The current study employs a transformation-based analytical technique, namely Riccati modified extended simple equation method (RMESEM) to construct and examine soliton phenomena in a prominent $(2+1)$-dimensional mathematical model namely Nizhnik-Novikov-Veselov system (NNVS), which has potential applications in exponentially localized structure interactions. The suggested RMESEM uses a variable transformation to turn the desired NNVS into a nonlinear ordinary differential equation (NODE). The resulting NODE is then assumed to have a closed-form solution, converting it into an algebraic system of equations. When the resulting algebraic system is dealt with RMESEM's strategy using Maple, a range of dark and bright soliton solutions in the form of rational, exponential, periodic, hyperbolic and rational-hyperbolic functions are revealed. Some 3D, contour and 2D graphs are plotted for visual representations of these soliton solutions that demonstrate their versatility. The findings deepen our understanding of the NNVS's dynamics, shedding light on its behavior and potential uses.

This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics.

In this research, we investigated the integrable fractional reduced spin Hirota-Maxwell-Bloch system with Katugampola fractional derivatives, a crucial model for analyzing the femtosecond pulses transmitted within an erbium-doped fiber. The modified unified method and the $(G^{\prime }/G)$-expansion techniques were employed to acquire analytical soliton solutions comprising bright soliton, mix dark-bright soliton, dark soliton, perturbed dark and bright soliton, multi-soliton, and periodic dark-bright soliton. The fundamental mechanics of the model was revealed by dynamically displaying some of the found solutions using 2D and 3D graphs. The $\rho$-derivative framework was used to analyze the effect of the space-fractional derivative on the supplied model, offering a more dynamic and applicable way to improve the accuracy of the findings. Additionally, a range of graphical representations were used to show the perturbed system's time series plots and the planar system's phase portraits under the Hamiltonian analysis to emphasize the model's significance and dynamic behavior in erbium-doped fiber. Our findings of this study are expected to have important ramifications for soliton theory, erbium-doped fiber, optical fibers, physical engineering, and nonlinear dynamics. Additionally, the study shows that the $(\frac{G^{\prime }}{G})$-expansion method and the modified unified approach are simple, robust, and efficient methods that produce many soliton solutions for a range of nonlinear fractional partial differential equations in the mathematical sciences.

This research study explores novel multitude of optical soliton solutions of the $(2+1)$-dimensional nonlinear paraxial wave equation by three ansatzes, namely the generalised Kudryashov-auxiliary method, tan-cot method, and tanh-coth method. The model under consideration is particularly applied to the study of the wave propagation in nonlinear materials such as Kerr media. Using the simulation tool Maple, we compute breather and other optical soliton solutions in the forms of Jacobian elliptic, hyperbolic, periodic breather, breather-interaction, dark and bright solutions for the selected nonlinear paraxial wave equation with the aid of the suggested techniques. We also provide a series of two- and three-dimensional plots that represent the dynamics and interaction of the identified optical soliton solutions by assigning numerical values to the involved free parameters. We also carry out dynamical analysis to determine the stability of the model to the variation in the parameters and initial conditions and to gain a clearer understanding of how the system is prone to chaos. The obtained outcomes are very vital in fiber optics, nonlinear optics, and communication systems. Finally, the techniques used provide clear-cut soliton solutions of nonlinear partial differential equations, which can enhance the study of nonlinear wave phenomena and provide new insights into the dynamics of other complex systems.