1,874 research outputs found
Hamiltonian Structures for the Generalized Dispersionless KdV Hierarchy
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.Comment: 16 pages, plain Te
Integrable Models and the Higher Dimensional Representations of Graded Lie Algebras
We construct a zero curvature formulation, in superspace, for the sTB-B
hierarchy which naturally reduces to the zero curvature condition in terms of
components, thus solving one of the puzzling features of this model. This
analysis, further, suggests a systematic method of constructing higher
dimensional representations for the zero curvature condition starting with the
fundamental representation. We illustrate this with the examples of the sTB
hierarchy and the sKdV hierarchy. This would be particularly useful in
constructing explicit higher dimensional representations of graded Lie
algebras.Comment: 13 pages, late
Supersymmetric Two Boson Equation, Its Reductions and the Nonstandard Supersymmetric KP Hierarchy
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 Schr\"odinger (sNLS) equations
as constrained, nonstandard supersymmetric Kadomtsev-Petviashvili (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 extended supersymmetric system.Comment: 44 pages, plain Te
Properties of an Alternate Lax Description of the KdV Hierarchy
We study systematically the Lax description of the KdV hierarchy in terms of
an operator which is the geometrical recursion operator. We formulate the Lax
equation for the -th flow, construct the Hamiltonians which lead to
commuting flows. In this formulation, the recursion relation between the
conserved quantities follows naturally. We give a simple and compact definition
of all the Hamiltonian structures of the theory which are related through a
power law.Comment: 11 pages, plain Te
A Nonstandard Supersymmetric KP Hierarchy
We show that the supersymmetric nonlinear Schr\"odinger equation can be
written as a constrained super KP flow in a nonstandard representation of the
Lax equation. We construct the conserved charges and show that this system
reduces to the super mKdV equation with appropriate identifications. We
construct various flows generated by the general nonstandard super Lax equation
and show that they contain both the KP and mKP flows in the bosonic limits.
This nonstandard supersymmetric KP hierarchy allows us to construct a new super
KP equation which is nonlocal.Comment: 18 pages, plain TeX, UR-1367, ER-40685-81
Bi-Hamiltonian Structure of the Supersymmetric Nonlinear Schrodinger Equation
We show that the supersymmetric nonlinear Schr\"odinger equation is a
bi-Hamiltonian integrable system. We obtain the two Hamiltonian structures of
the theory from the ones of the supersymmetric two boson hierarchy through a
field redefinition. We also show how the two Hamiltonian structures of the
supersymmetric KdV equation can be derived from a Hamiltonian reduction of the
supersymmetric two boson hierarchy as well.Comment: 13 pages, plain Te
Deformed Harry Dym and Hunter-Zheng Equations
We study the deformed Harry Dym and Hunter-Zheng equations with two arbitrary
deformation parameters. These reduce to various other known models in
appropriate limits. We show that both these systems are bi-Hamiltonian with the
same Hamiltonian structures. They are integrable and belong to the same
hierarchy corresponding to positive and negative flows. We present the Lax pair
description for both the systems and construct the conserved charges of
negative order from the Lax operator. For the deformed Harry Dym equation, we
construct the non-standard Lax representation for two special classes of values
of the deformation parameters. In general, we argue that a non-standard
description will involve a pseudo-differential operator of infinite order.Comment: Latex file, 15 page
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