Nonlinear Integrable PDEs and Non-commutative Geometry (Integrable Systems)

Integrable nonlinear PDEs such as KdV and its cousins have a large number of interesting exact solutions. Those usually come in families and are constructed by beautiful algebraic and geometric methods. It is one of the more recent discoveries that these solutions in many cases admit a natural interpretation in terms of noncommutative geometry. In such a geometry the coordinate functions may no longer commute under multiplication, so that xi xj differs from xj xi. This, on one hand, makes things harder since the usual geometric methods may not help and need to be carefully adapted. On the other hand, this point of view helps to reveal hidden symmetries of these PDEs and obtain nice new formulas for their solutions. This is an exciting area for further research with potential applications ranging from the theory of integrable Hamiltonian systems to the special function theory and combinatorics.

Integrable Systems

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.