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In this work, we use a monoatomic chain subject to an anharmonic potential with nearest neighbor couplings to derive the discrete nonlinear evolution equation. Thereafter, the modulational instability growth expression is derived by using the linear stability analysis. Our work displays the influence of the interaction potential and nearest neighbor couplings on the modulational instability and bandwidths. As a result, we show that the number of modulational unstable modes increases, while the number of modulational stable modes decreases when certain specific values are taken by the interaction potential. The same observation has been highlighted when the nearest neighbor couplings get high values. A numerical simulation of the continuous wave has been realized to confirm the fact that the modulational instability is sensitive to the nonlinear parameters due to the development of the modulated wave structures and trains of waves. Another particular result of this work has been displayed by driving the left and right ends of the chain into the lower and upper forbidden gaps. Supratransmission thresholds have been derived to get the corresponding driving amplitudes. It emerges that for the driving amplitude above the supratransmission threshold, the modulated waves occur in the lower forbidden frequency gap. At the same time, it results in a chaos-like motion in the propagation range of time in the upper forbidden gap. Such observations confirm the fact that in nonlinear structures, when the driving amplitude is considered above the supratransmission threshold, localized waves (gap solitons) are generated. [ABSTRACT FROM AUTHOR]

In this work, the modulation instability and solitonic rogue waves are addressed in the repulsive lattice formed by identical particles with nearest neighbor couplings. We use the multiple scale method to derive the extended nonlinear Schrödinger equation with fourth-order dispersion and cubic–quintic nonlinearity. To calculate the modulation instability growth rate, a linear stability analysis is used. Thereafter, we have demonstrated that both fourth-order dispersion and quintic nonlinearity can change the amplitude of the plane wave and bandwidth of the modulation instability in normal and anomalous dispersion regimes. The dynamics of the solitonic rogue waves have been pointed out to show an increasing amplitude with the variation of the free amplitude parameter of the chain of magnets. Via the numerical simulation, the modulated wave patterns have been exhibited to manifest the development of the modulation instability spectrum in the lattice, and the long-time evolution of the continuous waves has brought new features to show an increasing amplitude of the trains of pulses under the variation of the interaction interatomic parameters. These parameters are revealed to be a suitable tool to manipulate nonlinear objects in nonlinear media where higher-order dispersion competes with nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of coupled nonlinear waves and the modulated wave patterns in a single-wall carbon nanotube. We use an external on-site potential and the Brenner potential to derive the discrete equation of motion of the structure, where the nearest-neighbor interaction is included. Using the quasi-discrete multiple-scale approach, the coupled nonlinear Schrödinger equation is derived. The linearized technique is used to derive an expression of the modulation instability growth rate. For specific values of the inter-atomic interaction potential (i.e., nearest-neighbor parameters), the bandwidths and the modulation instability growth rate have been subjected to modification in amplitude. A particular attention has also been paid to the single and coupled excitation modes, where bright and dark solitons are pointed out in the lower and upper frequency gaps. Via numerical simulations, confirmations of the effects of inter-atomic interaction potential are given, where trains of waves are displayed to manifest the growth of the plane wave. We have also demonstrated that for strong values of the excitation wave number, wave patterns arise in the structure to confirm the fact that the continuous wave is sensitive to its variations. The prominent aspects of this work rely on the inter-atomic interaction potential effects on the bright and dark solitons and the modulation instability as an appropriate tool to control the nonlinear waves in single-wall carbon nanotubes. [ABSTRACT FROM AUTHOR]

In this work, we investigate diverse analytical solutions and modulation instability of the nonlinear Schrödinger equation with an anti-cubic nonlinear term. We use the traveling wave transformation and the New Generalized Extended Direct Algebraic Method to perform a variety of exact traveling wave soliton-like solutions. The obtained solutions are among other trigonometric function solutions, complex soliton solutions, and rational solutions. A particular behavior has been depicted in self-focusing and defocusing nonlinearity where dark soliton changes to super rogue waves and breather soliton is obtained. We then use the linearizing scheme to get the growth rate of modulation instability. For a specific value of the excitation wave number and a specific time of simulation, diverse waveforms of the modulated waves are obtained. We have demonstrated that the combined dispersion and nonlinear terms, as well as the anti-cubic nonlinear term, can develop modulation instability. To confirm the predictions based on the analytical results, we use the direct numerical simulation, from which we have displayed a modulation wave of the growth rate modulation instability. It results equally that the anti-cubic nonlinearity is an important object to control the propagation of the solitonic waves in the nonlinear systems. [ABSTRACT FROM AUTHOR]

