Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation
arXiv:1610.07760 · doi:10.1016/j.physd.2017.03.004
Abstract
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.
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- Localised Radial Patterns on the Free Surface of a Ferrofluid
- Snaking bifurcations of localized patterns on ring lattices
- A free boundary approach to the Rosensweig instability of ferrofluids
- Small-amplitude static periodic patterns at a fluid-ferrofluid interface
- On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions