  We establish the existence of spatially localised one-dimensional free
surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a
general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic
equations can be derived from a variational principle that allows one to
formulate them as an (infinite-dimensional) spatial Hamiltonian system in which
the unbounded free-surface direction plays the role of time. A centre-manifold
reduction technique converts the problem for small solutions near onset to an
equivalent Hamiltonian system with finitely many degrees of freedom.
Normal-form theory yields the existence of homoclinic solutions to the reduced
system, which correspond to spatially localised solutions of the
ferrohydrostatic equations.

Analysis of PDEs