Gelfand-Dikii Brackets for Nonstandard Lax Equations

Brunelli, J. C.; Das, Ashok; Huang, Wen-Jui

doi:10.1142/s0217732394002008

Cited by 22 publications

(63 citation statements)

References 18 publications

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“…(7). Such an observation, in fact, had led to the definition of the Gelfand-Dikii brackets in the case of the TB system [9]. However, the present derivation is quite general and holds even in super space since we have not assumed any specific property of ∂.…”

Section: Transformation To a Standard Systemmentioning

confidence: 84%

“…In fact, a modification of the Gelfand-Dikii brackets in the case of the Two Boson equation was already noted in ref. [9] and it is the purpose of this letter to extend such a generalization to superspace to include supersymmetric nonstandard integrable systems. The Hamiltonian structures for the supersymmetric Two Boson (sTB) have, of course, already been obtained directly [6][7].…”

Section: Introductionmentioning

confidence: 99%

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Gelfand–dikii Brackets for Nonstandard Supersymmetric Systems

Das

Panda

1996

Mod. Phys. Lett. A

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We show how a general nonstandard Lax equation (supersymmetric or otherwise) can be expressed as a standard Lax equation. This enables us to define the Gelfand–Dikii brackets for a nonstandard supersymmetric equation. We discuss the Hamiltonian structures for the nonstandard super KP system and work out explicitly the two Hamiltonian structures of the supersymmetric two-boson system from this point of view.

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Section: Transformation To a Standard Systemmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Gelfand–dikii Brackets for Nonstandard Supersymmetric Systems

Das

Panda

1996

Mod. Phys. Lett. A

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“…However, as was discussed by Olver and Nutku the Riemann equation has an additional Hamiltonian structure. Namely, the third-order [14] and the generalization of the Gelfand-Dikii brackets was performed in [25]. The derivation of the Poisson brackets for equations with nonstandard dispersionless Lax representation is an interesting and relevant problem and is also under investigation.…”

Section: Discussionmentioning

confidence: 99%

“…Here (E 2 ) ≥1 stands for the purely nonnegative (without p 0 terms) part of the Laurent polynomial obtained from E 2 . For dispersive systems the nonstandard Lax representation was introduced by Kupershmidt in [14] and the generalization of the Gelfand-Dikii brackets was performed in [25]. The derivation of the Poisson brackets for equations with nonstandard dispersionless Lax representation is an interesting and relevant problem and is also under investigation.…”

Section: Discussionmentioning

confidence: 99%

HAMILTONIAN STRUCTURES FOR THE GENERALIZED DISPERSIONLESS KdV HIERARCHY

Brunelli

1996

Rev. Math. Phys.

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We study from a Hamiltonian point of view the generalized dispersionless KdV hierarchy of equations. From the so called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and third Hamiltonian structures are calculated directly from the r-matrix approach. Since the third structure is not related recursively with the first two ones the generalized dispersionless KdV hierarchy can be characterized as a truly tri-Hamiltonian system.

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“…While the generalization of the Gelfand-Dikii brackets for the bosonic nonstandard Lax equations was obtained in ref. [25], the extension of these brackets to superspace is technically more difficult and not yet understood. Therefore, we will construct the brackets by supersymmetrizing the bosonic Hamiltonian structures (2.17), (2.18) and (2.20) in a direct way [10,13].…”

Section: Hamiltonian Structures For the Stbmentioning

confidence: 99%

Supersymmetric Two-Boson Equation, Its Reductions and the Nonstandard Supersymmetric Kp Hierarchy

Brunelli

Das

1995

Int. J. Mod. Phys. A

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In this paper, we review various properties of the supersymmetric Two Boson (sTB) system. We discuss the equation and its nonstandard Lax representation. We construct the local conserved charges as well as the Hamiltoniam structures of the system. We show how this system leads to various other known supersymmetric integrable models under appropriate field redefinition. We discuss the sTB and the supersymmetric nonlinear Schrödinger (sNLS) equations as constrained, nonstandard supersymmetric KadomtsevPetviashvili (sKP) systems and point out that the nonstandard sKP systems naturally unify all the KP and mKP flows while leading to a new integrable supersymmetrization of the KP equation. We construct the nonlocal conserved charges associated with the sTB system and show that the algebra of charges corresponds to a graded, cubic algebra. We also point out that the sTB system has a hidden supersymmetry making it an N = 2 extended supersymmetric system.

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Gelfand-Dikii Brackets for Nonstandard Lax Equations

Abstract: We generalize the construction of Gelfand-Dikii brackets to the case of nonstandard Lax equations. We also discuss the possible origin of Kac-Moody algebras present in such systems.

Cited by 22 publications

References 18 publications

Gelfand–dikii Brackets for Nonstandard Supersymmetric Systems

Gelfand–dikii Brackets for Nonstandard Supersymmetric Systems

HAMILTONIAN STRUCTURES FOR THE GENERALIZED DISPERSIONLESS KdV HIERARCHY

Supersymmetric Two-Boson Equation, Its Reductions and the Nonstandard Supersymmetric Kp Hierarchy

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