Instability Gain Research Articles

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.

Influence of fractional time order on W-shaped and Modulation Instability gain in fractional Nonlinear Schrödinger Equation

In this paper, we study the influence of fractional time derivative on W-shaped profile and Modulation Instability gain in fractional Nonlinear Schrodinger Equation (NLSE) which could be used to describe the propagation of pulses in random media, optical metamaterials and others nonlinear systems. We first imply the auxiliary equation method to set up bright, dark and W-shape optical solitons solutions under certain conditions. Thereafter, we graphically depict the obtained results which deeply show the potency of the fractional parameter order on the width and shape of the solitons. We then study the Modulation Instability (MI) gain spectra and we end up remarking that the MI gain sidebands and the MI gain shapes are mainly influenced by the fractional time derivative parameter compared to the previous works reported in nonlinear optic fibers (Wyller et al., 2002; Zhanga et al., 2017).

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.

M-shape and W-shape bright incite by the fluctuations of the polarization in a-helix protein

In this work, under consideration is the coupled system of nonlinear Schrödinger (NSL) and Boussinesq equations that serves as model to study the polarization fluctuation in α-helical protein. M-shaped and W-shaped solutions have been derived through the new modified Sardar sub-equation technique (SSEM). These solitons solutions define the polarization dynamics in the α-helical protein. The integration naturally leads to a constraint condition placed on the solitary wave variables which must hold for the solitary waves to exist. In addition, the standard linear stability analysis has permitted to study the behavior of the Modulation Instability (MI) gain spectra. Numerical simulation and physical interpretations of the acquired results are demonstrated. The derived structure of the acquired solutions give a rich platform to better understand the nonlinear dynamics in the α-helical protein.

Excitation of nonlinear beams: from the linear Talbot effect through modulation instability to Akhmediev breathers.

The smooth transition between stable, Talbot-effect-dominated and modulationally unstable nonlinear optical beam propagation is described as the superposition of oscillating, growing and decaying eigenmodes of the common linearized theory of modulation instability. The saturation of the instability in form of breather maxima is embedded between eigenmode growth and decay. This explains well the changes of beam characteristics when the input intensity increases in experiments on modulation instability and breather excitation in spatial-spatial experimental platforms. An increased accuracy of instability gain measurements, a variety of interesting nonlinear beam scenarios and a more selective and well-directed breather excitation are demonstrated experimentally.

Modulation instability in helicoidal spin-orbit coupled open Bose-Einstein condensates.

We introduce a vector form of the cubic complex Ginzburg-Landau equation describing the dynamics of dissipative solitons in the two-component helicoidal spin-orbit coupled open Bose-Einstein condensates (BECs), where the addition of dissipative interactions is done through coupled rate equations. Furthermore, the standard linear stability analysis is used to investigate theoretically the stability of continuous-wave (cw) solutions and to obtain an expression for the modulational instability gain spectrum. Using direct simulations of the Fourier space, we numerically investigate the dynamics of the modulational instability in the presence of helicoidal spin-orbit coupling. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold for amplitude perturbations.

A Direct Method for the Simultaneous Estimation of Self-Steepening and the Fractional Raman Contribution in Fiber Optics

We propose an original, simple, and direct method for the simultaneous estimation of the self-steepening parameter and the fractional Raman contribution in fiber optics. Our proposal is based on the dependence of the modulation instability gain on both parameters, as obtained from a linear stability analysis of the newly introduced photon-conserving generalized nonlinear Schrödinger equation (pcGNLSE), and requires only the CW or quasi-CW pumping of the waveguide under test and a few direct spectral measurements. Further, we demonstrate the feasibility of the estimation procedure by means of detailed simulations for typical waveguide parameters in relevant spectral ranges. Last, we discuss the range of applicability of the proposed method and compare its results, advantages, and disadvantages with a recently introduced method based on short-pulse dynamics.

