Component-trace identities for Hamiltonian structures

Abstract

We show that on a particular class of semi-direct sums of matrix Lie algebras, component traces of the matrix product can produce bilinear forms which are non-degenerate, symmetric and invariant under the Lie product. The corresponding variational identities are called componenttrace identities and provide tools in generating Hamiltonian structures of integrable couplings including the perturbation equations. An illustrative example of applying component-trace identities is given for the KdV hierarchy.

Nonlinear continuous integrable Hamiltonian couplings

“…The presented integrable couplings can also possess other integrable properties such as Hirota bilinear forms [24] and s-symmetry algebras [25]. All such analyses will enrich multi-component integrable equations (see, e.g., [15,26]) and help understand them better to work towards classification of integrable equations based on loop algebras.…”

“…In the variational identity (2.8), the expression on the left-hand side is the vector of variational derivatives with respect to all elements of u; U k denotes the partial derivative of U with respect to k, U u denotes the vector of partial derivatives of U with respect to all elements of u, and hÁ, Ái is a non-degenerate, symmetric and ad-invariant bilinear form over the Lie algebra consisting of square matrices of the form (2.3) (see [5,14,15] for general discussion). In what follows, we will make an application to the AKNS hierarchy to shed light on this generating scheme.…”

“…The set of all matrices above is closed under the matrix product, and so it constitutes a matrix Lie algebra, which is non-semisimple. The variational identities (see [5,14,15]) over this kind of Lie algebras can be used to furnish Hamiltonian structures for the corresponding continuous integrable couplings. We will illustrate such an idea of generating nonlinear continuous integrable Hamiltonian couplings by the AKNS hierarchy of soliton equations.…”

Bi-integrable couplings of a Kaup–Newell type soliton hierarchy and their bi-Hamiltonian structures

“…After completing the construction of the new KN type soliton hierarchy, we consider further work including corresponding bi-integrable couplings and bi-Hamiltonian structures via the variational identity [16,36]. Bi-integrable couplings of a soliton hierarchy u t ¼ KðuÞ ¼ Kðx; t; u; u x ; u xx ; .…”

“…. In addition, the Hamiltonian structures for soliton hierarchies can be established by the trace identity [8][9][10] or for integrable couplings by the variational identity [16,36].…”