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The Paraxial Model, a foundational concept in optics, provides a simplified yet effective approach to understanding the behavior of light in optical systems. This research investigates optical soliton solutions within the truncated time M-fractional Paraxial wave equation using the extended tanh method and the modified extended tanh method. Addressing the challenge of fractional order, we employ the shortened M-fractional derivative to eliminate it from the governing model, facilitating a detailed analysis. Through strategic manipulation of free parameters, soliton solutions are expressed in terms of hyperbolic and trigonometric functions, revealing a spectrum of soliton phenomena such as Lump soliton, rough wave, periodic soliton, and king wave for specific parameter values that are connected to the dispersal effect, the Kerr non-linearity effect and the diffraction effect of the governing equation. To enhance the understanding of these solutions, we utilize Matlab for graphical representations, providing a visual interpretation of the intricate dynamics. This study not only uncovers diverse soliton behaviors in the context of fractional wave equations but also offers valuable insights into their formation and complexity, bridging theoretical analysis with computational visualization.

The one-dimensional long water wave propagation in a nonlinear medium, including the dispersion process, is well simulated by the fractional-order modified equal-width (MEW) equation. This article establishes several recognized, standard, inclusive, and scores of typical exact wave solutions to the MEW equation using the double G'/G,1/G-expansion method. For specific parameter values, kink, periodic, periodic-singular, singular-kink, and other forms of solitons can be recovered from general solutions. The effect of the fractional parameter on wave forms has also been analyzed by depicting several graphs for different values of the fractional-order α. In order to illustrate the potential characteristics, three- and two-dimensional combined plots using Maple have been drawn. It has been established that the introduced approach is a potential tool for extracting new exact solutions to various nonlinear evolution equations (NLEEs) arising in engineering, science, and applied mathematics.

The Kundu–Mukherjee–Naskar (KMN) model is considered for describing the pulse propagation in optical fibres and communication systems. Two efficient approaches via the exp (−φ(ξ))-expansion and the improved F-expansion schemes are applied to the KMN model and in order to convert the KMN equation to an ODE, we test the complex wave conversions and present some optical soliton solutions, which are expressed in terms of hyperbolic, rational and trigonometric functions. We create three dimensional and two-dimensional images to explain the underlying optical wave properties of the KMN equation and compare our solutions with the ones in the literature. Our example shows that the two adopted methods provide an efficient mathematical tool for constructing new optical wave solutions to nonlinear partial differential equations arising in fibre optic communication systems.

The present paper applies the exp (−φ(ξ))-expansion and the extended tanh-function methods to the (2+1)-dimensional Heisenberg Ferromagnetic Spin Chain (HFSC) equation. The applied methods acquire some new exact traveling wave solutions to the HFSC equation, which are representing the hyperbolic, trigonometric, exponential and rational function solutions. All solutions exhibit distinct physical configurations, such as the periodic, dark and singular soliton solutions. Three dimensional (3D) and two dimensional (2D) cross sectional graphics of some obtained solutions are confirmed the periodic, dark and singular behaviors. Furthermore, the conformable derivative is also considered and discussed for aforesaid methods to the HFSC equation. As outcomes, some new optical solutions are also attained in terms of fractionality. Obtained new solutions ensured that aforesaid methods are the reliable treatment for seeking nonlinear phenomena to HFSC equation as well as any NLEEs arising in mathematical physics and fiber optics.

In this paper, we extract the enormous soliton solutions from the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation with the aid of enhanced (G′/G)-expansion method in nonlinear dynamics of magnets. All soliton solutions represent as standings of hyperbolic function solution and trigonometric functions solution with arbitrary parameters. The nonlinear wave prearrangement is analyzed and showed in 3D, 2D, and contour plots with explicit estimations of the free parameters plotted in this literature. The enhanced (G′/G)-expansion method is a suitable solution of nonlinear waves that improve the combination of magnetic models rises in the field of engineering as well.

Utilizing of illustrative scheming programming, the study inspects the careful voyaging wave engagements from the nonlinear time fractional modified Kawahara equation (mKE) by employing the advanced exp−φξ-expansion policy in terms of trigonometric, hyperbolic, and rational function through some treasured fractional order derivative and free parameters. The undercurrents of nonlinear wave answer are scrutinized and confirmed by MATLAB in 3D and 2D plots, and density plot by specific values of the convoluted parameters is designed. Our preferred advanced exp−φξ-expansion technique which is parallel to (G′/G) expansion technique is trustworthy dealing for searching significant nonlinear waves that progress a modification of dynamic depictions that ascend in mathematical physics and engineering grounds.

In this article, we implement the modified simple equation (MSE) and improve F-expansion method to find the exact solutions of the (2 ​+ ​1)-dimensional Heisenberg Ferromagnetic Spin Chain (HFSC) equation and construct traveling wave solutions in terms of hyperbolic functions, trigonometric functions and rational functions with arbitrary parameters. Some 2D and 3D graphics of analytical solutions obtained are presented in this paper. The methods are capable and highly effective mathematical tool for extracting exact traveling wave solutions to nonlinear evolution equations (NLEEs) arising in mathematical physics and engineering fields.

