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

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.