Symbolic Representation and Classification of N=1 Supersymmetric Evolutionary Equations

TL;DR

This paper extends symbolic representation techniques to N=1 supersymmetric differential polynomials, enabling classification of certain integrable supersymmetric evolutionary equations.

Contribution

It introduces a symbolic framework for supersymmetric polynomials and classifies all scalar λ-homogeneous N=1 supersymmetric evolutionary equations with nonzero linear terms.

Findings

Classified all scalar λ-homogeneous N=1 supersymmetric evolutionary equations with nonzero linear terms.

Developed symbolic operations for supersymmetric differential polynomials.

Provided a comprehensive description of integrable supersymmetric equations.

Abstract

We extend the symbolic representation to the ring of N=1 supersymmetric differential polynomials, and demonstrate that operations on the ring, such as the super derivative, Frechet derivative and super commutator, can be carried out in the symbolic way. Using the symbolic representation, we classify scalar $\lambda$-homogeneous N=1 supersymmetric evolutionary equations with nonzero linear term when $\lambda>0$ for arbitrary order and give a comprehensive description of all such integrable equations.