The Kuralay-Ma-Myrzakulov equation and the Akbota–Myrzakulov-Tolkynay-Zhaidary equation: Integrability, equivalence, reductions, and solutions

The Kuralay-Ma-Myrzakulov equation (KMME) and the Akbota- Myrzakulov-Tolkynay-Zhaidary equation (AMTZE) are studied. The integrability of these two equations is achieved by the existence of corresponding Lax pairs. The corresponding Lax representations are presented. Gauge equivalence between these two integrable equations is proved. It is shown that the KMME admits two integrable reductions, namely, the Manukure-Zhanbota equation and the Manukure-Zhaidary equation. Similarly, the AMTZE has two integrable reductions, namely, the Kairat-Kuralay-Myrzakulov-Shynaray equation (KKMSE) and the Wu-Zhang equation (WZE). From these results, it follows that the Manukure- Zhanbota equation is gauge equivalent to the KKMSE. At the same time, the Manukure-Zhaidary equation and the Wu-Zhang equation are gauge equivalent to each other. Some exact traveling wave solutions of the AMTZE are presented. These solutions demonstrate a variety of structures, including Jacobi elliptic, trigonometric, soliton, and rational forms. The results are illustrated through 3D and contour plots, which clearly depict the system's behavior during momentum propagation and help identify suitable parameter values. This graphical analysis offers important insights into the properties and dynamics of the soliton solutions derived from the integrable AMTZE equation.