Symmetry, reductions and exact solutions of the difference equation $u_{n+2}=au_n/(1+bu_nu_{n+1})$

TL;DR

This paper explores the solutions of a specific second-order difference equation using Lie symmetry methods to identify invariant solutions and understand its behavior.

Contribution

It introduces a symmetry-based approach to analyze and find exact solutions of the nonlinear difference equation.

Findings

Identification of Lie symmetries for the equation

Derivation of exact solutions using symmetry reductions

Insight into the structure of solutions through symmetry analysis

Abstract

We investigate the solutions of the second-order difference equation $u_{n+2}=(au_n)/(1+bu_nu_{n+1})$ using a group of transformations (Lie symmetries) that leaves the solutions invariant.