Effect of impurity on modulational instability of localized modes in a discrete quantum ferromagnetic spin chain

The modulational instability (MI) of localized modes in a discrete quantum ferromagnetic spin chain with the effect of impurity is studied in this paper. The model of Heisenberg ferromagnetic spin chain is considered and dynamical equation is derived by the use of Glauber's coherent state method along with Holstein-Primakoff (H-P) bosonic representation of spin operators. The dynamical equation is in the form of discrete nonlinear Schr$\ddot{o}$dinger (DNLS) equation which describes the complete dynamics of ferromagnetic spin chain. The MI technique is applied to the DNLS equation and obtains the dispersion relation which relates the frequency and wavenumber of the modulating perturbations. Using dispersion relation, we establish the stability criteria of the spin waves in ferromagnetic spin chain and explore how the presence of impurity supports the formation of localized modes as shown in the stability/instability profile. With the use of graphical illustrations of MI, the strength of impurity on the stability/instability of localized modes in ferromagnetic spin chain is discussed. Through numerical simulation, we explore how the role of impurity supports the excitation of localized modes in ferromagnetic chain and also analyze the short time and long time evolution of localized wave in ferromagnetic spin chain.