Nonlinear Modulational Instability Research Articles

On nonlinear wave structures, stability analysis and modulation instability of the time fractional perturbed dynamical model in ultrafast fibers

On nonlinear wave structures, stability analysis and modulation instability of the time fractional perturbed dynamical model in ultrafast fibers

Numerical simulation of freak wave generation and evolution process based on SPH method

With the occurrence of maritime accidents all over the world, the study of extreme waves has been promoted. These extreme waves are often called freak waves in the scientific community. At present, a variety of mechanisms are proposed to explain freak waves in the marine environment, such as directional focusing, dispersive focusing, and nonlinear modulation instability. At first, a hydrodynamic model is established based on the Smoothed Particle Hydrodynamics (SPH) method in the C# environment, and the regular wave generated by the piston wave generator is numerically simulated. Secondly, the simulation results were analyzed and visualized by using Paraview software, compared with physical experimental datas to verify the reliability of this model. Then, the generation and evolution of freak waves were numerically simulated through the movement of piston wave generators on both sides of a 2D numerical wave flume, and the influence of frequency and amplitude was analyzed by using directional focusing. The results show that if the frequency is smaller and the amplitude is larger, it is more likely to generate freak waves and wave-breaking phenomena. This study lays a foundation for the research of freak waves in the future.

Experimental tweaking of symmetry breaking in recurrent nonlinear modulational instability

Experimental tweaking of symmetry breaking in recurrent nonlinear modulational instability

Excitation of mirror symmetry higher-order rational soliton in modulation stability regimes on continuous wave background.

We study the relationship between the structures of the nonlinear localized waves and the distribution characteristics of the modulation stability regime in a nonlinear fiber with both third-order and fourth-order effects. On the background frequency and background amplitude plane, the modulation stability region consists of two symmetric curves on the left and right and a point on the symmetry axis. We find that the higher-order excitation characteristics are obviously different at different positions in the modulation stability region. Their excitation characteristics are closely related to the modulation instability distribution characteristics of the system. It is shown that asymmetric high-order rational solitons are excited at the left and right stable curves, and the symmetric one is excited at the stable points. Interestingly, the asymmetric higher-order rational solitons on the left and right sides are mirror-symmetrical to each other, which coincides with the symmetry of the modulation instability distribution. These results can deepen our understanding of the relationship between nonlinear excitation and modulation instability and enrich our knowledge about higher-order nonlinear excitations.

Generation mechanism and prediction of an observed extreme rogue wave

Rogue waves are individual ocean surface waves with crest height eta or trough-to-crest height H that are large compared to the significant wave height H_s of the underlying sea state: H/H_s>2.2 or eta /H_s>1.25. The physics of rogue wave generation and the potential of predicting the rogue wave risk are open questions. Only a few rogue waves in high sea states have been observed directly, but they can pose a danger to marine operations, onshore and offshore structures, and beachgoers. Here we report on a 17.6m high rogue wave in coastal waters with eta /H_s=1.98 and H/H_s=2.9 which are likely the largest normalized heights ever recorded. Simulations of random superposition of Stokes waves in intermediate water depth show good agreement with the observation. Non-linear wave modulational instability, a well known cause for rogue waves in laboratory settings, did not contribute significantly to the rogue wave generation. A parameter obtained from a routine spectral wave forecast provides a practical risk prediction for rogue waves. These results confirm that probabilistic prediction of oceanic rogue waves based on random superposition of steep waves are possible and should replace predictions based on modulational instability.

Influence of perturbation amplitude and perturbation frequency on Fermi-Pasta-Ulam-Tsingou recurrence phenomenon

Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomenon refers to the property of a multimode nonlinear system returning to the initial states after complex stages of evolution. The FPUT recurrence phenomenon closely links with modulation instability (MI) by employing the perturbed continuous waves as the initial condition. When the perturbation frequency is located inside the MI spectra, then the perturbed CWs are unstable and the perturbations will grow up with evolution. This nonlinear MI evolution results in the FPUT phenomenon. In this work, we explore in detail the effects of perturbation amplitude and perturbation frequency on the FPUT recurrence phenomena numerically, which has never been studied systematically, to the best of our knowledge. Using the results of our studies, we find that the perturbation amplitude can significantly affect the FPUT phenomenon. Firstly, the number of FPUT cycles is very sensitive to the perturbation amplitude. Large (small) perturbation amplitude can result in much more (much less) FPUT cycles. Secondly, very irregular (regular) FPUT wave evolution together with the corresponding spectra evolution can be observed at relatively large (small) values of perturbation amplitude, where the unequal (equal) distances are observed between adjacent maximum wave amplitudes spatially in the background of optical fibers. In contrast, the effects of perturbation frequency on the FPUT cycles are relatively minor, and the maximum FPUT cycles are observed at perturbation frequencies around the optimal modulation frequency generating the peak MI gain. However, the perturbation frequency can drastically affect the number of high-order sidebands excited at the distances of periodic maximum wave amplitude formation. We find that larger perturbation frequency leads to much fewer high-order sidebands. According to our studies, for observing FPUT conveniently and observing more FPUT cycles, the perturbation amplitude of the input signal should be as large as possible and the perturbation frequency should be around the optimum modulation frequency. We should also emphasize that the large perturbation amplitude results in irregular FPUT patterns with unequal distances between adjacent maximum wave amplitude formations spatially in the background of optical fibers, and large perturbation frequency results in much less high-order sidebands. Our results will provide very helpful information for the FPUT observation in experiment, and should arouse the interest of the readers in nonlinear physics.

