UK Nonlinear News,
Nonlinear Research at Heriot-Watt
There is a large research group at Heriot-Watt in applied
nonlinear systems. This comprises J.M. Ball,
K.J. Brown, J. Carr,
D.B. Duncan, J.C. Eilbeck, R.N. Hills, R.J. Knops, A.A. Lacey,
M. Levitin, O. Penrose, D.G. Roxburgh, B.P. Rynne and K. Zhang.
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
The department has strong links with the International Centre for Mathematical Sciences (ICMS), a joint
venture of Edinburgh and Heriot-Watt Universities, founded in 1990
to provide a centre of excellence in the Mathematical Sciences.
The department has received a number of EC HCM grants for EC fellows in
nonlinear areas, and coordinates an HCM network in `Reaction-Diffusion Equations' containing a total of 6 sites with a budget of 400,000
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
The work of the nonlinear systems group is especially relevant to the
areas of modelling, simulation and
prediction of complex systems. One example is the work with Unilever
on the dynamics of populations of mixed micelles modelled by
coagulation fragmentation equations.
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
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
- 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.
Chris Eilbeck / Heriot-Watt University / email@example.com
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Last Updated: 1st May 1996.