Solitons are particle-like solutions to nonlinear partial differential equations. In particular, the relativistically invariant sine-Gordon equation phi_xx-phi_tt=sin(phi) admits n-soliton solutions. There exist relativistic integrable n-particle systems of Calogero-Moser type that are intimately related to the sine-Gordon solitons. Specifically, the n-soliton solutions can be expressed in terms of the space- and time-translation generators of the n-particle systems. This yields soliton space-time trajectories that make it possible to follow the individual solitons during their collisions. At this classical level the particle-soliton correspondence is in great shape.
At the quantum level the corresponding particle-soliton equivalence has thus far only been established for the case n=2. This involves a new special function, which may be viewed as a 'relativistic' generalization of the Gauss hypergeometric function. Specializing to the sine-Gordon coupling, it yields a unitary eigenfunction transform for the n=2 particle Hamiltonians which encodes the bound-state spectrum and scattering. These are identical to those for the 2-body subspace of the sine-Gordon quantum field theory.
It is a major research project to establish the equivalence for n>2. To achieve this aim, various subproblems need to be tackled. They vary in difficulty, and for some of them there are very promising leads.
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.