In this paper, the modulation instability is analyzed in a novel model of coupled pairs of chains with coupling coefficient parameters. We derive the coupled system of the nonlinear equation by using the Lindstedt–Poincaré perturbation and the multi-scale method. From the linear stability analysis, an expression for the modulation instability growth rate is established, and we assessed the effects of the coupling strength on the modulation instability spectrum in acoustic and optical modes. In acoustic metamaterials, it appears that modulation instability develops and results in the formation of wave trains. We also demonstrated that the modulation instability bandwidth can be reduced or expanded depending on the effectiveness of the coupling coefficient as well as the angular frequency. Through the numerical experiments, we showed the propagation of the modulated wave patterns, manifesting the existence of the modulation instability growth rate in the structure. It results from this investigation that conventional materials where microstructural configurations are designed to exhibit unusual properties are attractive to modulated wave structures. In future works, soliton dynamics, rogue waves, and localized modes will be investigated to give rise to several applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the propagation of the modulated waves (MWs) patterns with the effects of the external field (EF) in a photorefractive crystal. To do so, we use the coupled nonlinear Schrödinger equations and we apply the auxiliary equation method to point out four families of solutions such as bright, dark, combined bright-dark, and bright-bright soliton. We pointed out the effects of EF on the modulation instability growth rate. Throughout the numerical simulation, we show that EF can generate MWs patterns when the short and open circuit condition is considered in the case of bright soliton. However, when we consider dark soliton solution as an initial condition and we have only obtained the MW pattern in the case of a short circuit. It results in this study of the propagation of the MWs patterns and chaos-like motion for specific cell indexes. [ABSTRACT FROM AUTHOR]

This work examines the effects of the higher-order dispersion (HOD) on modulated waves and modulation instability (MI) gain by using the perturbed nonlinear Schrödinger equation having the cubic quintic septic nonlinearity. We use the new generalized auxiliary equation method to underline the behavior of bright, dark soliton as well as the combined optical soliton solutions. Considering the constraint relation on the combined optical soliton solutions, we carry out the W-shaped profile. We observe that the obtained optical soliton solutions can spread without alteration and can preserve their shapes. Moreover, we enhance the obtained analytical results by using the Split-Step Fourier technic which yields points out stable optical solitons. We have obviously highlighted the effects of the HOD by means of the numerical simulation. Owing to the fact that the model has nonlinear and dispersions terms, we have shown how the MI growth rate and the modulation bands can expand under the influence of the HOD terms in normal and anomalous dispersion regimes. It is worth mentioning the appearance of the unstable zones when the value of the higher-order dispersion term increases and additional bands also emerges. Compared these results with some previous works (Kohl in Optik 203:163451, 2020; Yao in Res Phys 30:104825, 2021; Wang in Phys Lett A 372:417–423, 2008; Ismail in Appl Math Comput 209:425–429, 2009; Nestor in Eur Phys J Plus 135:380, 2020), additional bands of the MI have been pointed out while the MI gain spectra have shown new sides lobes. We expect that these relevant and concise results could probably help to improve the communication system through optic fibers. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the propagation of the modulated waves patterns in the cold bosonic atom zig–zag optical array where the first nearest neighbor and second nearest neighbor (SNN) are considered. We pointed out the forward and backward wave propagation and its unstable/stable zones. The effects of the NNs on modulation instability growth rate and MWs patterns have been highlighted. This analysis shows that the SNN can induce unstable zones from where the bullets appear. By considering the linear analysis, we have shown lower and upper gaps from where the group velocity vanishes and open the way to the nonlinear supratransmission matter. To get to this phenomenon, we have submitted one end of the lattice to the external harmonic forcing. Thus, we establish the expression of the threshold amplitude which is linked to the static breather. We have shown the propagation of bright soliton with the variation of the driven amplitude (DA). It results that nonlinear energy localization happening and BS acquires novel features for a large value of the DA. We can predict that the SNN and the DA are sources of energy for cold bosonic atoms in the zig–zag optical lattices. [ABSTRACT FROM AUTHOR]

