# Generating a chain of maps which preserve the same integral as a given map

## TL;DR

This paper extends duality concepts to difference equations, providing a method to generate chains of dual systems that preserve a given integral, with applications to various lattice-based systems.

## Contribution

It introduces a procedure to construct dual systems of difference equations based on integral relations, applicable to several important lattice models.

## Key findings

- Constructed dual systems for several 2D lattice equations.
- Identified conditions where dual systems do not exist.
- Applied the method to systems like NLS, mKdV, and Boussinesq.

## Abstract

We generalise the concept of duality to systems of ordinary difference equations (or maps). We propose a procedure to construct a chain of systems of equations which are dual, with respect to an integral $H$, to the given system, by exploiting the integral relation, defined by the upshifted version and the original version of $H$. When the numerator of the integral relation is biquadratic or multi-linear, we point out conditions where a dual fails to exists. The procedure is applied to several two-component systems obtained as periodic reductions of 2D lattice equations, including the nonlinear Schr\"{o}dinger system, the two-component potential Korteweg-De Vries equation, the scalar modified Korteweg-De Vries equation, and a modified Boussinesq system.