This work focuses on the study of the fractional parametric effect on the dynamics of solitary waves for the recently introduced integrable system known as the extended (3+1)-dimensional Kairat-II equation. This equation has a variety of applications and can be used to introduce enhanced algorithms in disciplines such as optical communications, plasma physics, differential geometric engineering, oceanography, and physics. The studied equation specifically demonstrates the relationships with the differential geometry of curves as well as aspects of equivalence. The recently developed fractional derivative, known as the \(\beta \) -derivative is under consideration for discussing the studied model and securing a variety of wave structures. The wave structures of different types, including mixed, dark, singular, bright-dark, bright, complex, and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques namely the modified generalized Riccati equation mapping method and the enhanced modified extended tanh-expansion approach. To obtain the desired exact solutions, the governing equation is transformed into an ordinary differential equation by the use of an appropriate wave transformation with \(\beta \) -derivative. The methodologies employed are recognised for their effectiveness, simplicity, and adaptability, enabling the incorporation of many types of soliton solutions in a unified framework. Furthermore, to visualize the solution behavior at different parameter values, we plot graphs with relevant parameters under the effect of \(\beta \) -fractional derivatives. The findings presented in this work can enhance the understanding of the nonlinear dynamic behavior of the given system and validate the effectiveness of the adopted methods. The results obtained are valuable for comprehending nonlinear science and its associated nonlinear higher-dimensional wave fields.
