Complete classification of rational solutions of PIV and its higher order generalizations

We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation (PIV) and its higher order generalizations known as the A_2n Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. Cyclic Maya diagrams are the key concept that allows to achieve a full classification. Finally, we establish the link with the standard approach to building rational solutions pioneered by the japanese school, which is based on applying the symmetry group of Bäcklund transformations on seed solutions. To conclude we will formulate a conjecture on an equivalent result for PV and its higher order generalizations. This is joint work with Yves Grandati (Université de Lorraine) and Robert Milson (Dalhousie University).