Dynamics of recurrent modulational instability near its separatrix in nonlinear fiber optics

Separatrix crossing stands for the continuation of the modulation instability into the strong depleted regime and is responsible for the symmetry breaking nature of the recurrence phenomenon. The near-separatrix dynamic for recurrent modulational instability is of great importance because it is closely associated with the formation of rogue breather structures. Here, we have developed the underlying phase-space structure of nonlinear modulation instability by treating the three-mode truncate model as a simple oscillator. This allowed us to reveal the high sensitivity of the switching dynamics characterized by inner or outer trajectories to the initial condition around a separatrix. Our results present a major step forward towards the complete understanding of recurrent modulational instability in a truly conservative setting and can provide a guidance to experimentally measure the involved recursive behaviors.

Optical soliton and weierstrass elliptic function management to parabolic law nonlinear directional couplers and modulation instability spectra

Optical soliton and weierstrass elliptic function management to parabolic law nonlinear directional couplers and modulation instability spectra

Ghost Interaction of Breathers

Mutual interaction of localized nonlinear waves, e.g., solitons and modulation instability patterns, is a fascinating and intensively-studied topic of nonlinear science. Here we report the observation of a novel type of breather interaction in telecommunication optical fibers, in which two identical breathers propagate with opposite group velocities. Under controlled conditions, neither amplification nor annihilation occurs at the collision point and most interestingly, the respective envelope amplitude, resulting from the interaction, almost equals another envelope maximum of either oscillating and counterpropagating breather. This ghost-like breather interaction dynamics is fully described by anN-breather solution of the nonlinear Schrödinger equation.

Dynamics of solitons with periodic loops intrinsic localized modes and modulational instability in a quantum 2D nonlinear square Klein-Gordon chain

Abstract We investigate the effects of dispersion and nonlinearity interactions in view of generation of nonlinear intrinsic localized modes (ILMs) and modulational instability (MI) of a quantum 2D Klein-Gordon chain. For this, using Glauber’s coherent state representation and multiple-scale method, we find that the system supports nonlinear excitations referred to bright, dark radial symmetric and bright, dark bilateral symmetric line non-traveling and traveling ILMs. We also find that moving bright and radial solution are evolving like localized periodic loops, which is very interesting in this work. By means of linear stability approach, the instability criteria, threshold amplitude, instability areas showing the generation of bright ILMs predicted to be unstable and growth rate of unstable modes in the system are obtained. Numerical investigations are performed to support the analytical analysis, and an excellent agreement with the analytical results has been found.

Modulation instability and rogue wave spectrum for the generalized nonlinear Schrödinger equation

We discuss modulation instability for the generalized nonlinear Schrödinger equation based on nonzero background wave frequency. First of all, we analyze the existence condition of modulation instability under different perturbed frequency. The influences of the background amplitude, background frequency and perturbed frequency on the modulation instability gain are researched, respectively. Also, we obtain the correspondences between several nonlinear excitations (Kuznetsov-Ma breather, general breather, rogue wave, bright soliton and plane wave) and modulation instability according to new parameters. Furthermore, by the Fourier transformation method, we perform spectrum analyses of the first-order and second-order rogue waves. The perturbed frequency of the rogue wave can affect the location and profile of the spectrum. And we find that the spectrum of the second-order rogue wave is jagged due to the collision of the rogue waves. These results would help us further understand the dynamics of rogue wave in complex systems.

Transformed nonlinear waves, state transitions and modulation instability in a three-component AB model for the geophysical flows

In this paper, we investigate a three-component AB model, which characterizes the baroclinic instability processes in the geophysical flows. Via the Darboux transformation, the breather solutions are derived. Then, we study the state transition and find that the breather solutions can be transformed into different kinds of stationary nonlinear waves, including the anti-dark soliton, multi-peak soliton, M-shaped soliton, W-shaped solitons and periodic waves. Moreover, by virtue of the second-order transformed solution, various nonlinear wave complexes are presented. Finally, we unveil the relationship between the modulation instability and state transition and show the existence regions for the transformed waves.

Modulational Instability in the Interaction of Fast and Slow Magnetosonic Modes

The present paper studies the magnetic filaments produced by the interaction between the high-frequency magnetosonic wave and low-frequency magnetosonic wave. These phenomena can be explained by using an envelope solitary wave solution of the nonlinear Schrodinger equation (NLSE). The analytical results from NLSE are in agreement with the numerical result from Ref. Modi et al. (Phys. Plasmas 21: 102305, 2014). Both the linear and the nonlinear modulational instability are studied for the system. Two critical points are obtained. The first critical point gives the regions of both the instability and the stability, while the second one shows that if the perturbed wave number is close to the second critical point, the perturbed wave may be the most stable one. It is also found that all the higher order perturbed waves can be neglected in the stability region when the perturbed wave number is larger than that of the first critical point.

