The New Combined Kairat-II-X Differential Equation: Diversity of Solitary Wave Structures via New Techniques
Usman Younas, Joriah Muhammad, Hajar F. Ismael, Tukur Abdulkadir Sulaıman, Homan Emadifar, Karim K. Ahmed
Abstract
This article examines the various dynamical behaviors of soliton solutions to the Kairat-II-X equation, which combines two significant models, the Kairat-II and Kairat-X equations. In this equation, the connections between the concept of equivalence and the differential geometry of curves are illustrated. Analytically studying the new combined Kairat-II-X differential equation is driven by the critical need to fundamentally understand the complex interactions and emergent behaviors arising from merging the distinct mathematical structures and physical principles embodied in the Kairat-II and X components. The recently developed integration tools, namely, the generalized (G’/G) method, generalized projective Riccati equation method, and Riccati modified extended simple equation method for extracting a diverse array of soliton solutions for the proposed model. The obtained solutions are expressed in the forms like kink, dark, bright-dark, singular, bright, complex, and combined solitons. Moreover, the hyperbolic, periodic, and exponential function solutions are discussed. A variety of graphs in three, two dimensional as well as contour graphs for explaining the physical dynamics of solutions are sketched with the assistance of suitable parameters. In contrast with existing methodologies, the techniques implemented offer a straightforward and simplified approximation of all solutions. This research makes a substantial contribution to the disciplines of nonlinear science and higher-dimensional nonlinear wave fields by clarifying the nonlinear dynamic characteristics of a system and validating the effectiveness of current methodologies.