Lax Matrix and Action-Angle Map for Calogero-Moser Systems of BC_n Type (Integrable Systems)

For nonrelativistic and relativistic integrable n-particle systems of trigonometric and hyperbolic Calogero-Moser type the Lax matrix is explicitly known, and has been used to construct the action-angle map in great detail. These systems correspond to the Lie algebra A_{n-1}. There also exist integrable n-particle systems generalizing the former to the Lie algebra BC_n. Physically speaking, this comes down to adding specific external field couplings. For these BC_n systems far less is known concerning Lax matrix and action-angle map. In the 'nonrelativistic' BC_n case, the Lax matrix is known, but no action-angle map has been constructed yet. In the 'relativistic' BC_n case, the action-angle map has been constructed for n=1, but in this case a Lax matrix is not even known for n=1. These results form a solid starting point for a further study of the open problems concerning Lax matrices and action-angle maps in the BC_n setting.

keywords: integrable systems, solitons, sine-Gordon equation, relativistic quantum mechanics, Calogero-Moser type systems, Lax matrices, action-angle maps


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.