Analysis of Karait-II-X equation: soliton solutions and modulational instability

This research presents a range of wave solutions for the recently introduced combined Kairat-II-X equation, utilizing the improved Riccati, the extended direct algebraic, and the generalized Kudryashov methods. This equation has become a cornerstone in the study of soliton dynamics, nonlinear wave phenomena, and plasma instabilities. By combining these methods with wave transformation and homogeneous balancing techniques, we derived concise, general solutions for the combined Kairat-II-X equation. The solutions include rational, exponential, trigonometric, and hyperbolic function solutions. Using appropriate values in some solutions, we visualize various soliton structures in 3D and 2D plots, revealing intriguing wave patterns. Additionally, we observed the evolution of rogue waves, such as the emergence of kink and multi-wave solitons at various time scales. The reported solutions will be very significant in equivalence and the curves’ differential geometry. Additionally, we studied the modulation instability of the combined Kairat-II-X equation to determine how the solutions of the studied model may grow exponentially without bound using the relevant parameters of the model.