Extended N = 2 supersymmetric matrix (1, s)-KdV hierarchies

Abstract

We propose the Lax operators for N = 2 supersymmetric matrix generalization of the bosonic (1, s)-KdV hierarchies. The simplest examples -the N = 2 supersymmetric a = 4 KdV and a = 5/2 Boussinesq hierarchies -are discussed in detail.E-Mail: 1) krivonos@thsun1.jinr.dubna.su 2) sorin@thsun1.jinr.dubna.su 1. Introduction. The existence of three different infinite families of N = 2 supersymmetric integrable hierarchies with the N = 2 super W s algebras as their second Hamiltonian structure is a well-established fact by now [1,2,3]. Their bosonic limits have been analyzed in [4], and three different families of the corresponding bosonic hierarchies and their Lax operators have been selected. Then, a complete description in terms of super Lax operators for two out of three families has been proposed in [5,6], and the generalization to the matrix case has been derived in [6].The last remaining family of N = 2 hierarchies is supersymmetrization of the bosonic (1, s)-KdV hierarchies [4]. We call them the N = 2 supersymmetric (1, s)-KdV hierarchies. As opposed to the bosonic counterparts of the former two hierarchies [4], the (1, s)-KdV hierarchy is irreducible (see [7] and references therein), i.e. its Lax operator cannot be decomposed into a direct sum of some more elementary components. This reduction property leads to a strong restriction of the original supersymmetric Lax operator: its bosonic limit should be irreducible. In other words, it should generate only a single operator component. This property is surely satisfied for a supersymmetric Lax operator which is a pure bosonic pseudo-differential operator with the coefficients expressed in terms of N = 2 superfields and their fermionic derivatives in such a way that it commutes with one of the two N = 2 fermionic derivatives. The Lax operator of this kind has in fact been observed in [4] for the N = 2 a = 5/2 Boussinesq hierarchy in the negative-power decomposition over bosonic derivative up to the ∂ −5 order. Quite recently, its closed analytic representation has been obtained in [8].The aim of the present letter is to present a new infinite class of reductions (with a finite number of fields) of N = 2 supersymmetric matrix KP hierarchy which includes the abovementioned family of N = 2 (1, s)-KdV hierarchies in the scalar case.

A Note on the Third Family of N = 2 Supersymmetric KdV Hierarchies

“…containing an infinite set of bosonic N = 2 superfields. Following [11], we associate with L the two pseudo-differential operators 1) The right-hand side is a ΨDO of order n−2, which proves the consistency of the flows (2.15). We used the notation ∂ p = ∂/∂t p .…”

“…Using this result, certain discrete invariance for the nonlinear evolution equations [11] can be extracted directly from the operator L. We shall see a simple example below.…”

“…Another consequence of Lax equations (2.16) is that they provide us with an infinite set of conserved quantities for the flows (2.14). Standard arguments coming from the study of KP type hierarchies [12] tell us that conservation laws are associated with res L k and resL k so that the quantities [7,11]…”

“…Actually, the bosonic limit of the two first N = 2 n-KdV hierarchies is composed of two decoupled KdV and non-standard KdV hierarchies [4,7] whereas the bosonic limit of the third one is the (1, n) KdV hierarchy [7] which is irreducible [8,9,10]. Recently, Krivonos and Sorin [11] gave a Lax representation for the third family of N = 2 KdV hierarchies.…”

Alternative dispersionless limit of N=2 supersymmetric KdV-type hierarchies

“…For one of the three known families of integrable supersymmetric hierarchies with N = 2, W n superalgebra as the second Hamiltonian structure, the Lax operators contain only bosonic operators of the forms [6]:…”