# Integrable boundary conditions for the Hirota-Miwa equation and Lie algebras

## TL;DR

This paper explores integrable boundary conditions for discrete Hirota-Miwa equations linked to affine Lie algebras, deriving Lax pairs, continuum limits, and conservation laws, advancing understanding of integrable systems and their algebraic structures.

## Contribution

It introduces boundary conditions compatible with integrability for Hirota-Miwa systems related to affine Lie algebras and derives associated Lax pairs and conservation laws.

## Key findings

- Boundary conditions compatible with integrability are established.
- Lax pairs for the systems are constructed.
- Conservation laws and higher symmetries are identified.

## Abstract

Systems of discrete equations on a quadrilateral graph related to the series $D^{(2)}_N$ of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equation by imposing boundary conditions compatible with the integrability property. The Lax pairs for the systems are presented. It is shown that in the continuum limit the quad systems tend to the corresponding systems of the differential equations belonging to the well-know Drinfeld-Sokolov hierarchies. The problem of finding the formal asymptotic expansion of the solutions to the Lax equations is studied. Generating functions for the local conservation laws are found for the systems corresponding to $D^{(2)}_3$. An example of the higher symmetry is presented.