On negative flows of the AKNS hierarchy and a class of deformations of bihamiltonian structure of hydrodynamic type

Abstract

A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parameterize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the one hand, two realizations of the first negative flow of the AKNS model and, on the other, two-component generalizations of Camassa-Holm and Dym type equations. The two-component generalizations of Camassa-Holm and Dym type equations can be obtained from the negative order Hamiltonians constructed from the Lenard relations recursively applied on the Casimir of the first Poisson bracket of hydrodynamic type. The positive order Hamiltonians, which follow from Lenard scheme applied on the Casimir of the second Poisson bracket of hydrodynamic type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS Hamiltonians give rise to charges conserved with respect to equations of motion of two-component Camassa-Holm and two-component Dym type equations.

20 pages, Latex

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• Deformations of N=2 super-conformal algebra and supersymmetric two-component Camassa-Holm equation
• Dressing chain for the acoustic spectral problem
• A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue
• Negative flows for several integrable models
• NLS-type equations from quadratic pencil of Lax operators: negative flows
• 3D-consistency of negative flows
• Miura-reciprocal transformations and localizable Poisson pencils
• Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows
• Integrable invariant Sobolev metrics on the Abelian extension of the diffeomorphism group of the circle and two-component generalizations of the Camassa-Holm equation