We use the fractional nonlinear Schrödinger equation (FNLSE) to describe the polarization mode in optical fiber with Self-Steepening, Self-Frequency Shift, and Cubic-quintic terms to analyze the effects of the fractional time parameter (FTP) on bright and dark solitons as well as breather-like solitons. We use the transformation hypothesis and auxiliary equations method to obtain three families of solutions such as combined bright soliton, dark solitons, and rational solitons. We have shown the effects of the fractional parameter (FP) on the W-shaped profile, bright and dark optical soliton solutions as well as the corresponding chirp component. It is observed that for small values of the FP, optical soliton shape is affected and the soliton is unstable. Moreover, one observes the effects of fraction time on Modulation Instability (MI) gain spectra and MI bands. For certain values of the FP, it is formed sides lobes and for specific small values of the FP, both stability zone increases and amplitude of the MI gain increase while the stability zones increase. To confirm the robustness of the analytical results, we have used a numerical investigation. One exhibits the formation of breathers-like soliton with stable amplitude for small values of the FTP. It results from this study that the FP is efficient and can be used as an energy source where soliton or breathers-like soliton are involved for communication in optical fibers. [ABSTRACT FROM AUTHOR]

Investigation of soliton solutions and Modulation Instability (MI) analysis to the (2+1)-Stochastic Chiral Nonlinear Schrödinger Equation (SCNLSE) having noise force and Brownian movement (BM) have been done in this work. We first use the transformation technic and auxiliary equation method (AEM) to establish three types of soliton solutions. Depending on the constraint relation set on the auxiliary equation parameters, it is obtained bright and dark soliton solutions. Thereafter, taking the same different parameters it is gained the combined bright-dark soliton, bright-bright soliton solutions. By employing suitable values of the used parameters together with the model parameters it is show out W-shaped soliton solutions. Throughout the paper it is clearly established the impact of the noise strength and BM on the obtained W-shaped soliton. To determine the stability or instability zones, it was used the linear stability tool. It is shown that by varying the different parameters the Modulation bands can grow instability and stability zones can also emerge. These results permit to consolidate the previous works (Albosaily et al. in symmetry 12:1874, 2020, Abdelrahman et al. in AIMS Math 6:2970, 2021) exposed on the effects of noise on the soliton and the MI gain is new in this context. [ABSTRACT FROM AUTHOR]

In this work, we examine optical solitons to the Radhakrishnan–Kundu–Lakshmanan equation (RKL) equation which describes the optical pulses in birefringent fiber (Raza and Javid in J Appl Anal Comput 10:1375–1395, 2020; Seadawy et al. in Opt Quant Electron 53:324, 2021) by using the New Generalized Auxiliary Equation Method (NGAEM). After some mathematical transformations, owing some constraint relations it is obtained two categories of soliton solutions. The first includes bright and dark optical solitons, while in the second class it is divulged the combined bright-dark and bright-bright optical solitons. Taking some suitable parameters of the model and the NGAEM, it is put up W-shaped optical solitons and diverse other solutions. Thereafter, we use the continuous waves as solutions of the model with small perturbations, to show the effects of the TOD, ellipticity angle and XPM on the Modulation Instability (MI) gain in normal and anomalous dispersion regime. It has been indicated that the third-order dispersion in normal/aanomalous dispersion regime can generate MI growth rate. At the same time, the ellipticity angle and such others parameters of the model play an important role during the MI growth rate (gain). Compared the obtained appropriate results in terms of analytical results and dynamics of the MI to Refs. (Raza and Javid 2020; Seadawy et al. 2021; Yepez-Martinez et al. in Chin J Phys 58:137–150, 2019; Drummond et al. in Opt Commun 78:137–142, 1990; Li et al. in Commun Theor Phys 65:231–236, 2016), they are new in our knowledge. [ABSTRACT FROM AUTHOR]

In this paper, we construct diverse solitary wave solutions for the nonlinear differential equation governing wave propagation in the low-pass nonlinear electrical transmission lines with conformable derivatives. However, by employing the new extended direct algebraic method and the improved Sub-ODE equation, we recovered W-shape bright soliton, dark soliton, periodic solutions, rational solutions and Weierstrass elliptic function solutions. The obtained results are new in nonlinear electrical transmission lines field. In addition, the acquired solitons are depicted with the appropriate parameters values of the methods and the nonlinear electrical transmission lines. The shape of the W-bright and dark soliton solutions points out the effect of the derivative order. Finally, the results indicate that the two integrations methods are a most applicable and forceful integration tools for emphasizing the soliton solutions. [ABSTRACT FROM AUTHOR]

