Pattern formation in continuous and discrete systems.

There is a great deal of interest in nonlinear differential and difference equations that exhibit homoclinic snaking. Briefly described, in spatially extended dynamical systems featuring subcritical Turing instabilities, there generally exists a parameter value (the Maxwell point) near which stable stationary fronts separate regions in space from where the solution is either uniform or spatially oscillating. In such cases, the solution curves of localised patterns 'snake' back and forth across a bifurcation diagram in a narrow region of parameter space. These snaking bifurcations arise in many experimental and theoretical contexts, including optics, convection, ferrofluids, Couette flow, buckling problems, neuroscience, and so forth.

Mathematical and numerical theories for snaking are generally well understood for the case of one-dimensional snaking on a line. However, the study of the analogous mechanisms governing two-dimensional snaking (and beyond) remains an open challenge. For this case of planar snaking, numerical observations demonstrate the existence of much more exotic solutions, including patterns that follow stripes as well as patches of hexagons, squares, and circular spots. In such cases, the snaking is much more complicated due to aspects such as overlapping pinning regions and isolas (closed curves of solutions). From a computational standpoint, there is much greater effort required in the analysis and description of these exotic bifurcation structures; analytical theory for their study remains highly limited.

In performing an asymptotic analysis near the critical Maxwell point, there is a complication that occurs. A standard multiple-scales analysis, which separates the slow spatial scale of the front from the fast oscillating patterns, is insufficient (in either 1D or 2D) to describe the pinning structure. In essence, this insufficiency occurs on account of a coupling between the fast- and slow scales that is determined via exponentially small terms. Recently, exponential asymptotics has proved successful in providing a theory to describe this coupling and hence the snaking. However, in contrast to the one-dimensional theory of homoclinic snaking, the extension of exponential asymptotics to the case of two- or higher-dimensional localised structures remains open.

A further previously unexplored research direction concerns the analysis of localised solutions for differential equations that, by construction, are not continuously differentiable arbitrarily many times, which is a key assumption in the exponential asymptotics calculations carried out in the literature to date. Despite being well-motivated by physical examples, how this non-smoothness affects the scaling laws and asymptotic analysis of these localised states is currently entirely open.
The goal of this PhD will be to develop computational and analytical tools for studying higher-dimensional snaking. There will be a significant emphasis on three interrelated subproblems: (i) a detailed numerical analysis of one- and two-dimensional snaking in discrete and continuum problems; this will extend the work that was initialised by Taylor & Dawes (2010) and modifications to the governing equations beyond what has been considered in the literature; (ii) a development of the asymptotic methods required for the study of snaking in ordinary differential and difference equations; (iii) a development of the asymptotic methods required for snaking in two-dimensional problems. In regards to a study of the asymptotic techniques, there will be significant opportunities to investigate other similar (but non-snaking) systems, particularly for travelling-wave problems or problems in dislocation theory.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.