Families of quasi-bi-Hamiltonian systems and separability

Abstract

It is shown how to construct an infinite number of families of quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton equations. Three explicit QBH structures are presented for the first three families of the constrained flows. The Nijenhuis coordinates defined by the Nijenhuis tensor for the corresponding families of QBH systems are proved to be exactly the same as the separated variables introduced by mean of the Lax matrices for the constrained flows.

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

“…This concept was first introduced in [4] in the particular case of systems with two degrees of freedom and it was quickly extended in [23,24] for a higher-dimensional systems. Some recent papers considering properties of this particular class of systems are [1,2,3,4,5,6,8,14,15,23,24,25,36].…”

A New Nonlinear Integrable Couplings of Solioton Hierarchy and its Hamiltonian Structure with Kronecker Product

“…[13][14][15][16][17][18][19][20][21][22]. In this paper in order to obtain new nonlinear integrable coupling system, we illustrate a new approach by the Kronecker product to nonlinear soliton equation hierarchy.…”

Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

“…Moreover, most of the examples constructed so far, belongs to separable systems, separated either through a bi-Hamiltonian formalism [12], [3] or through a spectral curve method [1], [2]. If additionally involutive constants of motion are quadratic forms of momenta, then we can again systematically construct related hydrodynamic systems which have a complete integral in the form (3.27).…”

“…In each case, we get a cofactor type hydrodynamic system. Just to demonstrate a vast universality of this approach, our second example is related to another soliton hierarchy, represented by the Jaulent Miodek spectral problem [26] (a special case of Antonowicz and Fordy spectral problem [27]) q i p i x = 0 1 α The bi-Hamiltonian (quasi-bi-Hamiltonian) representation was found in [3], [28] with Actually, the link is as follows:…”