A Comment on the Odd Flows for the Supersymmetric KdV equation

TL;DR

This paper analyzes odd flows in the supersymmetric KdV equation, revealing a new hierarchy related to the supersymmetric KP hierarchy and connecting it to super-W_{1+ ablafty} algebra, with implications for superintegrability and supergravity.

Contribution

It shows that only half of the introduced odd flows are in Lax form and introduces a new hierarchy linked to the supersymmetric KP hierarchy.

Findings

Half of the odd flows are in standard Lax form.

A new supersymmetric hierarchy is identified and related to the Jacobian supersymmetric KP hierarchy.

The algebra of symmetries is isomorphic to super-W_{1+ ablafty} algebra.

Abstract

In a recent paper Dargis and Mathieu introduced integrodifferential odd flows for the supersymmetric KdV equation. These flows are obtained from the nonlocal conservation laws associated with the fourth root of its Lax operator. In this note I show that only half of these flows are of the standard Lax form, while the remaining half provide us with hamiltonians for an SKdV-type reduction of a new supersymmetric hierarchy. This new hierarchy is shown to be closely related to the Jacobian supersymmetric KP-hierarchy of Mulase and Rabin. A detailed study of the algebra of additional symmetries of this new hierarchy reveals that it is isomorphic to the super-W_{1+\infty} algebra, thus making it a candidate for a possible interrelationship between superintegrability and two-dimensional supergravity.