Integrability, geometry and wave solutions of some Kairat equations

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In this paper, we study some Kairat equations. The relation between the motion of curves and Kairat equations is established. The geometrical equivalence between the Kairat-I equation and the Kairat-II equation is proved. We also proves that these equations is gauge equivalent to each other. Three types traveling wave solutions of the Kairat-II equation as well as its some integrals of motion are found. The techniques used in this paper can be adopted to study other integrable spin systems and nonlinear models. In particular, using these methods we study some Zhanbota equations.