The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems

Abstract

A new approach to Hamiltonian structures of integrable systems is proposed by making use of a trace identity. For a variety of isospectral problems that can be unified to one model ψx=Uψ, it is shown that both the related hierarchy of evolution equations and the Hamiltonian structure can be obtained from the same solution of the equation Vx=[U,V].

Finite-Dimensional Hamiltonian Systems from Li Spectral Problem by Symmetry Constraints

“…Starting from spectral problem (1.1), the Li hierarchy and its Hamilton structure were obtained by using the zero curvature equation and trace identity [7]. To find the symmetry constraint and useful formulae, we recall the Li hierarchy here.…”

Adjoint Symmetry Constraints Leading to Binary Nonlinearization

“…Similarly, applying the trace identity [16], the soliton hierarchy (4.14) has a bi-Hamiltonian formulation 15) where the Hamiltonian functionalsH l are defined bỹ…”

“…Therefore, the soliton hierarchy (3.14) is called the multicomponent AKNS soliton hierarchy. In order to generate the Hamiltonian structure of the multicomponent AKNS hierarchy (3.14), we apply the trace identity [16]:…”

On the construction of recursion operator and algebra of symmetries for field and lattice systems

“…Several methods were proposed of greater or lower generality, see for example refs. [1], [2], [3], [4]. Perhaps the best known is the one based on the implectic-symplectic factorization of the recursion operator, i.e.…”