Modulation instability of the ion-acoustic soliton in spin-polarized plasma under Bohm potential

The existence and stability criteria of one-dimensional ion-acoustic solitons in a quantum cold, over-dense plasma are investigated through theoretical and numerical analyses. Within the quantum magnetohydrodynamic dominant regime, distinct governing equations are employed for spin-up and spin-down electrons, with the Coulomb exchange interaction specifically considered between spin-down electrons. The Bohm potential affects both spin-up and spin-down electrons. The dispersion relations, derived up to third-order approximations, are obtained using the reductive perturbation method. Explicit expressions for instability are determined at the first-order correction, revealing extensive instability regions. The spin polarization ratio, Bohm potential coupling, soliton velocity, and electron number density influence these instability regions. By incorporating higher-order corrections, including second and third orders, a nonlinear Schrödinger-type equation for the electrostatic potential is derived. Based on this nonlinear equation, numerical instability analysis refines the first-order instability regions and reveals new instability regions that are not apparent in the first-order approximation. The analysis underscores the significant effects of quantum interactions, particularly the spin exchange effect and Bohm potential, on the instability region.