  We report parametric resonances (PRs) in the mean-field dynamics of a
one-dimensional dipolar Bose-Einstein condensate (DBEC) in widely varying
trapping geometries. The chief goal is to characterize the energy levels of
this system by analytical methods and the significance of this study arises
from the commonly known fact that in the presence of interactions the energy
levels of a trapped BEC are hard to calculate analytically. The latter
characterization is achieved by a matching of the PR energies to energy levels
of the confining trap using perturbative methods. Further, this work reveals
the role of the interplay between dipole-dipole interactions (DDI) and trapping
geometry in defining the energies and amplitudes of the PRs. The PRs are
induced by a negative Gaussian potential whose depth oscillates with time.
Moreover, the DDI play a role in this induction. The dynamics of this system is
modeled by the time-dependent Gross- Pitaevskii equation (TDGPE) that is
numerically solved by the Crank-Nicolson method. The PRs are discussed basing
on analytical methods: first, it is shown that it is possible to reproduce PRs
by the Lagrangian variational method that are similar to the ones obtained from
TDGPE. Second, the energies at which the PRs arise are closely matched with the
energy levels of the corresponding trap calculated by time-independent
perturbation theory. Third, the most probable transitions between the trap
energy levels yielding PRs are determined by time-dependent perturbation
theory. The most significant result of this work is that we have been able to
characterize the above mentioned energy levels of a DBEC in a complex trapping
potential.
