Exploring the Novel Wave Structures of the Kairat-X Equation via Two Analytical Methods

This paper aims to investigate the Kairat-X equation in the context of the ferromagnetic materials, optical fibers, differential geometry of curves, and equivalence aspects. Two efficient techniques are used to obtain new solutions: the modified extended tanh expansion method and the ( G′/G2 )-expansion function method. By applying these methods, the nonlinear ordinary differential form of the analyzed equation is obtained using the appropriate wave transform. The effective application of the proposed approaches has yielded a substantial number of analytical solutions for the model, including hyperbolic, bright-dark soliton, W-shaped soliton, and mixed-type trigonometric, rational, and trigonometric solutions. These methods are advantageous in deriving a wide variety of exact solutions; however, they can also present limitations in terms of computational complexity and the scope of applicable equations. Various graphical representations are given to enhance the understanding of the obtained solutions. To the best of our knowledge, all derived solutions are novel. Furthermore, the correctness of each solution has been verified using Maple software.