Nucleation of Ferro-solitons and Localised Ferro-patterns

In the presence of a vertically directed static magnetic field of sufficient strength, experiments involving a plate of ferrofluid (colloidal suspensions of ferromagnetic particles in a suitable carrier fluid) have been observed to form radially symmetric localised deformations of the surface and other patterns such as hexagons. The understanding of ferrofluids has led to many diverse applications such as liquid seals around spinning drive shafts in hard disks, radar absorbent paint used on aeroplanes and cancer detection/treatment.The ferrofluid experiments appear to be unique to soliton forming systems since they do not require a continuous input of energy to stabilise the solitons and the system is conservative, providing an excellent test bed for theories of localised pattern formation. Furthermore, unlike most soliton bearing systems, the ferrofluid experiment exhibits stable spots where there is no hysteresis between stripes and the quiescent state. A partial theoretical understanding of this phenomenon has recently been found by the PI in the context of reaction-diffusion systems near a Turing instability. However, spots are typically found to be unstable to hexagonal perturbations in reaction-diffusion systems with no hysteresis between stripes and quiescence. This is counter to what is observed in the ferrofluid experiment. Also, the ferrofluid experiment is a free-boundary problem where the analysis for reaction-diffusion systems cannot be applied.The proposed research holds promise of revealing a new mechanism for the creation of stable spots for applications in optical communication devices. In addition, there is a potential for industrial beneficiaries: magnetically controlled bearings and positioning systems may benefit from the ability to localise parts of the ferrofluid. It is also of interest to predict the properties required of a ferrofluid for localised hexagon patches to be observed experimentally for the first time; Ferrofluids are very expensive and so accurate predictions of the required susceptibility coefficients to observe hexagon patches are critical.The aim of the proposal is to provide theoretical insight for experimentalists working on ferrofluids (working with Dr. Reinhard Richter, Bayreuth) and other soliton bearing systems while also providing new analytical techniques for free-boundary problems (such as the ferrofluid problem) in collaboration with Prof. Bjorn Sandstede. With this in mind, the research programme is split into two projects; project A will carry out the analysis of axisymmetric ferrosolitons and project B will carry out the numerical investigation of general localised ferro-patterns.

The proposed research will impact on various research communities due to its inherent interdisciplinary approach with experimentalists and potential to address aspects of fundamental as well as applied research. Stationary spatially-localised patterns are found in chemical processes, granular media, viscous fluids, plasmas, combustion, liquid crystals, cylinder buckling, lattice systems, neural networks and solidification problems. From a theoretical perspective, the proposed research holds the promise of revealing the basic mechanism for the formation of axisymmetric localised patterns (spots) in a far larger region of existence than that known for reaction-diffusion systems. Research groups working on localised states will take advantage of the theoretical results on the emergence of spots, as they will be able to single out regions in parameter space where localised patterns are expected to occur. This is likely to occur on a medium time-scale (three to five years from the beginning of the proposal). On a longer time-scale, understanding the consequences of instabilities in free-surface problems like the ferrofluid and Faraday experiments could lead to the discovery of new types of structures that have not been observed to date. In addition, there is a potential for industrial beneficiaries: magnetically controlled bearings and positioning systems may benefit from the ability to localise parts of the ferrofluid. This is a key part of the proposal where the theoretical work is anticipated to feed into new experiments that may lead to new structures being observed that will have industrial applications. To ensure that beneficiaries will be able to take advantage of the research programme, we identify four activities. For each of them, we list the main objectives and collaborators. 1. Workshops and minisymposia (PI, Richter, Sandstede) - To disseminate results of the programme and engage with industry. 2. Website (PI, Sandstede) - To stimulate interactions with a wider network of beneficiaries and advertise numerical continuation routines. 3. Numerical package (PI) - To provide beneficiaries with a starting point for other applications. 4. Educational activities (PI, Sandstede) - To broaden the impact on non-scientific audience. A detailed description of these initiatives and of my previous experience in transferring knowledge to third parties can be found in the Impact Plan written as part of the proposal.