Light axionic dark matter, motivated by string theory, is increasingly favored for the "no weakly interacting massive particle era". Galaxy formation is suppressed below a Jeans scale of ≃10^{8} M_{⊙} by setting the axion mass to m_{B}∼10^{-22} eV, and the large dark cores of dwarf galaxies are explained as solitons on the de Broglie scale. This is persuasive, but detection of the inherent scalar field oscillation at the Compton frequency ω_{B}=(2.5 months)^{-1}(m_{B}/10^{-22} eV) would be definitive. By evolving the coupled Schrödinger-Poisson equation for a Bose-Einstein condensate, we predict the dark matter is fully modulated by de Broglie interference, with a dense soliton core of size ≃150 pc, at the Galactic center. The oscillating field pressure induces general relativistic time dilation in proportion to the local dark matter density and pulsars within this dense core have detectably large timing residuals of ≃400 nsec/(m_{B}/10^{-22} eV). This is encouraging as many new pulsars should be discovered near the Galactic center with planned radio surveys. More generally, over the whole Galaxy, differences in dark matter density between pairs of pulsars imprints a pairwise Galactocentric signature that can be distinguished from an isotropic gravitational wave background.