On the non-abelian superalgebra spanned by the conserved quantities of N=1 supersymmetric Korteweg-de Vries equation

TL;DR

This paper constructs an infinite sequence of bosonic non-local conserved quantities for the N=1 supersymmetric KdV equation, revealing a non-abelian superalgebra structure that extends the known conserved quantities.

Contribution

It introduces a new class of bosonic non-local conserved quantities with even parity and dimension 1, expanding the understanding of the supersymmetric KdV's conserved algebra.

Findings

The algebra of conserved quantities closes in polynomials of local and non-local quantities.

The new bosonic non-local conserved quantities cannot be derived from known local and non-local conserved quantities.

The work fits these quantities into a supersymmetric cohomology framework.

Abstract

We obtain an infinite sequence of bosonic non-local conserved quantities for the N=1 supersymmetric Korteweg-de Vries equation. It is generated from a bosonic non-local conserved quantity of Super Gardner equation. In distinction to the already known one with odd parity and dimension 1/2, it has even parity and dimension 1. It fits exactly in the supersymmetric cohomology in the space of conserved quantities that we also introduce here. Using results from this cohomology we obtain the Poisson bracket of several non-local conserved quantities, including the already known odd ones and the new even ones. The algebra closes in terms of polynomials of local and non-local conserved quantities. We prove that the bosonic non-local conserved quantities cannot be expressed as functions of the already known local and non-local conserved quantities of Super KdV equation.