Research in Differential Equations

Motivating this work is the enormous current interest in the interaction between nonlinear analysis, partial differential equations, topological and variational methods, and functional analysis. The emphasis here tends to be on the mathematical aspects of questions from solid and fluid mechanics though not to the exclusion of studying mathematical structures which occur naturally in such investigations. There is a rich interaction between many of the research topics, which include Dynamical Systems, Mathematical Hydrodynamics, Nonlinear Waves and Reaction-Diffusion Systems. The staff in this area are Professor C.J. Budd, Dr G.R. Burton, Professor L.E. Fraenkel FRS, Professor V.A. Galaktionov, Dr A.T. Hill, Dr J. Sivaloganathan, Professor J.F. Toland FRS and Professor D. Vassiliev.

Dynamical Systems

Recent research has included the study of solitary wave solutions of the water wave problem. The overall structure of the solution set is that associated with the presence of a Smale horseshoe in a finite dimensional system. The solutions found correspond to waves of depression which can have an arbitrary, but finite, number of troughs. The main tools employed in this study are centre manifold theory, bifurcation theory and variational/topological methods. This approach has opened up many new and interesting problems.

Other bifurcation problems currently studied include non-smooth stick-slip dynamical systems. These systems arise in the study of impact oscillators.

See the sections on Control Theory, Industrial Applied Mathematics, Mathematical Biology and Numerical Analysis for further research involving dynamical systems.

Mathematical Hydrodynamics

New methods in analysis have shed light on some old problems in fluid motion. The work on nonlinear waves and dynamical systems are examples of this. Another is a rigorous and quite descriptive theory of steady vortices in two and three dimensions in an ideal fluid. Much has been proved in recent years, but each answer raises new and more difficult questions.

Nonlinear Waves

This work is concerned with mathematical questions of existence, uniqueness and regularity of solutions of the Euler equations for two-dimensional flows with a free surface. Questions of the bifurcation of small-amplitude waves and of the existence and behaviour of large-amplitude waves are under consideration. Recent research has been especially concerned with the existence of standing waves and with the effects of surface tension on the bifurcation of steady waves. [See Nash-Maer Theory]

This study has led to the investigation of model problems for water waves, and in particular to the discovery of a class of Hamiltonian systems, common in the theory of waves, where solitary and periodic waves and bores occur as stable phenomena. These systems are studied using a tool not previously associated with Hamiltonian systems, namely the topological degree.

Reaction-Diffusion Systems

These equations lie at the heart of many problems in physics, engineering, and mathematical biology. Central to our work is a study of the nature of these equations using a combination of rigorous techniques (such as comparison methods, bifurcation theory and the use of rescaling and energy arguments), formal asymptotic methods (such as matched asymptotic expansions and series solutions) and numerical techniques. We are particularly interested in problems where the solutions of the equations develop singularities in a finite time. This behaviour is typical of models of combustion and burning which provide examples of problems with finite time blow-up, extinction or quenching, all of which are abstractions of observed physical phenomena. Singularities also arise in certain theories of the nature of turbulence. Describing these singularities requires the development of novel analytic and numerical tools and leads to many new mathematical problems. A further area of research involves the construction of new exact solutions of these equations by determining invariant manifolds (and invariant spaces) of the solution operator or by looking for self-similar solutions invariant under various group actions.

Symmetry by way of the maximum principle

The celebrate work of Gidas, Ni and Nirenberg on positive solutions of nonlinear elliptic equations has been extended from classical to generalized solutions (in order to make it applicable to certain physical problems) and elementary proofs have been found for some basic theorems of this subject. [L. E. Fraenkel: An introduction to maximum principles and symmetry in elliptic problems (Cambridge University Press 2000).]See also Nonlinear Diffusion Equations & Reaction-Diffusion Systems.

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Last updated: 04/03/2002