The main intension of this paper is to extract new and further general analytical wave solutions to the (2 + 1)-dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation in the sense of conformable derivative by implementing the advanced exp(-ϕ(ξ))-expansion method. This method is a particular invention of the generalized exp(-ϕ(ξ))-expansion method. By the virtue of the advanced exp(-ϕ(ξ))-expansion method, a series of kink, singular kink, soliton, combined soliton, and periodic wave solutions are constructed to our preferred space time-fractional (2 + 1)- dimensional AKNS equation. An extensive class of new exact traveling wave solutions are transpired in terms of, hyperbolic, trigonometric, and rational functions. To express the underlying propagated features, some attained solutions are exhibited by making their three-dimensional (3D), two-dimensional (2D) combined, and 2D line plot with the help of computational packages MATLAB. All plots are given to show the proper wave features through the founded solutions to the studied equation with particular preferring of the selected parameters. Moreover, it may conclude that the attained solutions and their physical features might be helpful to comprehend the water wave propagation in water wave mechanics.

This study integrates the (1 + 1)-dimensional time M-fractional Lonngren-wave (tM-fLW) equation. This equation is useful to describe the phenomenon of the electric signals in telegraph lines at the source of the tunnel diode. To extract some soliton solutions, two reliable analytical techniques, the new form of modified Kudryashov’s and the simplest equation, are used to accomplish this job. By implementing these techniques, the obtained solutions are expressed as rational, exponential, trigonometric, and hyperbolic functions with some free constraints. For the special values of the constraints, we obtained w-shape wave, bell-shaped wave, linked rogue wave, and multi-bell wave solutions by the NMK method, and dark bell and bright bell-shaped wave, linked rogue wave, periodic rogue wave, and w-shape wave by the SE method. The graphical depiction analyzes these traits, demonstrating the validity and efficacy of the suggested techniques. The obtained solutions, which have never been done before and show a good balance between the nonlinear physical components, are what make this research novel. It is remarkable to perceive that the simplest equation technique and the new form of modified Kudryashov’s technique are relaxed, companionable, and authoritative mathematical tools to elucidate a non-linear model. The movement role of the waves is explored and the modulation instability analysis is used to discuss the stability analysis of the obtained solutions, confirming that all created solutions are accurate and stable.

This work explores diverse novel soliton solutions of two fractional nonlinear models, namely the truncated time M-fractional Chafee-Infante (tM-fCI) and truncated time M-fractional Landau-Ginzburg-Higgs (tM-fLGH) models. The several soliton waves of time M-fractional Chafee-Infante model describe the stability of waves in a dispersive fashion, homogeneous medium and gas diffusion, and the solitary waves of time M-fractional Landau-Ginzburg-Higgs model are used to characterize the drift cyclotron movement for coherent ion-cyclotrons in a geometrically chaotic plasma. A confirmed unified technique exploits soliton solutions of considered fractional models. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. Keeping special values of the constraint, this inquisition achieved kink shape, the collision of kink type and lump wave, the collision of lump and bell type, periodic lump wave, bell shape, some periodic soliton waves for time M-fractional Chafee-Infante and periodic lump, and some diverse periodic and solitary waves for time M-fractional Landau-Ginzburg-Higgs model successfully. The required solutions in this work have many constructive descriptions, and corporal behaviors have been incorporated through some abundant 3D figures with density plots. We compare the m-fractional derivative with the beta fractional derivative and the classical form of these models in two-dimensional plots. Comparisons with others' results are given likewise.

This work explores diverse novel soliton solutions due to fractional derivative, dispersive, and nonlinearity effects for the nonlinear time M-fractional paraxial wave equation. The advanced exp [-φ(ξ)] expansion method integrates the nonlinear M-fractional Paraxial wave equation for achieving creative solitonic and traveling wave envelopes to reconnoiter such dynamics. As a result, trigonometric and hyperbolic solutions have been found via the proposed method. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. For any chosen set of the allowed parameters 3D, 2D and density plots illustrate, this inquisition achieved kink shape, the collision of kink type and rogue wave, periodic rogue wave, some distinct singular periodic soliton waves for time M-fractional Paraxial wave equation. As certain nonlinear effects cancel out dispersion effects, optical solitons typically can travel great distances without dissipating. We have constructed reasonable soliton solutions and managed the actual meaning of the acquired solutions of action by characterizing the particular advantages of the summarized parameters by the portrayal of figures and by interpreting the physical occurrences. New precise voyaging wave configurations are obtained using symbolic computation and the previously described methodologies. However, the movement role of the waves is explored, and the modulation instability analysis is used to describe the stability of waves in a dispersive fashion of the obtained solutions, confirming that all created solutions are precise and stable.

With the assistance of representative calculation programming, the present paper examines the careful voyaging wave arrangements from the general (2+1)-dimensional nonlinear development conditions by utilizing the Improved F-expansion strategy. As a result, the used technique is effectively utilized and recently delivered some definite voyaging wave arrangements. The recently created arrangements have been communicated as far as trigonometric and hyperbolic capacities. The created arrangements have been returned into their relating condition with the guide of emblematic calculation programming Maple. Among the produced solutions, some solutions have been visualized by 3D and 2D line graphs under the choice of suitable arbitrary parameters to show their physical interpretation. The delivered arrangements show the intensity of the executed technique to evaluate the accurate arrangements of the nonlinear (2+1)-dimensional nonlinear advancement conditions, which are reasonably pertinent for using nonlinear science, scientific material science and designing. The Improved F-expansion method is a reliable treatment for searching essential nonlinear waves that enrich a variety of dynamic models that arise in engineering fields.