Modulation Instability Gain Research Articles

The effect of Lévy index coefficient on modulational instability and rogue wave excitation in nonlocal media with cubic–quintic nonlinearities

This paper explores the modulational instability (MI) of a plane wave and its behavior in the nonlinear Schrödinger equation (NLSE) with a fractional diffraction term quantified by its Lévy index coefficient and nonlocal cubic–quintic nonlinearities. First, we analyze the stability of the plane wave solution and examine how nonlocal nonlinearities and the Lévy index coefficient affect the MI gain. We observe that the stability in the fractional NLSE exhibits new features that differ from those in the standard NLSE. Specifically, when dealing with competing cubic and quintic nonlinearities, the interaction between nonlocality and the Lévy index coefficient α can eliminate MI for low values of α, unlike the classical NLSE with α=2, where we find the plane wave to be unstable. Besides the linear stability analysis, numerical simulations are performed to understand further the plane wave dynamics from its nonlinear stage in this model. The results reveal the generation of periodic chains of localized peaks. Guided by analytical predictions and using the plane wave solution subject to Gaussian perturbation, we numerically investigate the possibility of exciting rogue waves in the parameter spaces where MI exists. We find that the different combinations of signs of the cubic and quintic nonlinearities (focusing and defocusing) and the fractional diffraction term significantly impact the formation of rogue waves. These results may pave the way for the theoretical and experimental study of nonlinear phenomena in physical models with fractional derivatives and nonlocal nonlinearities.

Exploration on modulation instability in 𝒫𝒯-symmetric non-Kerr Bragg grating structures

In this paper, we study Modulation Instability (MI) and its correlated gain spectrum with Parity-Time ([Formula: see text])-symmetry. The cubic-quintic nonlinearities in the Bragg grating structure produce ultrashort pulses. The MI evaluation started with a linear stability approach, and we investigated the effects of various system functions regarding the instability gain spectra at the top and bottom sides of the Photonic BandGap (PBG), in addition to the anomalous and normal dispersion domains. The interaction of cubic and quintic (focusing and de-focusing) nonlinearity is highlighted, resulting in a new type of MI band. Due to the consequences of quintic nonlinearity, an additional MI band appears for the three dynamics associated with the gain/loss function for the edges of the bottom PBG. After enforcing the [Formula: see text]-symmetry for the anomalous dispersion system, the dent transformed into a split under the influence of the power in the de-focusing quintic effect. Overall, MI gain and bandwidths across the entire MI gain spectrum are usually increasing. As an outcome, we are confident that these studies will aid in the generation of gap solitons in [Formula: see text]-symmetry regimes.

Dynamic interplay: unveiling inelastic breather collisions and modulation instability enhancement in a periodically gained inhomogeneous fiber optic communication system across temporal frequencies

Examining the impact of inhomogeneity on the propagation of femtosecond ultrafast optical pulses in fiber, we delve into the realm of the modified Hirota nonlinear Schrödinger equation (NLS) with inhomogeneity of variable coefficients (MIH-vc). Employing the Hirota bilinear method, we derive two soliton solutions for the modified Hirota NLS equation and analyze the effect of variable coefficients. The dynamical properties of these soliton solutions come to light as we meticulously analyze the corresponding plots. In our exploration, a noteworthy revelation unfolds as we witness the inelastic collision between two breathers, unleashing profound changes in the trajectory of femtosecond pulses. Furthermore, we showcase a detailed modulation instability analysis, unraveling the gain spectrum for our theoretical model. Through graphical illustrations, we elucidate how inhomogeneous functions intricately shape the modulation instability (MI) gain spectrum. A groundbreaking observation surfaces as, for the first time, we discern the periodic gain enhancement in relation to Group Velocity Dispersion along the fiber and its dynamic interactions.

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

Propagation of solitary wave solutions to (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation arise in mathematical physics and stability analysis

This innovative study simulates the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation, which has anomalous applications in the field of mathematical physics. The current research has two basic pillars. Firstly, numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, periodic, combo, W-shape, mixed trigonometric, exponential, and rational, based on the modified sardar sub-equation (MSSE) method and the improved F-expansion method. Secondly, the stability analysis of the selected model is manifested to study modulation instability (MI) gain. Furthermore, for the physical demonstration of the acquired solutions in 3D and 2D, contour and density plots are depicted. The discovered results have a distinctive feature that has not been developed before and indicate a good balance between the nonlinear physical components. Also, the resulting structure of the acquired results can enrich the nonlinear dynamical behaviours of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.

