2 research outputs found
On negative flows of the AKNS hierarchy and a class of deformations of bihamiltonian structure of hydrodynamic type
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.Comment: 20 pages, Late