The main topic of this project is the study of new integrable systems, the links between seemingly different integrable systems, and the properties of exact solutions given by special functions. Integrable systems is a distinct class of dynamical systems characterized by predictable behavior due to the existence of sufficiently many conservation laws, such as conservation of energy, conservation of momenta, and numerous others that are not explicitly named. Among integrable systems, there is a class called `exactly solvable models´ where a complete set of exact solutions can be obtained. A prominent examples is the Kepler problem of planetary motion which has exact solutions expressed in terms of integrals and hence the name integrable systems.
Integrable systems have made significant contributions to our understanding of both mathematics and physics, such as 1) the so-called sine-Gordon equation which appear in the context of propagation in junctions between two superconductors (i.e. Josephson junctions) and scattering data for relativistic particles in physics context, and differential geometry in mathematical context and 2) the Calogero-Moser-Sutherland (CMS) models that appear in physical systems such as the fractional quantum Hall effect and topological insulators, and in mathematical subjects as combinatorics and representation theory.
In the beginning of the 1980s, Ruijsenaars found that behaviour of the solutions of the sine-Gordon could also be obtained from the solutions of relativistic generalization of the CMS models, which gave rise to a new family of integrable systems. Ruijsenaars found that the interaction of the models has rational, trigonometric, hyperbolic, and elliptic versions, as was known for the CMS models, and it was later shown that there is an underlying root system of a Lie algebra associated to the model. While Liouville integrability has been established for all these relativistic models, little is known about the exact solutions in the elliptic cases. The most general of these models is the so-called van Diejen model which is a relativistic version of the elliptic CMS model associated to the (non-reduced) BC root system.
Another mathematically natural generalizations of the CMS model were discovered by Chalykh, Feigin, Sergeev, and Veselov (CFSV) which was related to root systems of Lie super-algebras. These generalizations of CFSV type are often referred to as deformed CMS models. The deformed model also has relativistic generalizations of Ruijsenaars and van Diejen type that have been discovered in the last decade. These models are known to be Liouville integrable for the non-elliptic cases, where an algebraic approach using so-called double affine Hecke algebras are applicable, and exactly solvable in the trigonometric case related to A-type root system.
The Painlevé equations are a set of non-linear differential equations with the so-called Painlevé property. At the end of the 19th century, it was shown that the Painlevé VI equation can be mapped to either an isomonodromic deformation of a linear ordinary differential equation or to a non-autonomous version of the one-variable elliptic CMS model of BC-type. This is commonly referred to as the Painlevé-Calogero correspondence.
The objectives of this project are to construct and study the exact solutions of the elliptic CMS models, prove the integrability of the elliptic deformed CMS models, and to extend the Painlevé-Calogero correspondence to the multivariate CMS models.