Impact of weak nonlocal nonlinearity on modulational instability and chirp-free soliton trains in parabolic law optical fiber

This study considers the generation and evolution of chirp-free soliton trains in a focusing cubic-quintic nonlinear optical fiber media (by utilizing experimental parameters), under the influence of weak nonlocal nonlinearity. Analysis of modulational instability (MI) reveals that the MI gain exist only in the anomalous dispersion regime, with the quintic nonlinear term increasing the maximum gain and sideband frequencies. The nonlocal nonlinearity parameter generally suppresses the impact of MI, because the maximum gain is greatly reduced with increase in nonlocality. By using the subsidiary ordinary differential equation method, stationary periodic soliton trains are obtained; with the intensity of the optical beam greatly reduced and frequency increased as nonlocality is gradually stepped up. However, an increase in nonlocality instead reduces the frequency of the traveling optical signals; derived via an efficient transformation method. Our results strongly suggest that nonlocal nonlinearity has potential applications in the control of laser beams; as nonlinear periodic optical signals can be easily transformed to very weak quasi-plane waves.

Jacobian elliptic wave trains in fourth-order dispersive nonlinear Schrödinger equation and the modulational instability

We analytically and numerically explore the existence of explicit exact Jacobian elliptic solutions of a nonlinear Schrödinger model with fourth-order dispersion and weak nonlocality. We delineate the parameter domain in which these highly dispersive doubly periodic waves balanced by the weak nonlocality and Kerr nonlinearity exist. Additionally, we have studied the modulational instability (MI) of these waves with the aid of linear stability analysis. We observe that in the presence of fourth-order dispersion, the MI gain increases with increasing the nonlocality parameter.

Impact of viscothermal loss on modulation instability and rogue waves in left-handed nonlinear diffractive acoustic transmission line metamaterials

In this study, the transmission line approach is used to describe the studied acoustic metamaterial model. Through Kirchoff’s pressure and volume-velocity laws and using multiple scales method, nonlinear coupled Schrödinger equations are obtained. Then, the amplitude disturbance method is applied to these equations to obtain and plot the modulational instability gain curves. Analytically, the impact of viscothermal loss on the modulational instability gain is studied. The similarity technique is used to derive integrable Manakov’s equations. First and second-order rational rogue wavelike solutions of coupled nonlinear Schrödinger are deduced. The results indicate that the modulational instability gain and Rogue wave intensities depend on the viscothermal parameter. This parameter can be considered in the design of nonlinear acoustic metamaterials to minimize the damage caused by the dynamics of freak waves.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Modulation Instability and Dynamical Analysis of New Abundant Closed‐Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M‐Fractional Derivative

In this work, we study the modulation instability (MI) and closed‐form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV‐ZK) equation with a truncated M‐fractional derivative. The mKdV‐ZK equation can be used to describe the behavior of ion‐acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F‐expansion scheme and unified solver techniques for the mKdV‐ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three‐dimensional (3D) plots with density plots and two‐dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed‐form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F‐expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell‐shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena.

Parametric instability in the pure-quartic nonlinear Schrödinger equation

We study the nonlinear stage of modulation instability (MI) in the non-intergrable pure-quartic nonlinear Schrödinger equation where the fourth-order dispersion is modulated periodically. Using the three-mode truncation, we reveal the complex recurrence of parametric resonance (PR) breathers, where each recurrence is associated with two oscillation periods (PR period and internal oscillation period). The nonlinear stage of parametric instability admits the maximum energy exchange between the spectrum sidebands and central mode occurring outside the MI gain band.

