# B\"acklund transformations of $Z_n$-Sine-Gordon systems

## TL;DR

This paper constructs multi-component $Z_n$-Sine-Gordon and Sinh-Gordon systems from Lie algebra reductions, providing Bäcklund transformations to generate new solutions and explicitly analyzing cases for $Z_2$ and $Z_3$.

## Contribution

It introduces a unified algebraic framework for $Z_n$-Sine-Gordon systems and derives their Bäcklund transformations, including explicit examples for $Z_2$ and $Z_3$.

## Key findings

- Derived Bäcklund transformations for $Z_n$-Sine-Gordon systems.
- Explicit solutions and superposition formulas for $Z_2$ and $Z_3$ cases.
- Established Lax pairs for the systems.

## Abstract

In this paper, from the algebraic reductions from the Lie algebra $gl(n,\mathbb C)$ to its commutative subalgebra $Z_n$, we construct the general $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon systems which contain many multi-component Sine-Gordon type and Sinh-Gordon type equations. Meanwhile, we give the B\"acklund transformations of the $Z_n$-Sine-Gordon and $Z_n$-Sinh-Gordon equations which can generate new solutions from seed solutions. To see the $Z_n$-systems clearly, we consider the $Z_2$-Sine-Gordon and $Z_3$-Sine-Gordon equations explicitly including their B\"acklund transformations, the nonlinear superposition formula and Lax pairs.