  The momentum distribution and dynamical structure factor in a weakly
interacting Bose gas with a time-dependent periodic modulation in terms of the
Bogoliubov treatment are investigated. The evolution equation related to the
Bogoliubov weights happens to be a solvable Mathieu equation when the coupling
strength is periodically modulated. An exact relation between the time
derivatives of momentum distribution and dynamical structure factor is derived,
which indicates that the single-particle property strongly related to the
two-body property in the evolutions of Bose-Einstein condensates. It is found
that the momentum distribution and dynamical structure factor cannot display
periodical behavior. For stable dynamics, some particular peaks in the curves
of momentum distribution and dynamical structure factor appear synchronously,
which is consistent with the derivative relation.
