# Symmetries of the Hirota Difference Equation

## TL;DR

This paper derives and analyzes the continuous symmetries of the Hirota difference equation, revealing their role as 'times' in integrable systems and providing explicit examples and Lax pairs.

## Contribution

It introduces a dressing method to derive symmetries, explores their action on variables and scattering data, and connects them to integrable PDEs.

## Key findings

- Derived continuous symmetries commuting with shifts
- Presented their action on dependent variables
- Provided examples and Lax pairs for related equations

## Abstract

Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as "times" of the nonlinear integrable partial differential-difference and differential equations. Examples of equations resulting in such procedure and their Lax pairs are given. Besides these, ordinary, symmetries the additional ones are introduced and their action on the Scattering data is presented.