Modulation Instability Spectrum Research Articles

Wave patterns of the coupled nonlinear Schrödinger equations in photonic crystal fibers with four-wave mixing

Abstract In this paper, we examine the behavior of modulation instability within photonic crystals. The model employed is the coherent coupled nonlinear Schrödinger equation, incorporating weak birefringence and four-wave mixing, which arises at the edge of the optical mode. The linear analysis is used to derive the modulation instability spectrum. Throughout the modulation instability spectrum, we identify both stable and unstable modes, thereby confirming the breakdown of the plane wave. For certain four-wave mixing parameters, the amplitude of the modulation instability spectrum and its bandwidths expand, creating an opening for localized structures to emerge. Another aspect of this study has been demonstrated in normal and anomalous dispersion regimes where an increasing initial amplitude of the plane wave is fulfilled. Specifically, numerical simulations highlight the occurrence of Benjamin-Feir instability, where wave patterns emerge under the influence of four-wave mixing. Additionally, solitonic waves are generated, demonstrating the presence of Akhmediev breathers and other modulated structures, confirming that photonic crystals with four-wave mixing are conducive to these formations. The findings from this study could inform future research in the development of nonlinear photonic waveguides.

Rogue and solitary waves of a system of coupled nonlinear Schrödinger equations in a left-handed transmission line with second-neighbors coupling

The effects of the second-order neighbor interaction have been effective on the coherent localized waves and modulation instability spectrum. A coupled nonlinear Schrödinger equation is derived in the nonlinear left-handed electrical lattice, and the linear stability is used to formulate the modulation instability spectrum expression. Unstable modes are displayed to show symmetric lobes and increasing bandwidths under the influence of the neighbor coupling. The Benjamin-Feir instability has prospered in the network, and for weak perturbed wave numbers, the unstable modes increase. The numerical simulation is used to develop localized waves, including rogue waves with one crest and two humps, Akhmediev breathers, and other modulated structures. We notice that, for strong values of the perturbed wave number, an exponential growth of the continuous wave arises to confirm the fact that the nonlinear left-handed electrical lattice with coupling strength is opened to coherent localized waves. The long-lived nature of the obtained structures has also been demonstrated for specific times of propagation.

Modulation instability spectrum and rogue waves of the repulsive lattices

Modulation instability spectrum and rogue waves of the repulsive lattices

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, Analytical, and Numerical Studies for Integrable Time Fractional Nonlinear Model through Two Explicit Methods

The present work deals with the investigation of the time‐fractional Klein–Gordon (K‐G) model, which has great importance in theoretical physics with applications in various fields, including quantum mechanics and field theory. The main motivation of this work is to analyze modulation instability and soliton solution of the time‐fractional K‐G model. Comparative studies are investigated by β‐fraction derivative and M‐fractional derivative. For this purpose, we used unified and advanced exp(−ϕ(ξ))‐expansion approaches that are highly important tools to solve the fractional model and are used to create nonlinear wave pattern (both solitary and periodic wave) solutions for the time‐fractional K‐G model. For the special values of constraints, the periodic waves, lumps with cross‐periodic waves, periodic rogue waves, singular soliton, bright bell shape, dark bell shape, kink and antikink shape, and periodic wave behaviors are some of the outcomes attained from the obtained analytic solutions. The acquired results will be useful in comprehending the time‐fractional K‐G model’s dynamical framework concerning associated physical events. By giving specific values to the fractional parameters, graphs are created to compare the fractional effects for the β‐fraction derivative and M‐fractional derivative. Additionally, the modulation instability spectrum is expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third‐order dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to theoretical physics. This work provided the effect of the recently updated two fraction forms, and in the future, we will integrate the space–time M‐fractional form of the governing model by using the extended form of the Kudryashov method. Maple 18 is utilized as the simulation tool.

W-chirped solitons and modulated waves patterns in parabolic law medium with anti-cubic nonlinearity

In this work, investigation of the W-chirped solitons and modulation instability are addressed in nonlinear optic fibers where cubic-quintic and parabolic law parameters are used. For this reason, the modified cubic-quintic complex Ginzburg–Landau equation with parabolic law nonlinearity is employed. Using the sub-ODE method, the chirp component and combined chirped soliton solutions, including rational soliton solutions, are obtained. Using specific parameters of the nonlinear structure, one can identify dark and bright solitons. For a suitable constant parameter of the chirp component, the W-chirped bright as well as the combined bright-dark solitons are displayed via the 3D plots. The linear stability analysis of the plane wave is used to derive an expression of the modulation instability spectrum that is employed to show the effects of the nonlinear parameters in normal and anomalous dispersion regimes. It is pointed out that both cubic and quintic parameters can enlarge or reduce the bandwidth of the modulation instability. Through the numerical simulation of the continuous wave, it becomes apparent that for suitable optical fiber parameter values, modulated wave objects are formed to exhibit mild oscillations. For specific wavenumber excitation, another modulated wave object is depicted to show how the unstable mode can switch to the stable mode for the long-time evolution of the continuous wave. The obtained results could certainly open a new future for signal treatment in optical fiber, where parabolic law is considered.

