Unveiling Soliton Dynamics: Innovative Methods and Numerical Insights for the Shynaray-IIA Equation

This paper examines exact solutions to the Shynaray-IIA equation utilizing the modified extended tanh method with the Riccati equation and generalized Arnous method. The Shynaray-IIA equation is a system of integrable coupled differential equations that includes soliton solutions. The freshly revealed soliton solutions exhibit a variety of mathematical structures, such as trigonometric, hyperbolic and logarithmic functions, which enable a thorough examination of their underlying wave behaviors. By manipulating specific parameters, a variety of solutions can be achieved, such as periodic, dark, bright and singular solitons. Each pair of solitons demonstrates mathematical symmetry. The study employs visually captivating 3D, contour and 2D graphs that emphasize the distinctive characteristics of the solutions. Furthermore, utilizing the differential transform method, the numerical solutions for the obtained soliton solutions are constructed. The findings illustrate the efficacy of the proposed analytical methodologies compared to existing methods, providing wider applications in nonlinear optics, plasma physics and wave propagation in complex media. The in-depth exploration of soliton solutions not only enhances our understanding of basic wave phenomena but also provides valuable perspectives for addressing nonlinear equations in various scientific areas.