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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]

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]

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]

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]

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 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]