Hamiltonian structure of multicomponent nonlinear Schrödinger equations in difference form

It is shown that the difference analogs of the mutticomponent nonlinear Schrödnger equations solvable by the inverse scattering technique for discrete block Zakharov–Shabat system are Hamiltonian systems. Using the generally employed formulation of a linear problem, this requires the introduction of complicated nonlinear and nonlocal Poisson brackets between the elements of the potential. An equivalent formulation of the linear problem is found for which the corresponding Poisson brackets for the potentials are canonical. The entire treatment is based on the method of expansion with respect to the “squares” of solutions of the linear problem.