Analytic investigation of the complex nonlinear (2+1)-dimensional Akbota-Myrzakulov-Tolkynay-Zhaidary equation via the new efficient improvement method

This paper investigated a new, complex, triple-integrable system in two spatial dimensions called the Akbota-Myrzakulov-Tolkynay-Zhaidary equation (AMTZE). The AMTZE originates from reducing a larger multi-component geometric spinning system. Aiming to extract some new exact solutions to the updated family of Myrzakulov-type integrable models, we reduce the triple system to a third-order ordinary differential equation (ODE) using a traveling wave transformation. Thereafter, we solved this equation with the modified generalized Kudryashov method. This method uses a rational solution built from an auxiliary Riccati-type equation. A balancing rule controls the form and the shape of the solution. We found several families of exact solutions for the complex-valued functions $ q(x, y, t) $, $ u(x, y, t) $, and $ r(x, y, t) $. These include bright solitons, singular solitons, and mixed wave shapes written in the form of $ \tanh $ and $ \coth $ functions. We emphasize that among the six solution families obtained, only Family 1.1 yields real frequencies $ \omega $ and thus corresponds to standard propagating traveling waves. The remaining five families give imaginary frequencies and represent non-propagating modes. We presented how the free parameters change the height, width, and speed of the waves. Three-dimensional plots, contour plots, and time-evolution plots made these changes easy to visualize. Our results demonstrated that the method works well on this kind of higher-order, three-dimensional triple system. This work also provided a clear distinction between physically propagating and non-propagating solution branches. All the generated solutions have been verified and satisfy the associated model.