Bosonization, Singularity Analysis, Nonlocal Symmetry Reductions and Exact Solutions of Supersymmetric KdV Equation

TL;DR

This paper explores the bosonization of the supersymmetric KdV equation, analyzing its singularity structure, establishing a Bäcklund transformation, and deriving exact solutions through symmetry reductions, revealing richer solution structures due to Grassmann parameters.

Contribution

It introduces a novel bosonization approach for the supersymmetric KdV system, proves its Painlevé property, and develops a generalized method for finding exact solutions.

Findings

Proved Painlevé property of the bosonized system.

Established a Bäcklund transformation and residual symmetry.

Derived exact solutions using symmetry reduction techniques.

Abstract

Assuming that there exist at least two fermionic parameters, the classical N= 1 supersymmetric Korteweg-de Vries (SKdV) system can be transformed to some coupled bosonic systems. The boson fields in the bosonized SKdV (BSKdV) systems are defined on even Grassmann algebra. Due to the intrusion of other Grassmann parameters, the BSKdV systems are different from the usual non-supersymmetric integrable systems, and many more abundant solution structures can be unearthed. With the help of the singularity analysis, the Painlev\'e property of the BSKdV system is proved and a B\"acklund transformation (BT) is found. The BT related nonlocal symmetry, we call it as residual symmetry, is used to find symmetry reduction solutions of the BSKdV system. Hinted from the symmetry reduction solutions, a more generalized but much simpler method is established to find exact solutions of the BSKdV and then the SKdV systems, which actually can be applied to any fermionic systems.