Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras $A^{(1)}_{N-1}$

TL;DR

This paper investigates integrable discrete exponential systems associated with affine Lie algebras, providing Lax pairs, conservation laws, and recursion operators, specifically focusing on systems related to the algebra $A^{(1)}_{N-1}$.

Contribution

It derives Lax pairs, conservation laws, and recursion operators for discrete systems linked to affine Lie algebras, advancing understanding of their integrability properties.

Findings

Lax pairs constructed for arbitrary N

Recursion operator for N=3 found, non-weakly nonlocal

Higher symmetries identified in characteristic directions

Abstract

The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been suggested earlier in \cite{GHY} which coincide with the corresponding Drinfeld-Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been approved by numerous examples. In the present article we study those systems from this class which are related to the algebras $A^{(1)}_{N-1}$. We found the Lax pair for arbitrary $N$, briefly discussed the possibility of using the method of formal diagonalization of Lax operators for describing a series of local conservation laws and illustrated the technique using the example of $N=3$. Higher symmetries of the system $A^{(1)}_{N-1}$ are presented in both characteristic directions. Found recursion operator for $N=3$. It is interesting to note that this operator is not weakly nonlocal.