On the exploring of the new complex and mixed hyperbolic properties of the nonlinear Kairat-II-X models

Abstract In this paper, we apply the powerful sine-Gordon expansion method (SGEM) and the rational sine-Gordon expansion method (RSGEM), combined with computational tools, to derive new traveling wave soliton solutions for the Kairat-II and Kairat-X equations, two significant nonlinear models in wave dynamics. These models play a crucial role in understanding nonlinear evolution equations with applications in mathematical physics, fluid dynamics, optical fibers, and plasma systems. Our study introduces a diverse set of solutions, including novel complex, rational, hyperbolic, and mixed trigonometric-hyperbolic traveling waves, each revealing intricate nonlinear behaviors such as wave stability, energy transfer, and wave-particle interactions. The graphical representations of these solutions, including three-dimensional (3D), two-dimensional (2D), and contour plots, illustrate the dynamic properties and physical implications of the derived wave structures. Key findings indicate that the SGEM and RSGEM methods efficiently generate solutions that capture localized wave formations, solitonic interactions, and rational wave structures that contribute to understanding the dispersion and stability of nonlinear waves. Our results demonstrate that the RSGEM, as an extension of the SGEM, provides an enriched framework for exploring more intricate wave dynamics by incorporating rational trigonometric functions. The physical significance of the parametric dependencies of these solutions is also examined. In addition, several simulations are presented, including contour surfaces and revolutionary wave behaviors, based on different parameter selections. Revolutionary surfaces, defined by height and radius as independent variables, are extracted to further illustrate wave dynamics. A detailed conclusion summarizing the findings is included at the end of this paper.