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The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative, which is commonly used to describe the propagation of wide pulses in nonlinear optical fibers. By employing the complete discriminant system of fourth-order polynomials, various types of explicit solutions are systematically classified, which include periodic solutions, the trigonometric functions, the double-period solutions, and the elliptic function solutions. Additionally, a series of 2D, 3D, and contour plots are generated to visually depict the spatial distribution and evolution of various solutions. This not only advances the development of nonlinear equations in theory but also provides valuable guidance in practical applications.

This paper obtains cubic–quartic optical solitons of generalized Kudryashov’s law of refractive index. The included perturbation terms are with maximum intensity. The retrieved soliton solutions are with the aid of F -expansion, exp-expansion and Riccati equation methods. Finally, the conservation laws of the model are also recovered and listed.

The paper investigates the coupled system of NLSE for the generalized Kudryashov’s equation in magneto-optic waveguides, in which the trial equation method and the complete discriminant system for a polynomial are employed to obtain a rich set of exact solutions, including rational solutions, solitary wave solutions, triangular function solutions, and elliptic function solutions. Numerical simulations are also conducted on the solutions under specific parameters to visually depict the structure and characteristics.

In the current work, the modified simplest equation approach and modified ( G ′ / G 2 ) -expansion approach are implemented to extract soliton solutions and other exact solutions for generalized Kudryashov’s equation with truncated M-fractional derivative. Various types of solutions are extracted such as dark soliton solutions, bright-singular soliton solutions, singular soliton solutions, combo singular solitons solutions, combo dark-singular soliton solutions, periodic solutions and rational solutions. Moreover, to emphasise the impact of truncated M-fractional derivative on the behaviour solutions for the presented problem, the 2D, 3D and contour representations of some obtained solutions are produced. The introduced methods are more reliable, applicability and simple as compared to many other methods to solve the nonlinear fractional partial differential equations.

In this study, using the Kudryashov R function method and the generalized Kudryashov method, the ( 1 + 1 ) -dimensional resonant nonlinear Schrödinger equation with dual-power law nonlinearity has been effectively examined for finding optical soliton solutions. The Kudryashov R function technique has numerous advantages that make symbolic computing considerably simpler, particularly while dealing with strongly dispersive nonlinear equations. The generalised Kudryashov method is noteworthy due to its capacity to address a wide range of complex nonlinear ordinary differential equations (NLODEs) observed in diverse engineering, scientific and mathematical fields. Newly generated nonlinear Schrödinger equation is demonstrated by the resonant nonlinear Schrödinger equation used to describe nonlinear optical phenomena. In order to achieve the goal, the governing model was first transformed into a NLODE, and then the solution sets and solution functions were derived based on the definitions of the suggested approaches. Solitons are localized wave forms that retain their shape and stability as they propagate across optical fiber. Optical solitons are characterized by their ability to maintain their shape and amplitude during propagation, even when encountering other solitons. Using the suggested approaches, the singular, dark, kink-antikink and bright soliton solutions from the governing equation have been extracted. Under the appropriate selection of parameter values, 3D, 2D and contour graphs are shown to illustrate the physical characteristics of the obtained results. All generated solutions are demonstrated to be analytically stable by the analysis of modulation instability, which also reveals the stability and movement of the waves. These solutions have extensive implications in the field of telecommunications and nonlinear fiber optics and assist in understanding the physical phenomena underlying the equation. These methods are new and standardised, and they can be applied to solve a variety of mathematical and physical problems.

This study investigates the stochastic nonlinear Schrödinger equation (SNLSE) with spatial noise intensity. Exact solutions, including trigonometric, rational, hyperbolic, periodic, dark, kink, anti-kink, and exponential forms, are achieved by utilizing the logarithmic transformation, the (G′G,1G)–expansion method, and the generalized Kudryashov method (gKM). These solutions are important for applications in nonlinear optical fibers, signal processing, communication, and engineering sciences. The effects of multiplicative noise on these solutions are analyzed through 3D, and contour visualizations by utilizing Mathematica 11. Moreover, the nonlinear dynamics of the system are analyzed by using phase portraits within bifurcation theory, describing chaotic behavior induced by external forces. Chaotic trajectories are further identified using 2D and 3D plots, time series analysis, and Lyapunov exponents. The model’s sensitivity under varying initial conditions is also examined.

Nonlinear evolution equations form the most fundamental theme in mathematical physics. The search for exact solutions of nonlinear equations has been of interest in recent years. In this paper, we obtain exact solutions of the nonlinear Jaulent–Miodek hierarchy and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff equation by using the generalized Kudryashov method. All calculations in this study have been made and checked back with the aid of the Maple packet program.

This article deals with the study of ultrasound propagation, which propagates the mechanical vibration of the molecules or of the particles of a material. It measures the speed of sound in air. For this reason, the third-order non-linear model of the Westervelt equation was chosen to be studied, as the solutions to such problems have much importance for physical purposes. In this article, we discuss the exact solitary wave solutions of the third-order non-linear model of the Westervelt equation for an acoustic pressure p representing the equation of ultrasound with high intensity, as used in acoustic tomography. Moreover, the non-linear coefficient B/A (being a part of space-dependent coefficient K), has also been investigated in this literature. This problem is solved using the Generalized Kudryashov method along with a comparison of the Modified Kudryashov method. All of the solutions have been discussed with both surface and contour plots, which shows the behavior of the solution. The images are prepared in a well-established way, showing the production of tissues inside the human body.

This study discusses a new version of the ( 2 + 1 ) -dimensional Kadomtsev-Petviashvili equation. This equation is used to model the behavior of nonlinear waves in various fields like ferromagnetic media, ion-acoustic waves in plasma physics, and fluid dynamics. It is instrumental in modeling surface and internal waves in straits or channels. The main goal of the research is to determine the exact solutions for this equation and analyze their physical characteristics. We obtain exact solutions using two improved techniques, namely the modified extended tanh-function and the modified generalized Kudryashov methods. These techniques investigate various exact solutions, such as exponential, rational, hyperbolic, and trigonometric. Besides, the sensitivity and the stability analysis of the model are presented. Additionally, three- and two-dimensional contour plots are created to present the physical behavior of the exact solutions.

In this paper we explore the new analytical soliton solutions of the truncated M-fractional nonlinear ( 1 + 1 ) -dimensional Akbota equation by applying the exp a function technique, Sardar sub-equation and generalized kudryashov techniques. Akbota is an integrable equation which is Heisenberg ferromagnetic type equation and have much importance for the analysis of curve as well as surface geometry, in optics and in magnets. The obtained results are in the form of dark, bright, periodic and other soliton solutions. The gained results are verified as well as represented by two-dimensional, three-dimensional and contour graphs. The gained results are newer than the existing results in the literature due to the use of fractional derivative. The obtained results are very helpful in optical fibers, optics, telecommunications and other fields. Hence, the gained solutions are fruitful in the future study for these models. The used techniques provide the different variety of solutions. At the end, the applied techniques are simple, fruitful and reliable to solve the other models in mathematical physics.