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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]

By employing the Modified Sardar Sub-Equation Method (MSEM), several solitons such as W-shape bright, dark solitons, trigonometric function solutions and singular function solutions have been obtained in two famous nonlinear evolution equations which are used to describe waves in quantum electron–positron–ion magnetoplasmas and weakly nonlinear ion-acoustic waves in a plasma. These models are the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation and the (3+1)-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation, respectively. Comparing the obtained results with Refs. 32–34 and Refs. 43–46, additional soliton-like solutions have been retrieved and will be useful in future to explain the interaction between lower nonlinear ion-acoustic waves and the parameters of the MSEM and the obtained figures will have more physical explanation. [ABSTRACT FROM AUTHOR]

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, bifurcation analysis, and soliton solutions are investigated in this paper. • A generalized nonlinear Schrodinger equation with quartic dispersion and cubic-quintic nonlinearity is considered. • The effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. [ABSTRACT FROM AUTHOR]

• The nonlinear discrete equation is studied. • The rational w- shape solitons are retrieved. • The modulation analysis is presented. • The obtained solutions are presented by 2D and 3D figures. This paper examines envelope solitons, rational W-shape solitons to a Nonlinear Electrical Transmission Line (NETL) with Josephson junction (JJ) that consist of the N t h cells of circuits. By employing the reductive perturbation approach in the semi-discrete approximation, we obtain the nonlinear Schrödinger equation (NLSE). From this equation, the frequency ranges of propagation of bright and dark solitons were obtained. As in most of the works, the NLSE obtained is well known as the seat of rogue waves and Peregrine solitons. The nonlinearity provided by the JJ, combined with the dispersion, made it possible to study the Modulation Instability (MI) gain spectrum. The 3D and 2D graphical representations illustrated the instability zones and the W-rational solutions have followed by using numerical simulation. The results obtained in this paper are of a very capital contribution for the study of rogue waves in nonlinear transmission lines or other physical problems. [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]