Optical soliton to multi-core (coupling with all the neighbors) directional couplers and modulation instability

n this research paper under consideration is the nonlinear multi-core coupler (coupling with all neighbors) with the parabolic law nonlinearity and optical metamaterials parameters. The integration scheme, namely the New sub-ODE method, is used to extract bright, dark, trigonometric function solutions, Jacobian elliptic function solutions and Weierstrass elliptic function solutions. To better understand the physical phenomena of the gained solutions, some physical explanation of the presented solutions under suitable values for the physical parameters via the 3D, 2D and contour simulations is given. The confrontation between the nonlinear and dispersion terms has permitted to study the behavior of the modulation instability (MI) gain spectra. The results obtained in this article are more general than those contained in the following papers [41–43] and will help to explain the virtue of the nonlinear optical birefringent.

Impact of fractional effects on modulational instability and bright soliton in fractional optical metamaterials

In this paper, we have derived the fractional nonlinear Schrödinger equation modeling pulse propagation in metamaterials under nonlocal time evolution. We have investigated the influence of the fractional effects on both modulational instability (MI) gain and bright soliton. A negative group velocity (GV) is one of the necessary characteristic signatures of a left-handed material. The negativity of GV is observed for some fractional parameters α. MI occurs when the signs of first and second order dispersion are opposed. The results show that the fractional parameters can control the MI gain as well as the amplitude and the width of pulse.

Theory of filter-induced modulation instability in driven passive optical resonators

We present the theory of modulation instability induced by spectrally dependent losses (optical filters) in passive driven nonlinear fiber ring resonators. Starting from an Ikeda map description of the propagation equation and boundary conditions, we derive a mean field model - a generalised Lugiato-Lefever equation - which reproduces with great accuracy the predictions of the map. The effects on instability gain and comb generation of the different control parameters such as dispersion, cavity detuning, filter spectral position and bandwidth are discussed.

Modulational instability induced generation of solitary wave profile of an anisotropic-ferromagnetic nanowire with asymmetric Dzyaloshinskii-Moriya interaction

We study the nanoscale magnetization evolution of a one dimensional ferromagnetic nanowire with the inclusion of next-nearest-neighbor interaction and Dzyaloshinkii-Moriya (DM) interaction. The system is modelled using Heisenberg Hamiltonian in the quasi classical approximation of spin operators. The dynamics of the system is found to be governed by generalized discrete nonlinear Schrodinger equation. Further we investigate analytically the modulational instability in ferromagnetic nanowire system, where the effect of next-nearest neighbour (NNN) interaction and DM interaction are taken into an account to show graphically the effect of interactions on Modulational Instability (MI) gain spectra.

Modulation instability in nonlinear chiral fiber

Abstract In this paper, the rigorous derivations of generalized coupled chiral nonlinear Schrödinger equations (CCNLSEs) and their modulation instability analysis have been explored theoretically and computationally. With the consideration of Maxwell’s equations and Post’s constitutive relations, a generalized CCNLSE has been derived, which describes the evolution of left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) components propagating through single-core nonlinear chiral fiber. The analysis of modulation instability in nonlinear chiral fiber has been investigated starting from CCNLSEs. Based on a theoretical model and numerical simulations, the difference on the modulation instability gain spectrum in LCP and RCP components through chiral fiber has been analyzed by considering loss and chirality into account. The obtained simulation results have shown that the loss distorts the sidebands of the modulation instability gain spectrum, while chirality modulates the gain for LCP and RCP components in a different manner. This suggests that adjusting chirality strength may control the loss, and nonlinearity simultaneously provides stable modulated pulse propagation.

The effects of fourth-order dispersion on modulation instabilities in two-core optical fibers with asymmetric CW state

The effects of fourth-order dispersion (FOD) on modulation instabilities in the twin-core optical fibers with the asymmetric CW state are studied systematically. When the product of the group velocity dispersion and FOD is negative, in the anomalous dispersion regime, two new MI bands in the high frequency are generated with one band either having the equivalent gain with that of original band or dominating the MI depending on the FOD value; in the normal dispersion, a new dominant MI band in the high frequency is always generated, and the original MI bands vanish in the presence of FOD at total power above a critical value. When the product of the group velocity dispersion and FOD is positive, the main effect of FOD is to shift the MI band to the low frequency and reduce the bandwidth simultaneously with minor effects on the instability gain. We also find that the effects of linear coupling coefficient dispersion can be significantly suppressed in the presence of FOD especially for the case of negative product of the group velocity dispersion and FOD. Finally, the effects of FOD on MI are verified numerically by launching continuous waves with the wave propagation method. Our results should provide very benificial information for successfully observing MI in TCFs or MI related applications with TCFs.

