  Light Axionic Dark Matter, motivated by string theory, is increasingly
favored for the "no-WIMP era". Galaxy formation is suppressed below a Jeans
scale, of $\simeq 10^8 M_\odot$ by setting the axion mass to, $m_B \sim
10^{-22}$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, $\omega_B= (2.5{\rm
\, months})^{-1}(m_B/10^{-22}eV)$, would be definitive. By evolving the coupled
Schr\"odinger-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 $\simeq 150pc$, 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 $\simeq 400nsec/(m_B/10^{-22}eV)$. 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.
