On a system of second-order difference equations

TL;DR

This paper derives explicit solutions for a specific system of second-order difference equations using Lie symmetry analysis, providing a method to solve such nonlinear discrete systems explicitly.

Contribution

It introduces a novel application of Lie symmetry analysis to second-order difference equations, yielding explicit solutions for a complex nonlinear system.

Findings

Explicit formulas for solutions of the difference system.

Identification of non-trivial symmetries via Lie analysis.

Methodology applicable to similar nonlinear difference equations.

Abstract

We obtain explicit formulas for the solutions of the system of second-order difference equations of the form $x_{n+ 1} = \frac{x_n y_{n-1}}{y_n (a_n + b_n x_n y_{n - 1})}, \quad y_{n+1} = \frac{x_{n - 1} y_n}{x_n (c_n+d_n x_{n-1} y_n)}$, where $(a_n)_{n\in \mathbb{N}_0},\; (b_n)_{n\in \mathbb{N}_0}, \;(c_n)_{n\in \mathbb{N}_0}$ and $(d_n)_{n\in \mathbb{N}_0}$ are real sequences. We use Lie symmetry analysis to derive non-trivial symmetries and thereafter, exact solutions are obtained.