It is a remarkable observation that the soliton equations, such as the Korteweg-de Vries (KdV) equation, have a precise and exact analogue in the theory of difference equations. Usually, applying 'brute-force' discretisation techniques to an integrable PDE such as the KdV equation will destroy its beautiful properties, and sophisticated techniques are needed to retain integrability. In Leeds we investigate all aspects of integrable partial difference equations (PÄEs), their exact solutions, initial value and boundary problems, and their symmetry reductions (leading to discrete analogues of the Painleve transcendents). The research has led to the discovery of new mathematical structures and solutions in terms of nonlinear special functions.
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.