Infinite sequence of new conserved quantities for N=1 SKdV and the supersymmetric cohomology

TL;DR

This paper constructs an infinite sequence of new non-local conserved quantities for the N=1 Super KdV equation using a Gardner transformation, revealing their structure within supersymmetric cohomology.

Contribution

It introduces a novel sequence of conserved quantities for N=1 SKdV and explores their algebraic and cohomological properties under supersymmetry transformations.

Findings

New non-local conserved quantities for N=1 SKdV are constructed.

Conserved quantities form a structure fitting into supersymmetric cohomology.

Extension of superfield rings makes local conserved quantities exact.

Abstract

An infinite sequence of new non-local conserved quantities for N=1 Super KdV (SKdV) equation is obtained. The sequence is constructed, via a Gardner trasformation, from a new conserved quantity of the Super Gardner equation. The SUSY generator defines a nilpotent operation from the space of all conserved quantities into itself. On the ring of $C_\downarrow^\infty$ superfields the local conserved quantities are closed but not exact. However on the ring of $C_{NL,1}^\infty$ superfields, an extension of the $C_\downarrow^\infty$ ring, they become exact and equal to the SUSY transformed of the subset of odd non-local conserved quantities of the appropriate weight. The remaining odd no-local ones generate closed geometrical objects which become exact when the ring is extended to the $C_{NL,2}^\infty$ superfields and equal to the SUSY transformed of the new even non-local conserved quantities we have obtained. These ones fit exactly in the SUSY cohomology of the already known conserved quantities.