# Reduction of lattice equations to the Painlev\'e equations: P$_{\rm IV}$ and P$_{\rm V}$

## TL;DR

This paper establishes a new link between ABS lattice equations and Painlevé equations, enabling the construction of difference-differential Lax representations for P_IV and P_V, advancing the understanding of integrable systems.

## Contribution

It introduces a novel relation between ABS equations and Painlevé equations, and constructs their difference-differential Lax representations.

## Key findings

- New relation between ABS and Painlevé equations
- Construction of difference-differential Lax representations for P_IV and P_V
- Enhanced understanding of integrable lattice systems

## Abstract

In this paper, we construct a new relation between ABS equations and Painlev\'e equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painlev\'e equations.