Remanent State in One-Dimensional Micromagnetics

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Abstract
A first integral is found for Brown's nonlinear equations in one dimension. When the external field is zero, another first integral can be found, which enables complete integration of the equations. For a unidirectional anisotropy with an easy direction perpendicular to the plane of the film, one of the integration constants is not determined by the boundary conditions, as if indicating a possibility of continuum of different remanence values for different histories. However, when a small field in the plane of the film is introduced as a perturbation, this degeneracy is removed, and the magnetization can change only in the plane defined by the field and the direction of anisotropy. There are still many discrete possible values for the remanence, each of which determines uniquely the susceptibility at that state.