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Fractional differential equations are being used to define numerous physical phenomena instead of conventional ordinary or partial differential equations. The secret is to get more generalization of the analysis, hence its solutions. The applications vary between electrical circuit modeling, oscillating circuits, shallow water behavior, viscous fluids, solution transportation, control theory and ultrafast optics. Sine–Gordon Expansion (SGE) Method is being employed hereafter to fully investigate the space–time conformable fractional Vakhnenko–Parkes equation and its modified version. Based on SGE method, the fractional models are transformed into an equivalent ordinary differential equation. The solutions are very important in studying the behavior of ultrafast optics in fibers or waveguides and the wave propagation throughout fiber optics. The solutions are formatted as solitons and are illustrated graphically. The illustrations show that the traveling waves inside fiber optics behave like solitons with traveling peak values according to the wave velocity. One other mode of propagation is to travel in kink shapes. Additionally, the ultrafast optical waves may propagate in shockwaves mode if the waves propagate at ultra-high speeds. [ABSTRACT FROM AUTHOR]

This paper investigates (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system equations using the improved Sadar subequation method. It uncovers analytical solutions, including dark solitons, bright solitons, and periodic waves. The dynamic behavior of these solutions is illustrated through 2D and 3D plots by adjusting parameters. The results highlight the effectiveness and simplicity of the proposed methods, providing a versatile approach to obtaining various traveling wave solutions. These findings are expected to advance the shallow water theory of ideal fluids. [ABSTRACT FROM AUTHOR]

The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is investigated in this study. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is an extension of Calogero–Bogoyavlenskii–Schiff equation that describes the movement of Riemann waves along y-axis while long waves moves along the x-axis. The dynamics of Riemann waves is one of the most significant applications including tsunami in rivers, internal waves in oceans and magento-sound waves in plasmas. Finding new precise solutions with the assistance of a relatively new extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique is the primary objective of this effort. The suggested techniques are important tools in the fields of mathematical physics. Successful extraction of hyperbolic, rational, and trigonometric function solutions are achieved by using the proposed analytical methods. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is studied for the first time using extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique in this work and novel solutions are observed. 3D plots, contour plots and 2D plots are used to depict the dynamics of the extracted solutions. [ABSTRACT FROM AUTHOR]