Modulation Instability Gain Spectrum Research Articles

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.

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.

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.

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.

Asymmetric modulation instability in nonlinear metamaterial waveguides

In this paper, we explore the formation of asymmetric modulation instability spectrum in the metamaterials as a result of the joined effect of self-steepening and intra-pulse Raman scattering. In general, the modulation instability gain spectrum is symmetric about the zero perturbation frequency. Here we observe asymmetric modulation instability gain spectrum and the asymmetry depends on the sign of the refractive index of the medium. When the refractive index of the medium is negative the band with high modulation instability gain is observed at positive detuning frequency regime in contrast to the case of positive refractive index regime where the band with high modulation instability gain is present at a negative detuning frequency regime. This preponderance is attributed to the opposite directionality of phase velocity and energy flow in a negative index material. Our study provides additional ways to tune soliton and ultrashort pulses utilizing engineering freedom of metamaterials.

Modulational instability in PT -symmetric Bragg grating structures with saturable nonlinearity

We investigate the nontrivial characteristics of modulational instability (MI) in a system of Bragg gratings with saturable nonlinearity. We also introduce an equal amount of gain and loss into the existing system, which gives rise to an additional degree of freedom, due to the concept of $\mathcal{PT}$ symmetry. We obtain the nonlinear dispersion relation of the saturable model and discover that such dispersion relations for both the conventional and $\mathcal{PT}$-symmetric cases contradict the conventional Kerr and saturable systems by not displaying the typical signature of loop formation in either the upper branch or lower branch of the curve drawn against the wave number and detuning parameter. We employ a standard linear stability analysis to study the MI dynamics of the continuous waves perturbed by an infinitesimal perturbation. The main objective of this paper is twofold: We investigate the dynamics of the MI gain spectrum at the top and bottom of the photonic band gap followed by a comprehensive analysis carried out in the anomalous and normal dispersion regimes. As a result, this perturbed system driven by the saturable nonlinearity and gain or loss yields a variety of instability spectra, which include the conventional sidebands, monotonically increasing gain, the emergence of a single spectrum in either of the Stokes wave-number regions, and so on. In particular, we observe a remarkably peculiar spectrum, which is caused predominantly by the system parameter, though the perturbation wave number boosts the former. We also address the impact of all the physical parameters considered in the proposed model, which include the coupling coefficient, dispersion parameter, and saturable nonlinearity on the phenomenon of MI for different $\mathcal{PT}$-symmetric regimes ranging from the unbroken one to the broken one in greater detail.

Non-collisional dynamics of nonautonomous three solitons through tailoring of modulated coefficients and modulation instability gain spectra

In this paper, the generalized nonautonomous nonlinear Schrödinger equation with non-isospectral parameter is investigated from the point of view of soliton management through modulation of dispersion and nonlinearity. Based on the obtained analytic soliton solutions, the effect of modulated coefficients such as dispersion and nonlinearity on the control of soliton transmission is studied. Especially, controllable soliton interactions are investigated through suitably tailoring the inhomogeneous profiles. Additionally, modulation instability (MI) has been analyzed graphically through linear stability analysis (LSA) and MI gain spectrum is analyzed for various forms of modulated coefficients. Through this study, we confirm that the modulated coefficients in the theoretical model under investigation affect the amplitudes, widths and trajectories of the nonautonomous solitons. Thus, one can obtain the optical control system by choosing the different form of modulated coefficients for the specific problem. Results obtained in this paper would be worthy in the field of soliton management for improving the capacity of optical fiber communication system.

Explicit propagating electrostatic potential waves formation and dynamical assessment of generalized Kadomtsev–Petviashvili modified equal width-Burgers model with sensitivity and modulation instability gain spectrum visualization

