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This article studies dark, bright, trigonometric and rational optical soliton solutions to the perturbed nonlinear Schrödinger–Hirota equation (PNLSHE). Hence, we have examined two cases: first, restrictions have been done to the third-order (TOD) (γ) as constraint relation, and the coupling coefficients (σ) is obtained as well as the velocity of the soliton by adopting the traveling wave hypothesis. Second, the TOD and the coupling coefficients are non-zero value, sending back to the PNLSHE, which has been studied in refs. [4,10,16] recently. By employing two relevant integration technics such as the auxiliary equation and the modified auxiliary equation method, miscellaneous optical solitary wave is obtianed, which is in agreement with the outcomes collected by the previous studies [4,16]. These results help in obtaining nonlinear optical fibers in the future.

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]

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 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]

• The Coupled Nonlinear Schrödinger Equation (CNLSE) throughout Kerr law nonlinearity is considered. • The linear stability analysis is used. • The modulation instability analysis is presented. • The obtained results are showed by 2D and 3D Figures. 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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the modulation instability and the dynamics of solitary waves in a higher-order nonlinear Schrödinger equation. We use a powerful and robust method known as the modified sub-equation method to secure the solitary waves and other numerous solutions to this nonlinear model. Several constraint conditions that guarantee the existence of these solutions are highlighted. Utilizing a linearizing technique, we establish the modulation instability gain. The influence of the higher nonlinear component on the modulation instability is also discussed. Furthermore, we perform the numerical simulations of this model by using the split-step Fourier method. Finally, several solutions are shown in two and three dimensions to better explain the behavior of the considered model. The two-dimensional plots depict Peregrine soliton under the influence of the nonlinear dispersion term. These graphs are useful for understanding the dynamic properties of the obtained results. • New optical solitons for the HNSE were constructed. • The obtained solutions were found using the modified sub-equation technique. • The model's modulation instability is studied. • The physical meaning for the solutions is graphically investigated. [ABSTRACT FROM AUTHOR]

In this article, an amelioration of the approaches namely the new extended direct algebraic method for solving the nonlinear conformable fractional Schrödinger-Hirota equation (FSHE) is presented. By using the traveling wave transformation change the fractional equation into the ordinary differential equation, and with the aid of mathematical software Maple, a number of new optical solitons solutions for them are calculated. The obtained results show that the proposed method is powerful, effective, and straightforward technique to work out new solutions of various types of nonlinear fractional partial differential equations in applied sciences. [ABSTRACT FROM AUTHOR]