On the super-NLS equation and its relation with the N = 2 super-KdV equation within the coset approach

Krivonos, S.; Сорин, А.С.; Toppan, Francesco

doi:10.1016/0375-9601(95)00651-i

Cited by 35 publications

(49 citation statements)

References 21 publications

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“…The relation of the chirality preserving scalar Lax operators of Ref. [5] with the former ones (which have been introduced in [10,4,6]) was observed recently [11]. For the Lax operator L s (3) the N = 2 and N = 1 residues 1 vanish since it does not contain the N = 2 fermionic derivatives acting as operators.…”

Section: Extended Matrixmentioning

confidence: 92%

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

Krivonos¹,

Сорин²

1999

Physics Letters A

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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.

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Section: Extended Matrixmentioning

confidence: 92%

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

Krivonos¹,

Сорин²

1999

Physics Letters A

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“…In addition, we require them to be globally chargeless (Q(H i ) = 0) with respect to a charge operator Q such that Q(H) = Q(H) = 0, Q(F ) = 1, Q(F ) = −1. The reason why we impose this condition is the desire to reproduce the conserved quantities of N = 2 NLS hierarchy in the limit H = H = 0; requiring this invariance amounts to the property that the reduced hamiltonians "live" on the quotient of N = 2 sl(2) ⊕ u(1) over its Cartan subalgebra u(1) ⊕ u(1), which is the cahracteristic feature of N = 2 NLS hierarchy [12]. For sure, this choice does not yield the most general set of N = 4 invariant hamiltonians one can construct (and does not even correspond to the most general hierarchy, see however the remarks in section 4), but it is the only choice which allows us to recover N = 2 NLS hierarchy.…”

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confidence: 99%

N = 4 super NLS-mKdV hierarchies

Ivanov¹,

Krivonos²,

Toppan³

1997

Physics Letters B

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N = 2 extension of affine algebra sl(2) ⊕ u(1) possesses a hidden global N = 4 supersymmetry and provides a second hamiltonian structure for a new N = 4 supersymmetric integrable hierarchy defined on N = 2 affine supercurrents. This system is an N = 4 extension of at once two hierarchies, N = 2 NLS and N = 2 mKdV ones. It is related to N = 4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks construction which relates N = 2 sl(2) ⊕ u(1) algebra to "small" N = 4 SCA. We also find the underlying affine hierarchy for another integrable system with the N = 4 SCA second hamiltonian structure, "quasi" N = 4 KdV hierarchy. It respects only N = 2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N = 2 affine algebra admitting a quaternionic structure possesses N = 4 supersymmetry and so can be used to produce N = 4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies. 1. Introduction. The integrable hierarchies of differential equations in 1 + 1 dimensions have been widely studied in the last several years both in the physical and in the mathematical contexts. There is an ever-growing evidence that their relevance should not be confined to the realm of pure mathematics, but rather their beautiful mathematical structures show themselves naturally when investigating physical problems. At first they appeared in connection with the discretized versions, via the matrix-models approach, of 2-dimensional gravity (see [1] and references therein). Recently they sprang up rather unexpectedly in the domain of 4-dimensional field theories as an underlying structure of the N = 2 super Yang-Mills theories in the Seiberg-Witten approach [2]. The remarkable relation of the KdV-type hierarchies, via their second hamiltonian structure, to the conformal algebras (Virasoro algebra and its extensions) establishes a link between such hierarchies and the string theories which has still to be fully appreciated.One is naturally led to investigate supersymmetric extensions of the above bosonic hierarchies. Besides N = 1 supersymmetric systems which were studied in a lot of papers starting from refs.[3], it is very interesting to consider the larger N cases. One of the main sources of interest in extended supersymmetric hierarchies with N > 1 is that they could lead to a sort of "unification or grand-unification" of known hierarchies. It could happen that seemingly unrelated bosonic or lower supersymmetric (N = 1, 2) hierarchies are different manifestations of a single "unifying" larger N supersymmetric hierarchy.During the last years, N = 2 supersymmetric hierarchies were a subject of many exhaustive studies. Much less was known about systems with higher N. In a series of papers [4][5][6][7] the first example of hierarchy with N = 4 supersymmetry, N = 4 super KdV system, was discovered and studied. It is related, through its second hamiltonian structure, to "small" N = 4 superconformal algebra, is bi-hamiltonian and admits a Lax-pair formulation. It encom...

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“…In recent years, there has been rapidly growing interest in various 1 + 1-dimensional soliton systems exhibiting invariance under (extended) supersymmetry and admitting Lax representations. For various contributions to this program see for instance [1,2,3,4,5,6] and references therein.Still, in our opinion there remains a need for a systematic approach to the problem. The purpose of this paper is to propose such a construction.…”

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confidence: 99%

Manifestly supersymmetric Lax integrable hierarchies

Aratyn¹,

Rasinariu²

1997

Physics Letters B

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A systematic method of constructing manifestly supersymmetric 1 + 1-dimensional KP Lax hierarchies is presented. Closed expressions for the Lax operators in terms of superfield eigenfunctions are obtained. All hierarchy equations being eigenfunction equations are shown to be automatically invariant under the (extended) supersymmetry. The supersymmetric Lax models existing in the literature are found to be contained (up to a gauge equivalence) in our formalism.1 Work supported in part by U.S. Department of Energy, contract Introduction. In recent years, there has been rapidly growing interest in various 1 + 1-dimensional soliton systems exhibiting invariance under (extended) supersymmetry and admitting Lax representations. For various contributions to this program see for instance [1,2,3,4,5,6] and references therein.Still, in our opinion there remains a need for a systematic approach to the problem. The purpose of this paper is to propose such a construction. The resulting formalism is manifestly supersymmetric, encompasses known supersymmetric models encountered in literature and provides a straightforward guidance for the future model building.The starting point is the SKP 2 hierarchy based on the Lax operator of an even parity and correspondingly involving only even time-flows [7]. The reduction scheme we propose generalizes to N = 1 supersymmetry the method which has previously been applied to the standard KP Hierarchy and resulted in the constrained KP hierarchy [8,9]. It is based on the symmetry constraints generated by pairs of conjugated eigenfunctions of the original hierarchy. It is an important feature of the formalism that the eigenfunctions remain eigenfunctions of the reduced hierarchy [9]. In the supersymmetric case the eigenfunctions are superfields and their flows provide a complete set of hierarchy equations. Due to the use of the manifestly covariant formalism the hierarchy equations are all invariant under supersymmetry. By analogy we generalize the method and obtain the N-extended supersymmetric Lax hierarchy models.We also introduce a gauge transformation to the nonstandard supersymmetric KP hierarchy. As in the bosonic case the gauge transformation induces an equivalence between standard and nonstandard Lax hierarchies. This mapping is used to gauge some of the supersymmetric nonstandard Lax hierarchies present in the literature to our models.The KP hierarchy and its reduction. The KP hierarchy consists of a set of multi-time evolution equations in the Lax form

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On the super-NLS equation and its relation with the N = 2 super-KdV equation within the coset approach

Cited by 35 publications

References 21 publications

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

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

N = 4 super NLS-mKdV hierarchies

Manifestly supersymmetric Lax integrable hierarchies

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