  In this paper, reduced-order models (ROMs) are constructed for the
Ablowitz-Ladik equation (ALE), an integrable semi-discretization of the
nonlinear Schr\"{o}dinger equation (NLSE) with and without damping. Both ALEs
are non-canonical conservative and dissipative Hamiltonian systems with the
Poisson matrix, depending quadratically on the state variables and with
quadratic Hamiltonian. The full-order solutions are obtained with the energy
preserving midpoint rule for the conservative ALE and exponential midpoint rule
for the dissipative ALE. The reduced-order solutions are constructed
intrusively by preserving the skew-symmetric structure of the reduced
non-canonical Hamiltonian system by applying proper orthogonal decomposition
with the Galerkin projection. For an efficient offline-online decomposition of
the ROMs, the quadratic nonlinear terms of the Poisson matrix are approximated
by the discrete empirical interpolation method. The computation of the
reduced-order solutions is further accelerated by the use of tensor techniques.
Preservation of the Hamiltonian and momentum for the conservative ALE, and
preservation of dissipation properties of the dissipative ALE, guarantee the
long-term stability of soliton solutions.
