The Gardner Category and Non-local Conservation Laws for N=1 Super KdV

TL;DR

This paper develops an algebraic framework using the Gardner category to derive non-local conserved quantities for N=1 Super KdV, providing new tools for analyzing supersymmetric differential equations.

Contribution

It introduces a Gardner category and algebraic methods to systematically obtain non-local conserved quantities in supersymmetric KdV equations.

Findings

Defined a fermionic substitution semigroup and Gardner category.

Derived algebraic expressions for non-local conserved quantities.

Extended the framework to rapidly decreasing superfields.

Abstract

The non-local conserved quantities of N=1 Super KdV are obtained using a complete algebraic framework where the Gardner category is introduced. A fermionic substitution semigroup and the resulting Gardner category are defined and several propositions concerning their algebraic structure are proven. This algebraic framework allows to define general transformations between different nonlinear SUSY differential equations. We then introduce a SUSY ring extension to deal with the non-local conserved quantities of SKdV. The algebraic version of the non-local conserved quantities is solved in terms of the exponential function applied to the $D^{-1}$ of the local conserved quantities of SKdV. Finally the same formulas are shown to work for rapidly decreasing superfields.