Hamiltonian and Poisson structures for the multicomponent semi-discrete integrable systems

Establish Hamiltonian and Poisson structures for the twelve-component semi-discrete nonlinear integrable system on a quasi-one-dimensional lattice and for the associated six-component reduction, including the derivation of a Hamiltonian function and a Poisson bracket (potentially spatially nonlocal) that reproduces their equations of motion via Hamilton’s equations and satisfies skew-symmetry and the Jacobi identity.

Background

Although the paper discusses conservation laws and sketches a prospective Hamiltonian density, the fully consistent Hamiltonian and Poisson formulations are not established. Appendix A outlines challenges, notably the apparent need for a nonlocal symplectic structure, which complicates verification of the Jacobi identity.

A rigorous Hamiltonian and Poisson framework would formalize the canonical structure, facilitate analytical and numerical studies, and clarify the physical interpretation (e.g., charge transport) of the coupled subsystems.