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The nonlinear Schrödinger equation (NLSE), which governs the propagation of pulses in optical fiber while including the effects of second, third, and fourth-order dispersion, is crucial for a comprehensive understanding of pulse propagation in optical communication systems. It assists engineers and scientists in optimizing and controlling the behavior of ultra-short pulses in complex and real-world optical systems. In this study, we solve the generalized NLSE for the pulse envelope E (z , τ) with ν -time derivative by employing the Sardar subequation method (SSM). We obtain new soliton solutions corresponding to the relevant parameters of this technique. Additionally, conditions depending on the parameters of optical pulse envelope E (z , τ) are provided for the existence of such soliton structures. Furthermore, the solitary wave solutions are expressed in the form of generalized trigonometric and hyperbolic functions. The dynamic behaviours of the solutions are revealed with specific values of the parameters that satisfy their respective existence criteria. The results indicate that SSM demonstrates high reliability, simplicity, and adaptability for use with various nonlinear equations. [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, we studied a nonlinear wave equation that models the propagation of optical solutions in a weakly nonlocal and parabolic competing nonlinear medium. The exact traveling wave type solutions formulated in hyperbolic functions, rational and trigonometric functions multiplied by some exponential functions to the governing equation are determined explicitly by the extended form of the Kudryashov method. We have examined the behavior of the modulation instability (MI) growth rate. To substantiate the stability of the obtained dark and bell-shaped solitons, we use the split-step Fourier method. In addition, the conservation laws describing significant physical concepts of this equation are examined. Compared to the obtained results with Younis et al. (J Nonlinear Opt Phys Mater 24(04):1550049, 2015), Zhou et al. (Optik 124(22):5683–5686, 2013; Proc Rom Acad Ser A 16(2):152–159, 2015) and Akinyemi et al. (Optik 230:166281, 2021), we have shown the propagation of the solitonic waves which sometime tilt from right to left with stable shape. [ABSTRACT FROM AUTHOR]

Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

In this paper, we obtained the exact solutions of the (1 + 1) -dimensional nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) and the seventh-order Sawada–Kotera–Ito (S-K Ito) equations with the help of the (w / g) -expansion method specially the (g ′ / g 2) -expansion and (g ′) -expansion methods. Soliton solutions found for the given equations are in the form of hyperbolic, trigonometric and rational solutions. All obtained solutions were checked. 3D and 2D graphs of some solutions were given and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we considered the nonlinear Schrodinger equation modeling cubic optical solitons in a polarization-preserving fiber with Kerr law. Using the improved t a n (φ / 2) -expansion method, a powerful and effective method, we constructed exact solutions including the hyperbolic function solution, the trigonometric function solution, the exponential solution and the rational solution with free parameters. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The obtained solutions are entirely new and can be considered as generalization of the existing results in the ordinary derivative case. [ABSTRACT FROM AUTHOR]

This papers presents new exact analytical solutions of a generalized Ito equation having three nonlinear terms, third- and fifth-order derivative forms that model the dynamics of traveling waves in nonlinear dispersive systems. With the help of Riccati equation method, we obtain different kinds of exact traveling wave solutions containing dark, singular, trigonometric, rational and other form of waves solutions that are more general than classical ones existing in the literature. Despite the originality of the new results obtained, the method used here is very efficient, powerful and can be extended to other types of nonlinear equations and more. Moreover, the behaviors of traveling waves solutions are portrayed graphically by selecting suitable values for the physical parameters. [ABSTRACT FROM AUTHOR]

This work addresses the analytical investigation of the prey–predator behavior modeled by nonlinear evolution equation systems with fractional derivative order. Through the New Extended Algebraic Method (NEAM), we unearthed diverse types of soliton solutions including bright, dark solitons, combined trigonometric function solutions, and singular solutions. Besides the results obtained in the work of Khater, some new complex soliton solutions are also unearthed. The NEAM can also be used like the synthesis of the two mathematical tools. [ABSTRACT FROM AUTHOR]

This article explores the abundant solitary wave solutions of the conformable coupled Jaulent–Miodek (JM) equations appearing in applied physics. The aforesaid coupled equations belong to the family of shallow-water wave equations. Two recent modified integration schemes are used for the first time to produce a novel solitary wave, trigonometric and other solutions with some free parameters in the conformable derivative sense. In particular, the modified Kudryashov and ( G ′ G 2 ) -expansion schemes are used to illustrate the wave propagations through aforesaid solutions of the JM equations. Furthermore, a comparison is made with some recent results and the dynamics of the obtained solutions are displayed for the reader via soft computation. The outcomes reveal that the methods are effective and provide a direct way of finding novel solutions. [ABSTRACT FROM AUTHOR]

In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV--Zakharov--Kuznetsov (fractional mKdV--ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV--ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given. [ABSTRACT FROM AUTHOR]

This study focuses on acquiring new complex hyperbolic and trigonometric function solutions to the generalized conformable fractional Gardner equation by using two distinct integration schemes, namely: the modified Kudryashov method and the hyperbolic function method with the help of symbolic computation package. We generate new closed-form solitary wave solutions to the governing model. Implemented schemes are pragmatically effective for solving various types of nonlinear partial differential equations with integer or fractional order arising in various fields of applied sciences. [ABSTRACT FROM AUTHOR]

In this paper, new periodic fractional trigonometric functions with the period 2 π are presented. We have generalized the Floquet system to the fractional Floquet system. The fractional derivatives are described with the use of modified Riemann-Liouville derivative. Moreover, the stability analysis of fractional Floquet system is introduced. [ABSTRACT FROM AUTHOR]