  By combining inelastic neutron scattering and numerical simulations, we study
the quasi-one dimensional Ising-like quantum antiferromagnet
BaCo$_{2}$V$_{2}$O$_{8}$ in a longitudinal magnetic field applied along the
magnetic anisotropy axis, which is also the chain direction. The external field
closes the excitation gap due to the magnetic anisotropy, inducing a transition
from the N\'eel ordered state to an incommensurate longitudinal spin density
wave phase. If the field is increased further, another transition into a
transverse antiferromagnetic phase takes place at 9 T due to the competition
between longitudinal and transverse correlations. We numerically and
experimentally show that the model of XXZ chains connected by a weak interchain
interaction well reproduces this transition. We also calculate the dynamical
susceptibility and demonstrate that it agrees quantitatively with inelastic
neutron scattering measurements. In contrast to the abrupt change of magnetic
ordering, the spectra do not change much at the transition at 9 T, and the spin
dynamics can be described as a Tomonaga-Luttinger liquid. We also refine the
modeling of BaCo$_{2}$V$_{2}$O$_{8}$ by including a four-site periodic term
arising from the crystal structure which enables to account for an anomaly of
the magnetic susceptibility appearing at 19 T as well as for the anticrossing
observed in the inelastic neutron scattering spectra.
