# Lattice paths and consistent systems of difference equations

#### Dr Pavlos Xenitidis, Liverpool Hope University

Difference equations defined on an elementary quadrilateral of the lattice constitute probably the most well known class of discrete integrable systems. Their integrability can be established in various ways with the most rigorous one provided by the existence of symmetries, i.e. evolution type differential-difference equations compatible with them. Even though difference equations admit only one hierarchy of symmetries in one direction, the same hierarchy may be compatible with many other (higher order) difference equations. A new interesting characteristic is that they can also define symmetries of overdetermined  consistent systems of difference equations.

In this talk I am going to present two novel hierarchies of systems of partial difference equations and discuss their integrability properties. They are hierarchies of consistent systems of $N$ difference equations, $N \ge 2$, with the $i$-th equation being defined on a stencil of $i \times (N+1-i)$ quadrilaterals, $i=1,\ldots,N$. A nice and simple method for the construction of these hierarchies will be given which employs lattice paths connecting the origin with the lattice points $(i,N+1-i)$. The integrability of these systems will be established by the derivation of the lowest order symmetries in both lattice directions which are related to the Bogoyavlensky and the Sawada-Kotera lattices, respectively. Finally I will discuss how these hierarchies generalise two well-known quad equations and how one hierarchy can be viewed as a degeneration of the other.

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