Invariant solutions of the supersymmetric sine-Gordon equation

TL;DR

This paper performs a detailed symmetry analysis of the N=1 supersymmetric sine-Gordon equation, deriving invariant solutions including algebraic, hyperbolic, and periodic types through symmetry reduction methods.

Contribution

It provides a comprehensive classification of symmetries and invariant solutions for the supersymmetric sine-Gordon equation in different formulations.

Findings

Derived Lie superalgebras of symmetries for the supersymmetric sine-Gordon equation.

Classified subgroups with codimension 1 orbits in the symmetry analysis.

Obtained explicit algebraic, hyperbolic, and periodic solutions.

Abstract

A comprehensive symmetry analysis of the N=1 supersymmetric sine-Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables. Next, we consider the super-sine-Gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables. In each case, a Lie (super)algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed. The method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model. Several types of algebraic, hyperbolic and doubly periodic solutions are obtained in explicit form.