# B\"acklund-Darboux Transformations and Discretizations of $N=2\; a=-2$ Supersymmetric KdV Equation

## TL;DR

This paper develops Darboux and Bäcklund transformations for the N=2 a=-2 supersymmetric KdV equation, enabling the construction of integrable semi-discrete and discrete systems with continuum limits.

## Contribution

It introduces new transformations and superposition formulas for the supersymmetric KdV equation, facilitating discretizations while preserving integrability.

## Key findings

- Constructed Darboux and Bäcklund transformations for the equation.
- Derived super semi-discrete and full discrete integrable systems.
- Analyzed continuum limits of the discrete systems.

## Abstract

The $N=2 \;a=-2$ supersymmetric KdV equation is studied. A Darboux transformation and the corresponding B\"acklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated B\"acklund transformation. The B\"{a}cklund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems. The continuum limits of these discrete systems are also considered.