Clout of fractional time order and magnetic coupling coefficients on the soliton and modulation instability gain in the Heisenberg ferromagnetic spin chain

In this paper we utilize the auxiliary equation method to show the leverage of the fractional derivative parameter and the Magnetic Coupling Coefficients (MCC) on the procured soliton solutions and the Modulation Instability (MI) gain in Heisenberg ferromagnetic spin. The pedestal of the work is set on the ($2+1$)-fractional time order dimensional Heisenberg ferromagnetic spin chain equation which demonstrate the propagation of nonlinear waves in Heisenberg ferromagnetic spin chain system. The results collected have enclosed bright-dark soliton, dark soliton and rational solutions. Compared the acquired results with Zhao et al. (2016) [26], Akbar (2021) [27], Hosseini et al. (2021b) [28], further soliton-like solutions have been drop in and the marvel of the fractional parameter has been exposed across the spatiotemporal and plot evolution of the bright and dark soliton solutions. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the MCC jointed with the fractional time derivative order. The collected results are new in literature in our knowledge and will give the pipe to the solitary waves theory in Heisenberg ferromagnetic spin including inhomogeneity parameters.