  Magnetic materials host a wealth of nonlinear dynamics, textures, and
topological defects. This is possible due to the competition between strong
nonlinearity and dispersion that act at the atomic scale as well as long-range
interactions. However, these features are difficult to analytically and
numerically study because of the vastly different temporal and spatial scales
involved. Here, we present a pseudo-spectral approach for the Landau-Lifshitz
equation that invokes energy and momentum conservation embodied in the magnon
dispersion relation to accurately describe both atomic and continuum limits.
Furthermore, this approach enables analytical study at every scale. We show the
applicability of this model in both the continuum and atomic limit by
investigating modulational instability and ultrafast evolution of magnetization
due to transient grating, respectively, in a 1D ferromagnetic chain with
perpendicular magnetic anisotropy. This model provides the possibility of
grid-independent multiscale numerical approaches that will enable the
description of singularities within a single framework.
