Dispersionless Limit of Integrable Models

Nonlinear dispersionless equations arise as the dispersionless limit of well know integrable hierarchies of equations or by construction, such as the system of hydrodynamic type. Some of these equations are integrable in the Hamiltonian sense and appear in the study of topological minimal models. In the first part of the review we will give a brief introduction to integrable models, mainly its Lax representation. Then, we will introduce the dispersionless limit and show some of our results concerning the two-component hyperbolic system of equations such as the polytropic gas and Born-Infeld equations.Comment: 25 pages, 4 figures, Te

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 $N=2$ 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 $n$-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

Davey-Stewartson Equation from a Zero Curvature and a Self-Duality Condition

We derive the two equations of Davey-Stewartson type from a zero curvature condition associated with SL(2,{\bf R}) in $2+1$ dimensions. We show in general how a $2+1$ dimensional zero curvature condition can be obtained from the self-duality condition in $3+3$ dimensions and show in particular how the Davey-Stewartson equations can be obtained from the self-duality condition associated with SL(2,{\bf R}) in $3+3$ dimensions.Comment: 9 pages, UR-1332, ER-40685-78