Lie Symmetries, Soliton Solutions, Conservation Laws, and Stability Analysis of the Combined Kairat‐II‐X Equation

This study explores the newly proposed model known as the Kairat‐II‐X (K‐II‐X) equation, which integrates the Kairat‐II and Kairat‐X equations. The K‐II‐X equation provides a framework for describing the connections between the differential geometry of curves and the concept of equivalence aspects. We apply the Lie symmetry analysis method to investigate the vector field and symmetry reductions of the equation. Using these symmetry reductions, we obtain the exact traveling wave solutions by employing the extended Sinh–Gordon equation expansion method. Furthermore, the conservation laws for the model are derived through the multiplier method. Finally, the modulation instability of the model is examined through standard linear stability analysis. The analytical techniques and the results reported herein are novel and offer deeper insight into the structure and dynamics of this integrable system.