This paper shows out optical soliton solutions of the twin-core couplers with parabolic law nonlinearity with optical metamaterials parameters. It is used the improved new sub-ODE method (SOM) to found dark optical soliton, trigonometric function solutions, bright optical soliton and Jacobian elliptic function (JEF) solutions. Besides, physical explanation of the obtained bright and dark optical soliton solutions have been done and the influence of some parameters of the model have been set out. As the two-core coupler is favorable for modulation instability (MI), it has been studied the steady state of the results. More importantly, we have checked the effects of the optical metamaterials parameters on the formation of the Modulation bands in normal and anomalous dispersive regime. The results obtained in this paper set out Weierstrass Elliptic Function solutions compared to Mirzazadeh et al. (J Nonlinear Opt Phys Mater 24:1550017, 2015), Vega-Guzman et al. (Acta Phys Pol A 133:167–178, 2018) and will be helpful in the feature in optical birefringent fibers. [ABSTRACT FROM AUTHOR]

Throughout this paper, we investigate the propagation of solitary waves in metamaterials. Indeed, we focus our attention on a nonlinear evolution equation, namely the nonlinear Schrödinger equation (NLSE) that models the dynamic of waves in such materials. Investigating solitons structures of such an equation, we make use for the purpose, of a mathematical tool which is a new generalized extended direct algebraic method (NGEDAM). As a results, rich solution structures are constructed analytically to which new solutions are provided to complete the ones already obtained elsewhere. In addition, the modulation instability (MI) analysis has been studied by employing the linearizing technic. We highlight the effects of the self-steepening (SS) associated to metamaterials parameters on MI bands. The obtained results have set up the W-shaped profile of the optical soliton solutions, which will certainly improve the communication over the optical fibers in diverse modes. [ABSTRACT FROM AUTHOR]

In this paper, we show out the chirped and the corresponding chirp with their stability to the perturbed Chen–Lee–Liu equation with self-phase modulation and nonlinear dispersions. By employing the traveling-wave technique, we establish directly the nonlinear ordinary differential (NODE) equation. Based on the parameters of the obtained NODE, several cases of the exact traveling wave have been obtained with their corresponding chirp. We, also note that the group velocity dispersion, the self-phase modulation and the nonlinear dispersion terms have impacted the gain spectrum of the modulation instability (MI). The one important aspect is that without computer codes all these results were obtained. These results will be certainly helpful in optical fibers. [ABSTRACT FROM AUTHOR]

In this work, the modified Sardar sub-equation method is used to construct optical soliton solutions to the coupled nonlinear Schrödinger equation (CNLSE) having third-order and fourth-order dispersions describing short-pulse propagation in two-core fibers. Several solutions are determined including W-shaped bright, dark soliton solutions, singular soliton solutions, periodic function solutions and the combined complex soliton solutions. It should be noted that this integration scheme is very powerful mathematical tools for obtaining exact optical soliton solutions of nonlinear evolution equations. Under suitable values for the physical parameters, some representative wave structures are graphically displayed. In addition, the linear stability technic is used to analyze the modulation gain spectra in the birefringence associated with an ellipticity angle. In our knowledge, these obtained results are new in the context of nonlinear birefringent optical fibers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Based on Darboux matrix method, a generalized Darboux transformation of a generalized nonlinear Schrödinger equation possessing higher-order terms which refer to a femto second pulse propagation in a nonlinear fiber optic is constructed. This nonlinear Schrödinger equation can model the propagation of soliton in a Heisenberg spin chain. Using the above-generalized Darboux transformation, we calculate the first, two and third-order rogue wave solutions and give general formulae for generating Nth-order ones. We discuss the effect of higher-order terms contained in the above equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the generalized Benjamin-Bona-Mahony equation which better describes long waves with arbitrary power-law nonlinearity. As a result, we obtain exact travelling wave soliton solutions, such as anti-kink soliton solution, bright soliton solution, dark soliton solution and periodic solution. These solutions have many free parameters such that they may be used to simulate many experimental situations. The main contribution, in this work, is to not apply the computer codes for construction of exact solutions and not consider the integration constants as zero, because they give all variants for solutions. [ABSTRACT FROM AUTHOR]