Solitonic rogue waves induced by the modulation instability in a split-ring-resonator-based left-handed coplanar waveguide

Modulation instability and nonlinear excitation modes in lower and upper frequency bands are investigated in the nonlinear left-handed coplanar waveguide, where nonlinear capacitance (voltage-dependent) is used. The multi-scale method is employed to derive a system of two coupled nonlinear Schödinger equations governing the evolution of the kink and bright solitons of these modes. A modulation instability is investigated to show the effects of the left-handed and right-handed influences on the modulation instability gain and the intensity of the plane wave. In the lower and upper frequency modes, the breathing structures have strong values for the left-handed component. At the end, the numerical simulation of the continuous wave is used to show the development of the modulated waves and localized objects. Another relevant localized structure has been obtained to display the second-order breathers and Akhmediev breathers of types A and B under the variation of the perturbed wave number. Both linear inductances and perturbed wave numbers are tools for controlling the amplitude of the localized structures, and modulation instability is sensitive to these parameters.

Effects of coupling coefficient dispersion on the Fermi-Pasta-Ulam-Tsingou recurrence in two-core optical fibers

In a two-core optical fiber (TCF), the linear coupling coefficient between the two cores in general varies with the optical wavelength, a phenomenon known as coupling coefficient dispersion (CCD). Here we study the effects of CCD on the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena in a TCF. Modulation instability tends to be a precursor in the formation of breathers. The subsequent decay of breathers and re-emergence of modulation instability lead to FPUT type periodic patterns. For the symmetric or antisymmetric continuous-wave (CW) state of a TCF, CCD does not affect the modulation instability gain spectra. However, if asymmetric perturbations are applied to a symmetric CW state, CCD can drastically modify FPUT. In the anomalous dispersion regime, without CCD, only FPUT dynamics without energy transfer between the two cores occurs. With CCD effects, FPUT dynamics with strong energy transfer between the two cores can arise. As CCD increases, FPUT dynamics without energy transfer never vanishes, but the corresponding number of FPUT cycles can be drastically reduced to only one. In the normal dispersion regime, no FPUT dynamics with energy transfer between the two cores occurs. The major CCD effect is to significantly decrease the FPUT cycles, and only one FPUT cycle can be observed at relatively large magnitude of CCD values. As the magnitude of CCD increases further, more FPUT cycles arise again. Moreover, larger CCD will lead to higher (lower) propagation speed of the wave patterns in the anomalous (normal) dispersion regime respectively. To enhance theoretical insight, we perform a Floquet analysis for the linearized stability equations with periodic coefficients based on the FPUT dynamics in the normal dispersion regime. The monodromy matrix and eigenvalues can be computed explicitly. Larger eigenvalues correlate with stronger instability and more rapid disintegration of the periodic patterns, agreeing well with the trends of the numerical simulations.

Modulation instability in long distance φ-OTDR system

Recent trends in long distance phase-sensitive optical time-domain reflectometer (φ-OTDR) have led to a proliferation of studies on modulation instability (MI). However, most of these studies overlook the inaccuracies of MI gain spectrum based on the linear stability analysis of the nonlinear Schrödinger equation in the case of long distance over about 10 km. Here, we conducted a comprehensive exploration of MI characteristics in the φ-OTDR system, including the effects of input power, detection distance, and background noise on the MI evolution process. Experiments confirmed the presence of an intrinsic spectral component of MI surrounding the signal in long-distance detection, which has not been reported previously. Further, we proposed a fitting expression using numerical simulation to determine the optimal input power for the φ-OTDR system. Compared to traditional threshold formulas, it does not require complex theoretical derivations and demonstrated significantly higher accuracy, particularly in systems with ultra-long detection distances. The deviation was less than 10%, demonstrating its potential as a more precise and simplified method of determining optimal input power. This could help improve the efficiency and reliability of these systems, leading to further advancements in the field of MI research.

Dynamical behaviour of solitons and modulation instability analysis of a nonautonomous [formula omitted]-dimensional nonlinear Schrödinger equation

