In this paper, we present an innovative approach to acquire the exact solutions of the Shynaray-IIA equations (S-IIAE), by using the improved modified Sardar sub-equation method (IMSSEM). The S-IIAE are nonlinear and coupled partial differential equations that arise in various fields of physics and engineering such as optical fibers and ferromagnetic materials. The IMSSEM is applied to S-IIAE; we successfully derived exact solutions that accurately described the wave propagation behavior of the system under consideration. The obtained solutions include rational, trigonometric, and trigonometric hyperbolic function solutions. The obtained solutions are concise and offer a deeper insight into the dynamics and characteristics of the S-IIAE. Moreover, some of the new solutions to S-IIAE are plotted in different dimensions through which bright, anti-kink and bright solitary wave structures are established. The results of the study also indicated that the proposed method is a valuable approach for achieving analytical solutions to a wide range of nonlinear partial differential equations.

In this paper, we investigate the (2+1)-dimensional complex modified Korteweg-de Vries (CmKdV) system using the conformable derivative. The CmKdV system is a beneficial model in the field of nonlinear wave theory such as fluid flow, optical communication, and plasma physics. Explicit solutions are constructed, including periodic, solitary, and shock waves form using the Jacobi elliptic function expansion method. The solutions obtained are visually presented in various dimensions using Mathematica, providing a clear physical understanding of the effects of the conformable fractional derivative. This research enhances understanding of soliton behavior in complex nonlinear systems and demonstrates the effectiveness of combining conformable derivatives with analytical methods, while also providing new insights into the dynamics and diverse forms of propagating fluid waves.

This study examines a novel use of the Jacobi elliptic function expansion method to solve the Shynaray-IIA equation, a significant nonlinear partial differential equation that arises in optical fiber, plasma physics, surface symmetry geometry, and many other mathematical physics domains. This kind of solution has never been attained in research prior to this study. Numerous properties of a particular class of solutions, called the Jacobi elliptic functions, make them useful for the analytical solution of a wide range of nonlinear problems. Using this powerful method, we derive a set of exact solutions for the Shynaray-IIA equation, shedding light on its complex dynamics and behaviour. The proposed method is shown to be highly effective in obtaining exact solutions in terms of Jacobi elliptic functions, such as dark, bright, periodic, dark-bright, dark-periodic, bright periodic, singular, and other various types of solitons. Furthermore, a detailed analysis is conducted on the convergence and accuracy of the obtained solutions. The outcomes of this study extend the applicability of the Jacobi elliptic function approach to a novel class of non-linear models and provide valuable insights into the dynamics of Shynaray-IIA equation. This study advances the creation of efficient mathematical instruments for resolving intricate nonlinear phenomena across a range of scientific fields.