWs, we prove that the three first-order RWs after splitting are the same when the modulation parameters tend to infinity via the large-parameter asymptotic analysis. The method adopted here can be extended to construct localized waves of other nonlinear systems. The results obtained are new and they may provide a new perspective or reference in understanding the related rogue wave phenomenon in fluids, plasmas, nonlinear optics and Bose–Einstein condensation.