PhD projects in Dynamics of Patterns and nonlinear PDEs
The research in dynamics of patterns and nonlinear PDEs focuses on the analysis of nonlinear partial differential equations and on furthering our understanding of the formation and dynamics of patterns in biological, chemical and physical systems. Theoretical work has been done on the bifurcation theory of patterns, the analysis of partial differential equations, and the description of slowly evolving patterns using amplitude and modulation equations. Applications studied include the Faraday experiment, Rayleigh-Benard convection, the dynamics of spiral waves in chemical reactions, surface chemistry, quasi-periodic patterns on the ocean surface, atmospheric flows, and sand transport and dune formation in deserts. More details, including staff members working in this area, can be found on the dynamics of patterns and PDEs pages.
Below are some examples of PhD projects in dynamics of patterns and nonlinear PDEs. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Granular matter (Supervisor: Dr Anne Skeldon)
The equations that govern the motion of fluids are well-established. Less well understood is the motion of particulate materials, such as sand, soil or powders: sometimes these flow like fluids--you can pour gravel out of a bucket; and sometimes they behave like a solid--you can make a castle out of sand but not out of water. This project is joint with Prof Ugur Tuzun in Chemical and Process Engineering at Surrey to develop data analysis techniques/understand the physics of granular media.
Faraday waves (Supervisor: Dr Anne Skeldon)
Patterns can be made to form on the surface of a container of fluid by shaking the container up and down. The shaking has to be at the right frequency and right amplitude for any patterns to form--if you move a container of fluid up and down very slowly then the surface of the fluid will remain flat. There are a wealth of experimental results showing a variety of different possible patterns. On the theoretical front, arguments based on symmetries and the way different patterns interact has led to some knowledge of the mechanisms that cause particular patterns to become dominant. Although these have been tested on mathematical models describing the fluid, there are still significant gaps in our understanding. The aim of this project is to investigate further the theory underlying pattern selection in the Faraday and related problems. This will require using a variety of different dynamical systems techniques both theoretical and computational.
Attractors for hydrodynamical problems in unbounded domains (Supervisor: Dr Sergey Zelik)
The concept of a global attractor plays a very important role in the modern theory of dissipative systems generated by PDEs. For the case of equations in bounded domains (like a square, disk or ball), this attractor is usually finite-dimensional. Thus, despite the infinite-dimensionality of the initial phase space, the limit dynamics reduced to the attractor occurs finite-dimensional and can be studied by the methods of the classical dynamical systems theory. The situation is more difficult for the case where the underlying domain is unbounded (like R2 or R3) since the dimension of the attractor is typically infinite and the finite-dimensional reduction is no more possible. Nevertheless, a reasonable theory has been recently developed for large class of such systems. The aim of this project is to extend/apply this theory to hydrodynamical problems in unbounded (cylindrical) domains. Such extension was problematic during a long time since even the well-posedness of the Navier-Stokes equation in 2D unbounded domains in the proper classes of spatially non-decaying solutions was not known. The situation is changed now due to recent results on the solvability and dissipativity of 2D and 3D Navier-Stokes equations in cylindrical domains.
Soliton Switching in Fibres (Supervisor: Dr Gianne Derks)
For the optical transmission of data across a cable, one can use two (or more) coupled fibre cables. Experiments have shown that if a certain type of signal is put at one end of the cable, it will go to the other end of this cable and hardly anything happens in the other cable. However, if one puts other types of signals on the cable, the signal will switch to the other cable. This gives a convenient way of sending data consisting of zeros and ones. In this project we will aim for a better understanding of this experimentally observed process by investigating the family of soliton-like solutions, especially issues like existence, stability, bifurcations and invariant manifolds will be investigated.
Patterns in Surface Chemistry (Supervisor: Dr Rebecca Hoyle)
Regular patterns arise naturally in many physical, chemical and
biological systems - from hexagonal convection cells on the surface of
the sun to stripes on a zebra's back. Constantly changing irregular
patterns of carbon monoxide (CO) and oxygen are seen during CO
oxidation on platinum crystals in the [100] orientation. Recently a
reaction-diffusion model is developed to reproduce this pattern
formation and created numerical simulations that show patterns made up
of moving CO and oxygen fronts. Possible PhD projects in this area
include: extending the model to include the formation of subsurface
oxygen at higher pressures or developing a similar model for the NO +
NH3 reaction on Pt{100}. These interdisciplinary projects are great
opportunities for Maths graduates to apply their skills in a new area,
or for Chemistry graduates with good maths and computing skills to
move into theory.
Methods for Obtaining Discrete Symmetries (Supervisor: Prof Peter Hydon)
Differential equations are widely used as models of real systems. To understand the behaviour of a system, it is usually necessary to find the symmetries of the model differential equation. Recent work at Surrey has led to the first systematic technique for obtaining discrete symmetries. The purpose of this project is to simplify and extend the technique, and to create a Maple package that will systematically obtain the discrete symmetries of a given differential equation. This project combines differential equation techniques, linear algebra, group theory and Maple programming. It is expected that the resulting package will be distributed with future releases of Maple, and will be used by those working in many areas of mathematics and theoretical physics.
Spiral waves and defect interaction (Supervisor: David Lloyd and Bjorn Sandstede)
Spiral waves have been found in many experiments such as the famous Belousov-Zhabotinsky reaction and the oxidation of carbon-monoxide on platinum surfaces. In the former reaction, period doubling of spiral waves has been observed, and a numerical simulation of a model problem is shown to the left. This bifurcation has been analysed in 2D. Of interest would be an analytic and numerical study of this phenomenon in 3D where the line defect that connects the core to the bottom boundary becomes a two-dimensional surface, which itself may exhibit a interested dynamics. More generally, the various line defects visible in the simulation may interact with each other, and it would be of interest to study the time and length scales involved in this interaction from a analytical and numerical perspective.
Multi-symplectic structures and nonlinear PDEs (Supervisor: Prof Thomas J. Bridges)
An understanding of nonlinear PDEs is one of the great challenges for
the twenty-first century. In this project the geometry of nonlinear
PDEs is studied. Multi-symplectic structure, which generalises
classical symplectic geometry, is used as a backbone for the analysis.
The project has three parts:
(I) Develop the geometry of the total exterior algebra bundle for smooth two-dimensional manifolds and construct Dirac operators on this bundle as a basis for canonical hyperbolic and elliptic PDEs.
(II) Use the calculus of variations to find solutions of multi-symplectic Hamiltonian PDEs, using the Mountain Pass Theorem and related functional analytical tools.
(III) Apply numerical methods to either part (I) or part (II).
