Bosonization of Supersymmetric KdV equation

TL;DR

This paper introduces a bosonization method for the classical supersymmetric KdV equation, transforming it into coupled bosonic equations, enabling the explicit construction of novel localized solutions and revealing the system's rich soliton structure.

Contribution

The paper presents a new bosonization approach for supersymmetric KdV equations, allowing explicit solutions and uncovering novel localized excitations not accessible by previous methods.

Findings

Derived coupled bosonic equations from supersymmetric KdV

Obtained new multi-soliton solutions with rich localized excitations

Identified limitations in reproducing traditional multi-soliton solutions

Abstract

Bosonization approach to the classical supersymmetric systems is presented. By introducing the multi-fermionic parameters in the expansions of the superfields, the $\mathcal {N}=1$ supersymmetric KdV (sKdV) equations are transformed to a system of coupled bosonic equations. The method can be applied to any fermionic systems. By solving the coupled bosonic equations, some novel types of exact solutions can be explicitly obtained. Especially, the richness of the localized excitations of the supersymmetric integrable system are discovered. The rich multi-soliton solutions obtained here have not yet been obtained by using other methods. Unfortunately, the traditional known multi-soliton solutions can also not be obtained by the bosonization approach of this paper. Some open problems on the bosonization of the supersymmetric integrable models are proposed in the both classical and quantum levels.