Criteria for the existence of maximal instability of wave for (1+1) higher order nonlinear Schrodinger equation for Heisenberg ferromagnetic system

We investigate the maximum modulational instability gain and soliton dynamics in a ferromagnetic spin chain governed by a (1 + 1)-dimensional higher-order nonlinear Schrödinger equation. By employing linear stability analysis, the modulational instability (MI) gain spectrum is derived, and the corresponding maximum instability gain is determined. In addition, exact soliton solutions are obtained using the Jacobi elliptic function expansion method, enabling a detailed examination of localized-mode dynamics. Particular emphasis is placed on the role of the perturbation parameter ϵ in controlling both the maximum MI gain and the evolution of solitons in the ferromagnetic spin system. To the best of our knowledge, the systematic influence of ϵ on the MI characteristics and soliton dynamics of the considered ferromagnetic model has not been previously reported. The parameter ϵ characterizes the contribution of multiple spin interactions and anisotropic effects within the ferromagnetic system. Since it appears explicitly in the higher-order dispersive and nonlinear terms of the governing equation, it plays an important role in determining the instability characteristics of nonlinear spin waves and the subsequent evolution of localized excitations. By considering different regimes of ϵ, the MI spectrum is systematically analyzed to identify the conditions associated with maximum instability. Furthermore, the dependence of soliton amplitude, shape, and propagation behavior on ϵ is investigated through analytical solutions and graphical representations. The results provide a comprehensive picture of how higher-order magnetic interactions influence the stability and nonlinear evolution of localized modes in ferromagnetic spin chains.