Abstract
The object of this study was the Nurshuak-Tolkynay-Myrzakulov (NTM) system with $ M $-fractional in both time and space. Using appropriate traveling-wave transformations, the fractional nonlinear system was reduced to a nonlinear ordinary differential equation. Exact analytical solutions were constructed using the generalized Kudryashov method and the Sardar subequation method. The obtained solutions contained hyperbolic, trigonometric, rational, and Jacobi elliptic (cnoidal) wave structures that described localized solitons, periodic waves and algebraically decaying profiles. The influence of the fractional-order parameter $ \eta $ on the amplitude and localization width and propagation characteristics of the nonlinear waves used two- and three-dimensional graphical representations. In the limit $ \eta\rightarrow 1, $ all solutions recovered the corresponding exact solutions of the classical integer-order NTM system. This was the first study of the $ M $-fractional NTM system and its exact analytical solutions.