The major motive of this study is to analyze the nonlinear integrable model which is generalized Kadomtsev–Petviashvili modified equal width-Burgers equation. It can be utilized extensively a weakly non-linear restoring forces, dispersion, small damping and nonlinear media with dissipation to narrate the long wave propagation in chemical theory. This article allocates the partial differential equation by traveling waves transformation into an ordinary differential equation. In order to acquire the analytical propagating structures, one of the generalized techniques, new extended direct algebraic methodology utilizers. As a consequence, we establish the mixed singular solution, singular solution, mixed shock-singular solution, mixed complex solitary-shock solution, mixed periodic results, mixed trigonometric results have been derived in the formation of a mixed periodic and periodic class, the mixed hyperbolic solution, plane solution, which is derived via Mathematica. The Chaos investigation is carried out to envision the dynamical insights of ocean wave integrable model. The sensitive analysis performed to verify the perceptiveness of model regarding parameters and initial conditions. Modulational instability gain spectrum developed and envisaged with appropriate parametric values and ensured the stability of the considered model. In addition, two-dimension, three-dimension, and contour surfaces are embellished to validate the physical properties of the derived solutions. The developed electro potential soliton structures can reveal the deep atomic insights. The dynamics of physical phenomenon can be controlled by fractional parameter.

The Enhancement of Energy-Carrying Capacity in Liquid with Gas Bubbles, in Terms of Solitons

A generalized (3 + 1)-dimensional nonlinear wave is investigated, which defines many nonlinear phenomena in liquid containing gas bubbles. Basic theories of the natural phenomenons are usually described by nonlinear evolution equations, for example, nonlinear sciences, marine engineering, fluid dynamics, scientific applications, and ocean plasma physics. The new extended algebraic method is applied to solve the model under consideration. Furthermore, the nonlinear model is converted into an ordinary differential equation through the next wave transformation. A well-known analytical approach is used to obtain more general solutions of different types with the help of Mathematica. Shock, singular, mixed-complex solitary-shock, mixed-singular, mixed-shock singular, mixed trigonometric, periodic, mixed-periodic, mixed-hyperbolic solutions are obtained. As a result, it is found that the energy-carrying capacity of liquid with gas bubbles and its propagation can be increased. The stability of the considered model is ensured by the modulation instability gain spectrum generated and proposed with acceptable constant values. Two-dimensional, three-dimensional, and contour surfaces are plotted to see the physical properties of the obtained solutions.

The role of septic saturation on the cross-phase instability in two-core birefringent oppositely directed PIM-NIM coupler

We examine the cross-phase modulational instability in a two-core birefringent coupler made of a positive and negative index material. The modified form of generalized nonlinear Schrödinger equation contains a nonlinear septic term, cross-phase modulation, intrapulse Raman scattering, self-steepening, and conventional type of saturation nonlinearity. We investigate normal and anomalous group-velocity dispersion regimes and cases of birefringent effects such as zero-birefringence, linear birefringence, and circular birefringence. We observe that the MI gain spectrum shows significant changes in the presence of cross-phase modulation with a nonlinear saturation effect and non-Kerr nonlinearity. In addition, the self-steepening brings out new sidebands and shifts the existing sidebands to bring them together, providing a new pathway to generate solitons or ultrashort pulses in the presence of nonlinear septic saturation. The intrapulse Raman scattering effect causes conventional and Raman bands, extending the modulational instability (MI) domain range. The saturable nonlinearity suppresses the MI gain and reduces the bandwidth of the spectrum for both normal and anomalous dispersion regimes. But, in the case of nonlinear saturation with non-Kerr nonlinearity, the MI gain spectra exhibit efficient modification in a two-core birefringent oppositely directed coupler (ODCs) with adjustable effects of SS, IRS, and cross-phase modulation. Therefore, manipulating MI and solitons in a two-core ODCs is better performed when the birefringence is linear rather than circular. Here we present how to control and generate the MI and solitons in birefringent ODCs, emphasizing the negative index material (NIM) channel.

Omnipresent instability criterion of birefringent Kundu-Eckhaus model with nonic nonlinearity

This paper intends to present an investigation of the ultra shortpulse propagation under the nonlinear birefringent Kundu-Eckhaus (KE) model. The dynamics of modulational instability(MI) is revealed with the linear stability analysis of the KE equation with higher-order, the cubic, and nonic nonlinearity effects. The role of cubic-nonic nonlinearity on MI gain spectra in KE fiber system with self and cross phase modulation is studied for the first time. The impact of nonic nonlinearity is shown to be much more prominent over its cubic-quintic counterpart. The presence of cubic-nonic combo nonlinearity lets the MI gain profile to be adjusted with the rate of self-phase modulation (SPM). This new MI gain growth dynamics controlled by the SPM parameter will be useful on soliton generation in birefringent KE fiber media.

Impact of higher order dispersions and nonlinearities on instability criterion of oppositely directed coupler in the presence of negative index material channel

This paper intends to investigate the modulational instability (MI) criterion of two core positive-negative index coupler channel setup from the obtained analytical dispersion relation for MI gain profile. The governing equation, the modified nonlinear Schrodinger equation (MNLSE) is studied for the role of higher order dispersion (HOD) and higher order nonlinearity (HON) on instability spectra of two core positive indexed coupler (PIM) and negative indexed coupler (NIM) arrangement. The HOD comprises of the second, third, fourth and fifth order linear dispersion terms and HON is inclusive of cubic, quintic, septic, cubic-quintic-septic-self-steepening (SS), and cubic-quintic-septic-intrapulse Raman scattering (IRS). Our study reveals that it is the 6th order linear dispersion term which brings in the considerable changes of MI gain spectrum in both normal and anomalous dispersion regimes. In particular, the combined effect of septic SS and IRS in two core directional coupler with NIM channel proves to be very crucial in the generation of instability gain spectra. This study has higher prospects in generation of solitary wave in two core directional system for optical fiber communication.

Specific optical solitons solutions to the coupled Radhakrishnan–Kundu–Lakshmanan model and modulation instability gain spectra in birefringent fibers

Specific optical solitons solutions to the coupled Radhakrishnan–Kundu–Lakshmanan model and modulation instability gain spectra in birefringent fibers

Influence of the next-nearest neighbor and the boson–boson interactions on U-shaped, W-shaped profile and modulation instability gain spectra in a zig–zag optical lattice

In this paper, we employ the auxiliary equation method (AEM) to show the leverage of the nearest neighbors hopping and the boson–boson interaction in a zig zag optical lattice on the stressed optical solitons (os) and the Modulation Instability (MI) gain of cold bosonic atoms. The pedestal of the work is set on the continuum approximation of a cold bosonic atoms in a zig zag optical lattice which has demonstrated in detailed in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589]. The results collected have enclosed bright, dark optical solitons and the combined bright-dark and bright-bright optical solitons. Compared the acquired results with Tala-Tebue et al. [Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44], further the os including U-shaped and W-shaped profile have been drop in and the powerful of the nearest neighbors hopping and the boson–boson interaction has been exposed across the spatiotemporal and plot evolution of the bright and dark os. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the model parameters. The collected results are new in literature in our knowledge and will improve the previous results acquired in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589; Zhao XH, Tian B, Liu DY, et al. Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct 100(2016):587–595; Uddin MF, Hafez MG, Hammouch Z, et al. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020. https://doi.org/10.1080/17455030.2020.1722331; Inc M, Aliyu AI, Yusuf A. Optical solitons to the nonlinear Shrdingers equation with spatio-temporal dispersion using complex amplitude ansatz. J Mod Opt 2017;64(21):2273–2280].

Attitude of the Modulation Instability gain in Oppositely Directed Coupler with the effects of the Intrapulse Raman Scattering and Saturable Function

To establish the effects of the Higher-Order Dispersion (HOD) and Raman Scattering (RS) on Modulation Instability (MI) in the presence of the Saturable Function (SF), it is used the Coupled Nonlinear Schrödinger Equation (CNLSE) which describes pulses propagating in a higher optical beam in oppositely guided coupler. It has been shown through the MI gain spectra the power of the SF in the presence of HOD and Raman effects (RE). By utilizing the HOD with the saturable nonlinearity (SN), it is formed three sides lobes and some new MI bands emerged. On the same way, it is observed that the RE have more potency in the presence of saturable parameter (SP) in the normal and anomalous dispersion regime. It is important to mention that by varying the SF, third-order dispersion (TOD) and Group Velocity Dispersion (GVD) in the absence of Raman parameter (RP) it is obtained three sides lobes which is an supplementary results compare to Alves et al. (2020), Guérin et al. (1995) and Mohamadou et al. (2010). These results could be important to stress the dynamics of the MI in optical fibers such as optical amplification of poor signal, supercontinuum generation, data transmission and signal processing system and very fast transmission signal and so on.