Modulation instability, bifurcation analysis and solitonic waves in nonlinear optical media with odd-order dispersion

In this paper, modulation instability, bifurcation analysis, and soliton solutions are investigated in nonlinear media with odd-order dispersion terms. A generalized nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity is considered. A linear analysis method is used to derive an expression of the modulation instability spectrum, and the effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. The results show that for negative values of the fourth order in a normal dispersion regime, the modulation instability vanishes. Another important aspect is revealed when the fourth-order dispersion gets positive and has high values; additional bands emerge to enlarge the bandwidths of the modulation instability. To further confirm the effects of the odd-order dispersion in the nonlinear structure, the bifurcation, phase portraits, and chaotic behaviors have been investigated to show how instability arises in the nonlinear structure. Using numerical simulation, in particular the Runge-Kutta algorithm, the sensitivity of nonlinear systems is pointed out to confirm an unstable behavior. This investigation confirms once again the fact that the modulation spectrum is sensitive to higher-order dispersion in normal and anomalous dispersion regimes. A traveling wave hypothesis is employed to lead to the direct integration of the nonlinear system, and some specific soliton solutions are extracted. For particular constraint conditions on the discriminant, bright and dark solitons as well as Jacobi elliptic function solutions emerge. The obtained results could be used to improve the transmission signal via the optical fiber and secure the data in communication systems.

Modulation instability and modulated wave patterns in a nonlinear electrical transmission line with the next-nearest-neighbor coupling

In this paper, we investigate modulation instability and solitary waves in a one-dimensional nonlinear electrical transmission line with second-neighbor interactions. Through the quasi-discrete multiple scale method, we derive coupled nonlinear Schrödinger equations. To investigate the modulation instability in the structure, we use the linear stability analysis, and an expression for the modulation instability spectrum is derived. From the analytical investigation, we show that the second neighbor coupling affects both the modulation instability growth rate and modulation instability bands. Furthermore, we show via single and coupled mode excitation that bright and dark solitary waves can propagate at the lower and upper cutoff frequencies. To evaluate the robustness of the analytical investigation, we use a direct numerical simulation of the continuous wave. We conclude that the second-neighbor couplings generate rogue waves in the structure during the long-time evolution of the plane wave. We also demonstrate that with a variation of the excitation wave number, the nonlinear electrical transmission line can support new objects as wave molecules for high values of the wave number. This feature is not yet observed in nonlinear electrical transmission lines and will be useful in many fields of physics.

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 vector exciton-polariton condensates with photonic spin-orbit coupling

The paper demonstrates the existence of modulational instability (MI) in nonlinear exciton-polariton condensates (EPCs). A set of coupled two-dimensional (2D) driven-dissipative Gross-Pitaevskii equations describing the spin-up and spin-down polaritons, containing the photonic spin-orbit (SO) coupling and the rate equations describing the density of incoherent excitonic reservoir density, is reduced to a system of two coupled 2D complex Ginzburg-Landau (2DCGL) equations, with saturable nonlinearities for complex polariton order parameters, thanks to the adiabatic approximation and small density fluctuations approximation. The analytical approach relies on the linear stability analysis of continuous waves (CWs) to derive an expression for the growth rate and conduct a parametric study of MI. The effect of different parameters on the growth rate spectrum is discussed with an emphasis on the photonic SO coupling and the pumping power. The predictions are verified against a direct simulation of the 2DCGL equations, and excellent agreement is found. The emergence of solitonic clusters manifests the evolution of MI. Phase diagrams on a MI spectrum for the CWs are presented against the magnetic field ($B$) and SO coupling strength ($\ensuremath{\sigma}$), based on which the dynamical behaviors of the emerging structures are debated, along with their response to changing $\ensuremath{\sigma}$ and $B$. The results suggest that the photonic SO coupling and the magnetic field constitute efficient tools for nonlinear mode selection and characterization via the MI process and are equally important in any eventual experimental realization of such a process in the studied EPC system.

