DISCRETE LAX PAIRS, REDUCTIONS AND HIERARCHIES

The term ‘Lax pair’ refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous contin-uous systems such as the Korteweg de Vries equation and the Painleve ́ equations all have integrable discrete analogues, which retrieve the original systems in the contin-uous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations