Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation
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.
Keywords
Cite
@article{arxiv.1610.07760,
title = {Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation},
author = {Mark D. Groves and David J. B. Lloyd and Athanasios Stylianou},
journal= {arXiv preprint arXiv:1610.07760},
year = {2018}
}