Quantitative relations between fundamental nonlinear waves and modulation instability

Nonlinear waves are ubiquitous in various physical systems, and they have become one of the research hotspots in nonlinear physics. For the experimental realization, observation and application of nonlinear waves, it is very important to understand the generation mechanism, and determine the essential excitation conditions of various nonlinear waves. In this paper, we first briefly review the experimental and theoretical research progress of nonlinear waves in recent years. Based on the exact nonlinear wave solutions and linear stability analysis results, we systemically discuss how to establish the quantitative relations between fundamental nonlinear waves and modulation instability. These relations would deepen our understanding on the mechanism of nonlinear waves. To solve the excitation conditions degenerations problem for some nonlinear waves, we further introduce the perturbation energy and relative phase to determine the excitation conditions of nonlinear waves. Finally, we present a set of complete parameters that can determine the excitation conditions of nonlinear waves, and give the excitation conditions and phase diagrams of the fundamental nonlinear waves. These results can be used to realize controllable excitation of nonlinear waves, and could be extended to many other nonlinear systems.

Experimental realization of Fermi-Pasta-Ulam-Tsingou recurrence in a long-haul optical fiber transmission system

The integrable nonlinear Schrödinger equation (NLSE) is a fundamental model of nonlinear science which also has important consequences in engineering. The powerful framework of the periodic inverse scattering transform (IST) provides a description of the nonlinear phenomena modulational instability and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. It associates the complex nonlinear dynamics with invariant nonlinear spectral degrees of freedom that may be used to encode information. While optical fiber is an ideal testing ground of its predictions, maintaining integrability over sufficiently long distances to observe recurrence, as well as synthesizing and measuring the field in both amplitude and phase on the picosecond timescales of typical experiments is challenging. Here we report on the experimental realization of FPUT recurrence in terms of an exact space-time-periodic solution of the integrable NLSE in a testbed for optical communication experiments. The complex-valued initial condition is constructed by means of the finite-gap integration method, modulated onto the optical carrier driven by an arbitrary waveform generator and launched into a recirculating fiber loop with periodic amplification. The measurement with an intradyne coherent receiver after a predetermined number of revolutions provides a non-invasive full-field characterization of the space-time dynamics. The recurrent space-time evolution is in close agreement with theoretical predictions over a distance of 9000 km. Nonlinear spectral analysis reveals an invariant nonlinear spectrum. The space-time scale exceeds that of previous experiments on FPUT recurrence in fiber by three orders of magnitude.

Quantitative approach to breather pair appearance in nonlinear modulational instability

We investigate the generation of symmetric and asymmetric pairs of breathers from localized disturbances in the nonlinear stage of modulational instability governed by the nonlinear Schrödinger equation. An asymptotic matching approach allows us to predict with great accuracy how their emergence and evolution depend on the parameters of the perturbation. The same approach turns out to be applicable to other classes of perturbations which, due to their different decays in time, do not give rise to breathers.

Study of soliton solutions of higher-order nonlinear Schrödinger dynamical model with derivative non-Kerr nonlinear terms and modulation instability analysis

Abstract In this work, soliton solutions of higher order non-linear Schrodinger equation (NLSE) with derivative non-Kerr non-linear terms are constructed by employing two analytical techniques, namely, exp ( - Φ ( ζ ) ) -expansion and improved simple equation methods. This dynamical model plays a key role in engineering and physics, and it can be equation of pulses promulgation behind ultra-short range in the system of optical communication. The constructed soliton solution helps researchers for understanding the physical phenomenon of this equation. The standard linear stability analysis is utilized and the stability of model is investigated which substantiate that all results are stable and exact. Graphically, the movements of some solutions are depicted at appropriate values of parameters. The achieved results shows simplicity, reliability and power of the current schemes.

Nonlinear Modulational Instability of Dispersive PDE Models

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schr\"odinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumption on the dispersive operator, we use a bootstrap argument to overcome the loss of derivative.

Growth rate of modulation instability driven by superregular breathers.

We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schrödinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.

Nonlinear space charge dynamics and modulational instability in the interaction of intense laser pulses with electron-positron plasmas

Nonlinear space charge dynamics and modulational instability in the interaction between ultrashort, intense laser pulses and electron-positron pair plasmas are investigated taking into account the relativistic ponderomotive force and the relativistic mass of electrons and positrons. By coupling Maxwell's equations and hydrodynamic model, the electron and positron density distributions and the dispersion relation for the modulational instability are obtained. Moreover, two coupled nonlinear equations for the scalar and vector potentials are derived and numerically solved. The results show that the growth rate of instability increases with the decrease in the electron and positron temperatures. Moreover, it is shown that when the temperatures of electrons and positrons are not equal to each other, the profiles of scalar potential are similar to bright-like or dark-like solitons.