# On Reductions of the Hirota-Miwa Equation

## TL;DR

This paper explores how to derive Lax pairs and presymplectic structures for reductions of the Hirota-Miwa equation, demonstrating Liouville integrability of related maps and connecting to discrete Toda and KdV equations.

## Contribution

It provides a systematic method to obtain integrable structures for reductions of the Hirota-Miwa equation, linking them to well-known integrable systems.

## Key findings

- Lax pairs and presymplectic structures are derived for the reductions.
- Certain maps are shown to be Liouville integrable.
- Connections to discrete Toda and KdV equations are established.

## Abstract

The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations.