The minimal N = 2 superextension of the NLS equation

Abstract

We show that the well known N = 1 NLS equation possesses N = 2 supersymmetry and thus it is actually the N = 2 NLS equation. This supersymmetry is hidden in terms of the commonly used N = 1 superfields but it becomes manifest after passing to the N = 2 ones. In terms of the new defined variables the second Hamiltonian structure of the supersymmetric NLS equation coincides with the N = 2 superconformal algebra and the N = 2 NLS equation belongs to the N = 2 a = 4 KdV hierarchy. We propose the KP-like Lax operator in terms of the N = 2 superfields which reproduces all the conserved currents for the corresponding hierarchy.

The solution of the N = 2 supersymmetric f-Toda chain with fixed ends

“…Let us proceed with the N = 2 super-NLS equation [2] ∂f ∂t = f ′′ + 2D(f f Df ), ∂f ∂t = −f ′′ + 2D(f f Df ), (2.1) where f (x, θ, θ) and f (x, θ, θ) are chiral and antichiral fermionic superfields, …”

N = 2 local and N = 4 non-local reductions of supersymmetric KP hierarchy in N = 2 superspace

“…In this section we consider a reduction of the N = 4 supersymmetric KP hierarchy which is inspired by the Lax operator L (27) and preserves its algebra of flows (8)(9)(10). Keeping in mind the results of the previous section and equation (27), let us introduce the following constraint on the operator L − 1 (4)…”

“…Let us study the secondary reduction of the N = 4 Toda chain hierarchy considered in the preceding sections. With this aim, we impose the following secondary constraint 9 on the Lax operator L (28): 9 See also refs. [41,42,37], where the similar reduction of the Manin-Radul [29] and Mulase-Rabin [30,31] N = 1 supersymmetric KP and KdV hierarchies has been discussed.…”

“…are identically satisfied as well. Using these relations, the involution (55) and the algebra structure (8)(9)(10) we are led to the conclusion that only half of the flows (43) are consistent with the reduction (117-118), and these flows are…”

The N = 2 supersymmetric Toda lattice hierarchy

“…First, we can extend the above correspondence to the a = 4 N = 2 super-KdV [8] hierarchy. Using the mapping [7] which connects the N = 2 super-NLS and the a = 4 N = 2 super-KdV hierarchies, as well as using transformations (44) and (45), one can derive the mapping…”