This study uses a semi-inverse variational technique to investigate the unknown dynamical behaviour of optical solitons in the solutions of a nonautonomous (1+1)-dimensional nonlinear Schrödinger (NLS) equation. This method produces the bright, anti-bright, and bell-shaped solitons that depend on the delicate balance among the time-dependent spatiotemporal dispersion (STD), group velocity dispersion (GVD), and self-phase modulation (SPM). It is obtained that the soliton solution exists only for non-zero SPM, although the soliton velocity and frequency do not depend on SPM. The soliton amplitude and intensity depends explicitly on time-dependent dispersion and self-phase modulation of the medium. It is found that the focusing medium has a remarkable impact on the intensity of the soliton in comparison to the defocusing, linear, and quadratic nonlinearity. The modulation instability (MI) analysis of the NLS equation has also been investigated using standard linear stability analysis. The explicit dispersion relation has been derived, and MI gain is discussed using different STD, GVD, and SPM coefficients as test functions of time t. The MI gain varies rapidly with the perturbation wave number at a fixed time. The 3D and contour plot of MI gain show that the stability of the NLS equation can be managed significantly by using the time-dependent spatiotemporal, group velocity dispersion, self-phase modulation, wave number, and initial incidence power.

Tunable modulation instability spectra in negative index materials

This paper investigates the formation of tunable modulation instability windows in metamaterials. We consider the influence of third-order dispersion, fourth-order dispersion, cubic nonlinearity, quintic nonlinearity, detuning parameter, self-steepening and higher-order nonlinear dispersions on the gain spectrum in addition to the basic requirement to observe modulation instability such as group velocity dispersion and self-phase modulation. As the metamaterials provide engineering freedom, all the material parameters are tunable, which leads to tunable modulation instability gain spectrum formation. This tunable gain spectrum provides additional ways to form solitons and ultrashort pulses with desired parameters.

Modulated wave patterns brought by higher-order dispersion and cubic–quintic nonlinearity in monoatomic chains with anharmonic potential

This work investigates the modulation instability and wave patterns in the extended nonlinear Schrödinger equation with higher-order dispersion and cubic–quintic nonlinearity. We use the one-dimensional monoatomic chain with anharmonic potential to derive the discrete model. From the multiple scale method combined with a quasidiscreteness approximation, we derive the cubic–quintic nonlinear schrödinger equation, and thereafter an expression of the modulation instability gain is obtained by using a linearized expression. One notices that the modulation instability growth is sensitive to the higher-order dispersion and nonlinearities terms. A split-step Fourier method is used to assess the analytical predictions. A long evolution of the continuous wave is shown to lead to the formation of the bright soliton, and Akhemediev breathers also emerge to manifest the modulation instability. We have also demonstrated that the excitation wave number generates the train of waves to confirm the fact that the continuous wave can grow exponentially with any value of the latter. We mention equally that the model of the extended cubic–quintic nonlinearity with complex envelope has opened new features of the modulated wave patterns in monoatomic chains.

Formation of Rogue Waves and Modulational Instability with Zero-Wavenumber Gain in Multicomponent Systems with Coherent Coupling.

It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Starting with the Bers-Kaup-Reiman system for three-wave resonant interactions, we identify a specific RW-building mechanism based on MI which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by MI in other multicomponent systems-in particular, in the massive Thirring model.

Optical soliton, linear stability analysis and conservation laws via multipliers to the integrable Kuralay equation

This paper studies the integrable Kuralay equation which is a kind of integrable spin system in (1＋1) dimensions. Several new exact solutions of different kinds such as optical solitons, solitary waves and elliptic function solutions of this equation are retrieved by utilizing two effective techniques, namely, the modified F-expansion method and the new extended auxiliary equation method. The modulation instability (MI) analysis of the model is studied by using the linear-stability analysis and the MI gain spectrum is acquired. Moreover, the multiplier approach is applied to retrieve the conservation laws. At last, physical interpretations of the obtained results are demonstrated.

Effects of the higher-order dispersion on solitary waves and modulation instability in a monomode fiber

In this paper, we investigate the modulation instability and the dynamics of solitary waves in a higher-order nonlinear Schrödinger equation. We use a powerful and robust method known as the modified sub-equation method to secure the solitary waves and other numerous solutions to this nonlinear model. Several constraint conditions that guarantee the existence of these solutions are highlighted. Utilizing a linearizing technique, we establish the modulation instability gain. The influence of the higher nonlinear component on the modulation instability is also discussed. Furthermore, we perform the numerical simulations of this model by using the split-step Fourier method. Finally, several solutions are shown in two and three dimensions to better explain the behavior of the considered model. The two-dimensional plots depict Peregrine soliton under the influence of the nonlinear dispersion term. These graphs are useful for understanding the dynamic properties of the obtained results.