Spin waves have been researched in quantum physics and solid state physics since the discovery of spin wave solutions to a magnetic lattice by Felix Bloch in the early 20th century. In this thesis, we analyze a one-dimensional ferromagnetic lattice from a theoretical standpoint. In particular, we find the dispersion relation for spin waves in a one-dimensional anisotropic lattice, and determine the stability of this solution with respect to modular perturbations. This thesis aims to further establish the pathways and conditions under which spin wave solutions are unstable and lead to localized mode solutions.