q-discrete Painleve equations, their hierarchies and properties

Abstract

The main objective of this thesis is to derive hierarchies of q-discrete Painelevé equations. Some of the important properties of these hierarchies will also be given, namely Lax pairs, Bäcklund transformations, solutions of their associated linear problems for special values of parameters and their symmetry groups. To construct these hierarchies, we apply a geometric reduction and a staircase method on a multi-parameteric generalized lattice modified Korteweg-de Vries equation. In addition, the property of consistency around the cube is used in order to find Bäcklund transformations. Starting with the base case of q-discrete second, third and fourth Painlevé equations on A_5 initial-values surface, new hierarchies of q-discrete third and fourth Painlevé equations are discovered, and we also rediscover the hierarchy of q-discrete second Painlevé equation. In this thesis, we provide the Lax pairs for each member in these hierarchies. Using the consistency around the cube, we also provide Bäcklund transformation for the entire hierarchy of q-discrete second and third Painlevé hierarchies. We generate a hierarchy of special solutions starting with seed solutions for q-discrete second and third Painlevé hierarchies. An assumption made is that particular parameter values would enable the ability to diagonalize the Lax pair. As a consequence, we found that the associated linear problem for the three hierarchies can be solved in terms of q-Gamma function. Furthermore, the hierarchy of q-discrete fourth Painlevé hierarchy can be reduced to one equation that can be linearlized to become Riccati equation which has hypergeometric special solutions. Finally, we investigated the affine Weyl group structure of the symmetry group for each hierarchy. In this thesis, we construct the explicit representation of the symmetry group for the first and second member of these hierarchies.