Exact results in the three-dimensional Landau-Ginzburg model of magnetic inhomogeneities in uniaxial ferromagnets: I. Continuous transitions

The Landau-Ginzburg model is applied to a three-dimensional uniaxial ferromagnet in the presence of magnetic inhomogeneities. The equation of motion for the order parameter is derived and its steady-state exact, analytical solutions are found using the method of symmetry reduction for partial differential equations. All translationally invariant solutions are studied in detail and their energies are calculated exactly. Some results on spherically and cylindrically symmetric solutions are presented, as well as on solutions invariant under subgroups involving dilations.