W-shaped profile and multiple optical soliton structure of the coupled nonlinear Schrödinger equation with the four-wave mixing term and modulation instability spectrum

We investigate diverse optical solitons and other solution for the Coupled Nonlinear Schrödinger Equation (CNLSE) having Four-Wave Mixing (FWM) and Kerr nonlinearity term. We stress the effects of different parameters of the model on the obtained optical soliton by employing the New Generalized Extended Direct Algebraic Method (NGEDAM). The solutions contain W-shaped profile of the bright optical soliton, dark optical soliton and diverse other solutions including the combined complex optical soliton. Interestingly, W-shaped optical soliton and dark optical soliton own a good argument for long communication and big data transmission originating from the self-steepening, Kerr nonlinearity effects and FWM. We also, emphasize the Modulation Instability (MI) gain spectra to narrow the impact of the FWM in birefringence. Thereafter, we involve the ellipticity angle effects in normal and anomalous dispersion regime on the MI gain spectra. Parametric conditions for the existence of the obtained optical soliton solutions and MI gain are given.

Modulation instability in one-dimensional bi-periodic optical lattice made of negative index metamaterials waveguide arrays

In this paper, the dispersion relation and modulation instability (MI) have been studied in one-dimensional bi-periodic optical lattice of negative index metamaterials waveguide arrays. It is found that the plateaus will be appeared on the dispersion curve when the coupling coefficient is around 0.25. The plateau will become wider with the increase of the nonlinear and longitudinal modulation depth. The phase shift has little effect on MI the gain spectrum when the values of coupling coefficient, nonlinear coefficient and longitudinal modulation depth coefficient are in the range that is conducive to the generation of plateaus. However, if the values of these coefficients are in the range that is not conducive to the generation of plateaus, the effect of phase shift on the MI gain spectrum will become significant. In addition, it is found that the inverse periodic modulation between waveguides causes the light intensity to change periodically during transmission.

Clout of fractional time order and magnetic coupling coefficients on the soliton and modulation instability gain in the Heisenberg ferromagnetic spin chain

In this paper we utilize the auxiliary equation method to show the leverage of the fractional derivative parameter and the Magnetic Coupling Coefficients (MCC) on the procured soliton solutions and the Modulation Instability (MI) gain in Heisenberg ferromagnetic spin. The pedestal of the work is set on the (2+1)-fractional time order dimensional Heisenberg ferromagnetic spin chain equation which demonstrate the propagation of nonlinear waves in Heisenberg ferromagnetic spin chain system. The results collected have enclosed bright-dark soliton, dark soliton and rational solutions. Compared the acquired results with Zhao et al. (2016) [26], Akbar (2021) [27], Hosseini et al. (2021b) [28], further soliton-like solutions have been drop in and the marvel of the fractional parameter has been exposed across the spatiotemporal and plot evolution of the bright and dark soliton solutions. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the MCC jointed with the fractional time derivative order. The collected results are new in literature in our knowledge and will give the pipe to the solitary waves theory in Heisenberg ferromagnetic spin including inhomogeneity parameters.

Solitary waves and modulation instability with the influence of fractional derivative order in nonlinear left-handed transmission line

The resolution of the reduced fractional nonlinear Schrödinger equation obtained from the model describing the wave propagation in the left-handed nonlinear transmission line presented by Djidere et al recently, allowed us in this work through the Adomian decomposition method (ADM) to highlight the behavior and to study the propagation process of the dark and bright soliton solutions with the effect of the fractional derivative order as well as the Modulation Instability gain spectrum (MI) in the LHNLTL. By inserting fractional derivatives in the sense of Caputo and in order to structure the approximate soliton solutions of the fractional nonlinear Schrödinger equation reduced, ADM is used. The pipe is obtained from the bright and dark soliton by the fractional derivatives order. By the bias of MI gain spectrum the instability zones occur when the value of the fractional derivative order tends to 1. Furthermore, when the fractional derivative order takes small values, stability zones appear. These results could bring new perspectives in the study of solitary waves in left-handed metamaterials as the memory effect could have a better future for the propagation of modulated waves because in this paper the stabilization of zones of the dark and bright solitons which could be described by a fractional nonlinear Schrödinger equation with small values of fractional derivatives order has been revealed. In addition, the obtained significant results are new and could find applications in many research areas such as in the field of information and communication technologies.