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.

Observation of modulation instability and rogue breathers on stationary periodic waves

We present both theoretical description and experimental observation of the modulation instability process and related rogue breathers in the case of stationary periodic background waves, namely cnoidal and dnoidal envelopes. Despite being well-known solutions of the nonlinear Schrodinger equation, the stability of such background waves has remained unexplored experimentally until now, unlike the fundamental plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we report on quantitative measurements of spontaneous modulation instability gain seeded by input random noise, as well as the formation of rogue breather solutions induced by a coherent perturbation. Our results confirm the generalization of modulation instability when more complex background waves are involved.

Dark solitons and modulational instability of the nonlinear left-handed transmission electrical line with fractional derivative order

This paper unearths solitary wave in a simplified electrical model practical of the nonlinear left-handed transmission electrical line having series capacitance and shunt inductance. The detailed study of the practical model digs out that by taken a good parameter of the lattice, the obtained equation from the evolution of the wave envelope can be reduced to a one-dimensional fractional nonlinear Schrödinger equation, which gives the pipe of the dark solitons propagation. By simulating transient circuits, the evolution of the soliton and modulational instability gain were demonstrated in the areas of perturbation frequency, pulsation, nonlinearity and signal strength with a continuous wave type input signal. It has also been shown that increased perturbation, nonlinearity and signal strength in simplified metamaterial models of short nonlinear transmission lines lead to the formation of Schrödinger dark solitons and increased instability gain in a continuous pulse structure. The establishment of Schrödinger solitons and the progressive distance of a point, i.e. the fractional parameter α = 1, could reduce the width of the signal bandwidth and the breaking of the symmetry of the instability gain in the simplified electrical model practical of the nonlinear left-handed transmission electrical line, so that the evolution of the system cannot be controlled due to the memory effect, which can find important practical applications in communication systems.

Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves

The core of this research is to investigate the system of equations for the ion sound and Langmuir waves through the Bernoulli sub-equation method. It is one of the most effective methods for solving nonlinear ordinary differential equations. To transform a given system to a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. Novel wave behaviors of the guiding system are acquired, such as kink-type singular solutions, wave solutions, periodic wave solutions, and periodic singular solutions. Besides, the modulation instability analysis of the given system is investigated based on the standard linear stability analysis and the modulation instability gain spectrum analysis. Furthermore, 2D, 3D and contour graphs of all acquired solutions are also plotted under the selection of appropriate parameter values.

Generation of density waves in dipolar quantum gases by time-periodic modulation of atomic interactions

We study the emergence of density waves in dipolar Bose-Einstein condensates (BEC) when the strength of dipole-dipole atomic interactions is periodically varied in time. The proposed theoretical model, based on the evolution of small perturbations of the background density, allows to compute the growth rate of instability (gain factor) for arbitrary set of input parameters, thus to identify the regions of instability against density waves. We find that among other modes of the system the roton mode is most effectively excited due to the contribution of sub-harmonics of the excitation frequency. The frequency of temporal oscillations of emerging density waves coincides with the half of the driving frequency, this being the hallmark of the parametric resonance, is characteristic to Faraday waves. The possibility to create density waves in dipolar BECs, which can persist after the emergence, has been demonstrated. The existence of a stationary spatially periodic solution of the nonlocal Gross-Pitaevskii equation has been discussed. The effect of three-body atomic interactions, which is relevant to condensates with increased density, upon the properties of emerging waves has been analyzed too. Significant modification of the condensate's excitation spectrum owing to three-body effects is shown.

Modulation instability in waveguides with an arbitrary frequency-dependent nonlinear coefficient.

In this Letter, we present, for the first time, to the best of our knowledge, the modulation instability (MI) gain spectrum of waveguides with an arbitrary frequency-dependent nonlinear coefficient ensuring strict energy and photon-number conservation of the parametric process. This is achieved by starting from a linear stability analysis of the recently introduced photon-conserving nonlinear Schrödinger equation. The derived MI gain is shown to predict some unique features, such as a nonzero gain extending beyond a zero-nonlinearity wavelength and a complex structure of the MI gain spectrum. Analytical results are shown to be in excellent agreement with numerical simulations.