Universal Equations and Integrability (Integrable Systems)
Many natural systems can be modelled or described by nonlinear partial differential equations, integro-differential equations, differential-difference equations or just fully discrete equations. It is very difficult, if at all possible, to find solutions of nonlinear equations, except using a computer. Rather surprisingly, sizeable classes of nonlinear systems are found to have an extra property, integrability, which changes the picture completely. For integrable equations there is a well developed theory, which is used to find multi-parametric exact solutions, to study the general Cauchy problem and the asymptotic behaviour of solutions, to find infinite hierarchies of symmetries and conservation laws. There is a deep reason why integrable equations often have fundamental and universal meaning. Multi-scaling asymptotic expansions provide a tool to study integrable and near-integrable equations. The properties of near-integrable equations are not yet well understood; it is a new and promising subject for research.
The Integrable Systems group in Leeds, one of the leading groups worldwide working on integrable systems, has been strengthened by three recent appointments, one at a senior level. Integrable Systems are systems that, albeit highly nontrivial and nonlinear, are amenable to exact and rigorous techniques for their solvability. They can take many shapes or forms: nonlinear evolution equations, partial and ordinary differential equations and difference equations, Hamiltonian many-body systems, quantum systems and spin models in statistical mechanics. A large number of mathematical techniques have been developed to unravel the rich structures behind these systems. The six permanent members of staff work with five postdocs and postgraduate students.
Leeds is a major centre for the theory of discrete and quantum integrable systems. This group represents a wide range of research activities into integrable nonlinear systems, their symmetries, solution techniques and the underlying mathematical structures, as well as more mathematical aspects of physical systems, for example quantum systems. The models comprise ordinary and partial differential and difference equations, dynamical mappings, discrete Painleve equations, Hamiltonian and many-particle systems and systems of hydrodynamic type. The theory and its specific models have wide-ranging applications, for example, in nonlinear optics, theory of water waves, integrable quantum field theory, statistical mechanics and combinatorics random matrix theory and nanotechnology.