Nonlinear Research at Heriot-Watt

General

Its basic research themes are the modelling of physical systems, the analysis of the resulting ordinary and partial differential equations, and the use of scientific computation to analyse the resulting models. The main areas of activity are Elasticity and the Calculus of Variations, Infinite-Dimensional Dynamical Systems, Reaction-Diffusion Equations, Statistical Physics and Phase Transitions, Nonlinear Waves and Solitons. Over many years this group has pioneered the use of analysis in modelling and has demonstrated that these methods give scientific information not available by other means. Examples of this include material microstructure, and the use of Statistical Physics in continuum modelling, such as phase-field models with applications to diffusion induced grain boundary motion.

International Centre for Mathematical Sciences

Research Grants

Members of the department are involved in a EC network `Phase Transitions and Surface Tension' and INTAS networks in `Exactly solvable 3D models of statistical mechanics' and `Nonlinear modes in quasi-one-dimensional polymers and high-temperature superconductors'.

Applied Nonlinear Research

In materials modelling, a key challenge for future years is to understand how to derive and reconcile models for the same system at different length-scales (e.g. atomic, continuum), for which the group's work on statistical physics and material microstructure is particularly pertinent.

We also place great value on our links with industry and other organisations. In recent years this has included collaboration with the British Geological Survey in modelling seismic wave interactions with complicated rock formations and with British Gas in modelling flow in their main transmission network.

We were recently awarded a Teaching Company Scheme grant for modelling and simulation of the welltest process used in oil reservoir exploration. We have also gained a variety of industrial projects for the research component of our MSc in the Mathematics of Nonlinear Models (a joint course run with Edinburgh University).

An interesting collaboration has been developed with Professor M Russell at the Rutherford Laboratory, who has been appointed to a visiting professorship at Heriot-Watt, in the area of solitons and breathers in solid state physics. This work has many potential applications in the areas of High Temperature Superconductivity and in spluttering technologies in material science.

Some staff research interests in Nonlinear Science

• J.M. Ball works on a nonlinear theory of martensitic phase transformations, in particular establishing and analysing a new mechanism for hysteresis based on the geometric incompatibility of parent and product phases.
• K.J. Brown studies global bifurcation theory for semilinear equations involving an indefinite weight function on all of Rⁿ - such equations arise in population genetics - and constructs new types of sub and supersolutions for such problems in the case where the weight function is small.
• J. Carr works on discrete lattice equations, and general systems of coagulation-fragmentation equations.
• D.B. Duncan works on numerical methods for soliton and other nonlinear o.d.e.'s and p.d.e.'s including coagulation-fragmentation equations.
• J.C. Eilbeck studies new hierarchies of integrable mth order n-particle Hamiltonian systems with interesting r-matrix structures, quantum solitons in non-integrable systems, and bifurcation problems in competition and predator-prey models.
• A.A. Lacey is currently working on the interpretation of experimental results in Modulated Differential Scanning Calorimetry, Free-Boundary Problems, especially with regard to singularities (e.g. corners in Hele-Shaw flow), and both local and non-local parabolic problems exhibiting blow-up.
• D.G. Roxburgh studies various elasticity problems which involve large nonlinear deformations and stresses; examples include membrane problems and the stability of deformed rubber blocks.
• B.P. Rynne is working on local and global bifurcation theory and, in particular, the structure of global bifurcating continua of solutions in nonlinear ordinary and partial differential equations. He is also working on generic properties of solution sets of such equations.
• O. Penrose works on mathematical model for diffusion-induced grain boundary motion, and on Monte Carlo methods for simulating spinodal decomposition in elastically anisotropic metals.
• K. Zhang studies the calculus of variations and elliptic systems; nonlinear elasticity and material phase transformations; application of harmonic analysis to elliptic systems and variational problems in mechanics, and the shape from shading problem in computer vision.