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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.

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.

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.

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.