This thesis deals with discrete Lax systems and integrable lattice equations (i.e., partial difference equations (PΔEs) ) associated with elliptic curves. We will be concerned with their derivation and integrability properties, as well as with certain reductions. In particular the construction of a new class of higher-rank elliptic type integrable system forms one of the core results, opening new avenues of investigation.
The primary integrable system of interest is Adler’s equation (nowadays often referred to as Q4), which is a lattice version of the Krichever-Novikov (KN) equation. For this equation we exhibit a new Lax pair, the compatibility of which yields the equation in its so-called 3-leg form and which forms a starting point for the investigation in this thesis. It is this particular Lax pair that is most readily generalized to higher-rank cases, in contrast to other known Lax pairs for Q4. In fact, the most general class of higher-rank Lax pairs
contains not only higher-rank versions of Q4 but also equations which are conjectured to be related to discrete versions of the Landau-Lifschitz (LL) equations. We will briefly treat the latter, but our main focus will be on the class of higher-rank systems related to Adler’s lattice equation.
Furthermore, by considering limits on the solutions, whereby the curve degenerates, we will propose higher-rank analogues of various equations in the well-known ABS list. Finally, we will set up a general scheme that corresponds to isomonodromic deformations on the torus, from which non-autonomous elliptic type difference equations can be derived.
