# On integrability aspects of the supersymmetric sine-Gordon equation

## TL;DR

This paper explores the integrability features of the supersymmetric sine-Gordon equation, including Lax pairs, Riccati equations, Bäcklund and Darboux transformations, and constructs super multisoliton solutions with geometric interpretations.

## Contribution

It provides a comprehensive analysis of the integrability properties of the supersymmetric sine-Gordon equation, including explicit solution construction and geometric characterizations.

## Key findings

- Constructed Lax pairs and zero-curvature representations.
- Derived super Riccati equations and auto-Bäcklund transformations.
- Built non-trivial super multisoliton solutions and geometric surface characterizations.

## Abstract

In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-B\"acklund transformation of the supersymmetric sine-Gordon equation. In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym--Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.