Evolution Of Rogue Waves Research Articles

Modulational instability, generation, and evolution of rogue waves in the generalized fractional nonlinear Schrödinger equations with power-law nonlinearity and rational potentials.

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ, the Lévy index α, and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W-shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena.

A coupled "AB" system: Rogue waves and modulation instabilities.

Rogue waves are unexpectedly large and localized displacements from an equilibrium position or an otherwise calm background. For the nonlinear Schrödinger (NLS) model widely used in fluid mechanics and optics, these waves can occur only when dispersion and nonlinearity are of the same sign, a regime of modulation instability. For coupled NLS equations, rogue waves will arise even if dispersion and nonlinearity are of opposite signs in each component as new regimes of modulation instability will appear in the coupled system. The same phenomenon will be demonstrated here for a coupled "AB" system, a wave-current interaction model describing baroclinic instability processes in geophysical flows. Indeed, the onset of modulation instability correlates precisely with the existence criterion for rogue waves for this system. Transitions from "elevation" rogue waves to "depression" rogue waves are elucidated analytically. The dispersion relation as a polynomial of the fourth order may possess double pairs of complex roots, leading to multiple configurations of rogue waves for a given set of input parameters. For special parameter regimes, the dispersion relation reduces to a cubic polynomial, allowing the existence criterion for rogue waves to be computed explicitly. Numerical tests correlating modulation instability and evolution of rogue waves were conducted.

Controllable Nonautonomous Rogue Waves in the Modified Nonlinear Schrödinger Equation with Distributed Coefficients in Inhomogeneous Fibers

Abstract Under investigation in this paper is an inhomogeneous modified nonlinear Schrödinger (MNLS) equation describing the ultrashort pulse dynamics with the distributed dispersion, self-phase modulation, self-steepening, and linear gain/loss. Nonautonomous rogue waves for the inhomogeneous MNLS equation are constructed via the modified Darboux transformation with the inhomogeneous parameters. The dynamical behaviours and main characteristics of the nonautonomous rogue waves in inhomogeneous fibers are discussed by analysing certain physical quantities. The inhomogeneous effects on the evolution of rogue waves are considered by virtue of the presented rational solutions with distributed coefficients. It is found that the group velocity dispersion (GVD) and linear gain/loss coefficients have effects on the trajectories and amplitudes of the rogue waves, respectively. Additionally, an intriguing composite rogue wave of the inhomogeneous MNLS equation is revealed, owing to the proper choice of the GVD coefficient. Our results could be useful for controlling the rogue waves in the dispersion-managed fiber system.