Observation of the noise-driven thermalization of the Fermi-Pasta-Ulam-Tsingou recurrence in optical fibers

We report the observation of the thermalization of the Fermi-Pasta-Ulam-Tsingou recurrence process in optical fibers. We show the transition from a reversible regime to an irreversible one, revealing a spectrally thermalized state. To do so, we actively compensate the fiber loss to make the observation of several recurrences possible. We inject into the fiber a combination of three coherent continuous waves, which we call Fourier modes, and a random noise. We enhance the noise-driven modulation instability process against the coherent one by boosting the input noise power level to speed up the evolution to the thermalization. The distributions of the Fourier modes power along the fiber length are recorded thanks to a multi-heterodyne time-domain reflectometer. At low input noise levels, we observe up to four recurrences. Whereas, at higher noise levels, the Fourier modes fade into the noise-driven modulation instability spectrum revealing that the process reached an irreversible thermalized state.

The effect of coherent coupling nonlinearity on modulation instability and rogue wave excitation

We study modulation instability (MI) in both anomalous and normal dispersion regimes of a coherently coupled system. It is found that there exist three types of MI spectra with distinct characteristics termed baseband, passband, and zero-baseband based on the instability analysis of the in-phase and out-of-phase CW solutions. The coherent coupling nonlinearity is the source of the passband and zero-baseband variants. Guided by analytical predictions, we investigate numerically the excitation of rogue waves on weakly perturbed in-phase and out-of-phase continuous wave solutions in the parameter space where different kinds of MI exist. Simulations provide supporting evidence that rogue waves can only emerge in regimes where baseband or zero-baseband MI occurs. Moreover, the peak intensity of rogue waves in the case of baseband MI is greater than in the zero-baseband case. Finally, a combination of analysis and numerics uncovers the parameter conditions necessary for the generation of rogue waves.

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.

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.

Omnipresent coexistence of rogue waves in a nonlinear two-wave interference system and its explanation by modulation instability

We investigate the coexisting rogue wave dynamics associated with two fundamental-frequency optical waves interacting in a quadratic nonlinear medium. Using the vector Chen--Lee--Liu nonlinear Schr\"odinger equation model, we obtain exact rogue wave solutions at first and higher orders on the more general periodic backgrounds. We unveil that the inherent self-steepening effect may allow an omnipresent rogue wave coexistence over a broad range of parameters in both the normal and anomalous dispersion regimes, in addition to allowing ultrastrong peak amplitudes. We also demonstrate that such universality of coexistence can be anticipated by the appearance of two peaks in the modulation instability spectrum. We numerically confirm the robustness of the coexisting Peregrine solitons against initial noise as well as their excitation from a turbulent wave field caused by modulation instability. We expect that these findings will shed light on the generation of extreme wave events resulting from the interference of multiple continuous-wave fields.

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

Effects of ellipticity angle on soliton solutions and modulation instability spectra in two-core birefringent optical fibers

Effects of ellipticity angle on soliton solutions and modulation instability spectra in two-core birefringent optical fibers

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.

MODULATION INSTABILITY AND OPTICAL SOLITONS OF RADHAKRISHNAN-KUNDU-LAKSHMANAN MODEL

This paper studies the solitons of Radhakrishnan-Kundu-Lakshma- nan (RKL) model with power law nonlinearity. The modified simple equation method and $ \exp(-\varphi({q}) $) method are presented as integration mechanisms. Dark, bright, singular and periodic soliton solutions are extracted as well as the constraint conditions for their existence. A prized discussion on the stability of these soliton profiles on the basis of index of the power law nonlinearity is also carried out with the help of physical description of solutions. The integration techniques have been proved to be extremely efficient and robust to find new optical solitary wave solutions for various nonlinear evolution equations describing optical pulse propagation. Moreover, using linear stability analysis, modulation instability of the RKL model is studied. Different effects contributing to the modulation instability spectrum gain are analyzed.

Raman peak shifts due to walk-off in non-instantaneous Kerr media with higher-order effects

We study analytically and numerically the modulation instability (MI) of two incoherently co-propagating optical pulses in nonlinear fiber, taking into account high-order nonlinearity, walk-off effects, delay response time, and cross-phase modulation in several dispersion regimes. We show that higher-order nonlinearity is responsible for superposition of the secondary MI spectrum. We realize that there is a peak located in the emerged Raman band that we call “Raman peak” due to the combined effects of third-order dispersion, delay response time, and the opposite sign of the fourth-order dispersion parameter. These peaks are shifted toward higher frequencies when the walk-off increases.