Exploring the Novel Soliton Solutions of the Kairat-X Equation via Two Analytical Methods

In this study, the Kairat-X equation is investigated within the framework of the differential geometry of curves and their equivalence directions. By employing an appropriate wave transformation, the governing nonlinear partial differential equation is reduced to a nonlinear ordinary differential equation. Two analytical techniques, namely the modified extended tanh expansion method (METEM) and the $$\left( \frac{G'}{G^2} \right) $$-expansion function method expansion function method, are applied to construct exact traveling wave solutions. As a result, a rich variety of new analytical solutions are obtained, including hyperbolic, kink-type, solitary wave, complex bright–dark soliton, singular periodic, as well as mixed trigonometric, rational, and trigonometric–rational solutions. The diversity of the derived solutions highlights the effectiveness of the employed methods in capturing different wave structures of the Kairat-X equation. Graphical illustrations are presented to demonstrate the physical behaviors and characteristics of the obtained solutions. All solutions are symbolically verified using Maple software, confirming their validity. To the best of our knowledge, the results reported in this work are new and contribute to the analytical understanding of nonlinear evolution equations arising in differential geometric contexts.