I investigated soliton phenomena in a prominent nonlinear fractional partial differential equation (FPDE) namely the conformable coupled Drinfeld-Sokolov-Wilson system (CCDSWS) using a novel variant of the novel extended direct algebraic method (EDAM), namely $r$+mEDAM. The conformable fractional derivatives are used to generalize the model due to the memory and hereditary features that are inherent in the fractional dynamics. The model was initially transformed into a more manageable system of integer-order nonlinear ordinary differential equations (NODEs) through the implementation of complex transformation. The obtained system of NODEs is further transformed into a system of algebraic equations, which yields, by solving new plethora of soliton solutions for CCDSWS in the form of generalized trigonometrical, exponential hyperbolical, and rational functions. Moreover, we employed 2D, 3D, and contour graphics to show the behavior of acquired solitons, making it abundantly evident that the obtained solitons take the shape of kink, anti-kink, bright, dark, bright-dark, and bell-shaped kink solitons within the framework of CCDSWS. The results confirmed the efficiency of the presented approach in finding solitonic solutions, which in its turn expands knowledge of nonlinear FPDEs. The aimed to theoretical and application perspectives in fractional solitons applicable in areas such fluid mechanics, plasma physic, optical communications, etc.

This investigation offers an innovative analytical strategy, namely the Riccati modified extended simple equation method (RMESEM), to establish and analyze soliton results of the (2+1)-dimensional dynamical generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation (GZK-BBME) in plasma physics. This equation models the physical phenomena of long waves with small and finite amplitude in magnetic plasma. Using a wave transformation, the employed transformative technique first converts GZK-BBME to a nonlinear ordinary differential equation (NODE). With the incorporation of the Riccati equation, a close-form solution is then assumed for the resultant NODE by RMESEM, which converts the NODE into a set of algebraic equations. The fresh plethora of soliton results in the form of rational, exponential, rational-hyperbolic and periodic functional cases are obtained by addressing this set of equations. Several contour, 3D, and 2D graphs are also employed to visualizes the dynamics of these constructed soliton results. These graphs demonstrate that the acquired solitons adopts the type of diverse kink solitons, including cuspon, dark, bright, lump-type, and dark-bright kinks. In addition, our proposed RMESEM shows the applications of the model by producing different traveling soliton results, providing qualitative information on the GZK-BBMEs and possible applications in dealing with other similar kinds of non-linear models.

In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ( $r+\frac{G^{\prime }}{G}$)-expansion method and the simple ( $\frac{G^{\prime }}{G}$)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.