Existence of Small Amplitude Localised Patterns on the Surface of a Ferrofluid

"We investigate the existence of localised radial patterns on the surface of a ferrofluid under the influence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded at their core and decay exponentially away in the far-field. For a finite-depth, infinite expanse of ferrofluid with a linear magnetisation law, we formulate the system as a free surface problem which can be expressed as a non-autonomous PDE. By decomposing onto a suitable set of basis vectors which is independent of the radius coordinate, the problem is reduced to an infinite system of radial ODEs. We construct localinvariant manifolds in the core and far-field regions, and introduce geometric blow-up coordinates in order to track stable solutions through the far-field; any solution that lies on the intersection of the core and far-field manifolds is a localised solution. Three classes of localised radial patterns are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift-Hohenberg equation; different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which thesesolutions emerge are identified.The existence of localised cellular patches in the Swift-Hohenberg equation, which acts as a toy model for the ferrofluid problem, is also investigated. We approximate the planar Swift-Hohenberg equation by a truncated Fourier-polar expansion with truncation order N; after projecting onto each Fourier mode, we arrive at a system of N + 1 radial ODEs. Applying the same techniques as seen for localised radial solutions, we establish the existence of localised cellular patches with a general 2k-lattice, subject to a matching condition that depends on both N and k. Numerical and analytical techniques are applied to investigate this matching condition for arbitrary k and N; small-N solutions are found numerically and continued in parameter space to larger amplitudes, where they might be observable in practice.