Lax-Darboux Schemes and Miura-type Transformations

This thesis is concerned with the study of integrable differential difference and partial difference equations. The thesis splits into two parts. In part I, we begin by studying Darboux transformations of Lax operators related to the tetrahedral reduction group. We construct a general Darboux matrix of a specified type, then use first integrals to derive interesting novel subcases. In the process, we derive Bäcklund transformations of an associated system of partial differential equations. Subsequently, we utilise the derived Darboux matrices to construct integrable discrete systems, their generalised symmetries and conservation laws. Furthermore, we consider integrable reductions and potentiations of the obtained systems, as well as Miura-type transformations. Moreover, we demonstrate how one can use the Bäcklund transformations to derive explicit solutions to the associated continuous system. Finally, we present results regarding the octahedral reduction group. This is the first time that semi and fully discrete systems have been associated to sl3(C) and sl4(C)-based automorphic Lie algebras. In part II, we show that for differential difference equations which possess a Lax pair of a particular type, one can construct Miura-type transformations by considering invariants of associated algebraic structures. We begin by introducing the objects required by the construction and discussing the general theory. Subsequently, we demonstrate the efficacy of the construction by deriving Miura-type transformations related to the Narita-Itoh-Bogoyavlensky lattice and the discrete Sawada-Kotera equation, some of which appear to be new. Furthermore, we discuss how the construction can be applied to partial difference equations and systems, providing examples of its successful application. In the case of systems, the derived Miura-type transformations also appear to be new.