Towards the construction of N = 2 supersymmetric integrable hierarchies

Abstract

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super W n second hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct hamiltonians we are led to a new definition of super-residues for degenerate N=2 supersymmetric pseudodifferential operators. We have found a new non-polinomial Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.

Bi-Hamiltonian structure of the N=2 supersymmetric α=1 KdV hierarchy

“…Actually, besides the N = 2 α = 1 KdV equation there are another two inequivalent N = 2 supersymmetric Hamiltonian equations with the N = 2 superconformal algebra as a second Hamiltonian structure (the N = 2 α = −2 and α = 4 KdV equations [3,1]), but the N = 2 α = 1 KdV equation is rather exceptional [4]. Despite knowledge of its Lax-pair description, there remains a lot of longstanding, unsolved problems which resolution would be quite important for a deeper understanding and more detailed description of the N = 2 α = 1 KdV hierarchy.…”

The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy

“…Let us start with a short summary of the main facts concerning the N = 2 supersymmetric (n, m)-GNLS hierarchy [1,6] which will be useful in what follows.…”

“…9 For other examples of N = 4 supersymmetric NLS-type integrable hierarchies, see the recent paper [13] 10 Let us recall that, for general values of the parameters n and m, the N = 2 supersymmetric (n, m)-GNLS hierarchy is invariant with respect to GL(n|m) supergrop [1]. g s = for the integrals, where L mGN LS and L GN LS are the Lax operators of the mGNLS and GNLS hierarchies 11 .…”

“…Although an analysis of this connection for the N=2 supersymmetric integrable hierarchies is still lacking, it is nevertheless a fact that the latter hierarchies have extremely rich and interesting structures. In [1] a large class of such hierarchies was introduced: the N = 2 supersymmetric (n, m) Generalized Nonlinear Schrödinger (GNLS) hierarchies. They were subsequently studied in a number of other papers, [2,3,4,5,6].…”

“…Bosonic limit of the N = 2 super (n, m)-GNLS Hamiltonian structure. To derive the bosonic limit, we set all fermionic components of the superfields F a and F a equal to zero and define the bosonic components as [1] …”