Two-Dimensional Solitons on the Surface of Magnetic Fluids

We report an observation of a stable solitonlike structure on the surface of a ferrofluid, generated by a local perturbation in the hysteretic regime of the Rosensweig instability. Unlike other pattern-forming systems with localized 2D structures, magnetic fluids are characterized by energy conservation; hence their mechanism of soliton stabilization is different from the previously discussed gain-loss balance mechanism. The radioscopic measurements of the soliton’s surface profile suggest that locking on the wavelength defined by the nonmonotonic dispersion curve is instrumental in its stabilization.

Figure 1

Sketch of the experimental setup. See text for details.

Figure 2

Surface reliefs as reconstructed from the radioscopic images for (a) $B=8.922\mathrm{mT}$, and (b), (c) 10.407 mT. Each color indicates a layer thickness of 1 mm.

Figure 3

The amplitude of the pattern for $r<11\mathrm{mm}$ vs the magnetic induction. The crosses (dots) mark the values for increasing (decreasing) induction, respectively. The solid (dashed) lines display the least square fit to the roots $A_{\pm }=[\gamma (1+ϵ)\pm \sqrt{{\gamma }^2(1+ϵ{)}^2+4ϵg}]/(2g)$, of the amplitude equation $ϵA+\gamma (1+ϵ)A^2-g{A}^3=0$ of Ref. 7 with $\gamma =0.281{\mathrm{mm}}^{-1}$ and $g=0.062{\mathrm{mm}}^{-2}$. The full circles (squares) give the amplitude of the localized spike initiated at $B=8.91\mathrm{mT}$ for increasing (decreasing) induction, respectively.

Figure 4

A single soliton surrounded by the unperturbed magnetic liquid. The magnetic induction generated by the Helmholtz coils amounts to $B=8.91\mathrm{mT}$. The amplitude of the local pulse which produced the soliton was $B_{+}=0.68\mathrm{mT}$ at the bottom of the vessel.

Figure 5

The filled squares mark the profile of one period of the hexagonal pattern, measured at $B=9.07\mathrm{mT}$ in the center of the vessel; $r<8.8\mathrm{mm}$. Azimuthally averaged height profiles of two different solitons, measured at the same induction, are depicted by open symbols (one) and a dashed line (the other).

Figure 6

Nine solitons at $B=8.91\mathrm{mT}$.

Figure 7

The surface energy of the liquid layer for increasing (open squares) and decreasing (circles) magnetic induction. The full circles mark the increase of $E_{\mathrm{s}}$ through the successive generation of solitons at $B=8.91\mathrm{mT}$ (see also inset). To reduce the influence of the perimeter spikes, only the area $r<0.88R$ was covered, where $R$ is the radius of the vessel.

Figure 8

The solitons can form moleculelike clusters: (a) di-, (b) tri-, (c) tetra-, (d) penta-, (e) hepta-, and (f) oligomers. The height is indicated by switching between black and white after each mm. Here $B=8.71\mathrm{mT}$; each panel covers the area of $(87\mathrm{mm}{)}^2$.