  Axion dark matter can be converted into photons in the magnetospheres of
neutron stars leading to a spectral line centred on the Compton wavelength of
the axion. Due to the rotation of the star and the plasma effects in the
magnetosphere the signal is predicted to be periodic with significant time
variation - a unique smoking gun for axion dark matter. As a proof of principle
and to develop the methodology, we carry out the first time domain search of
the signal using data from PSR J2144$-$3933 taken as part of the MeerTIME
project on MeerKAT telescope. We search for specific signal templates using a
matched filter technique and discuss when a time-domain analysis (as is
typically the case in pulsar observations) gives greater sensitivity to the
axion-coupling to photons in comparison to a simple time-averaged total flux
study. We do not find any candidate signals and, hence, impose an upper limit
on the axion-to-photon coupling of $g_{a\gamma\gamma}<4\times 10^{-11}\,{\rm
GeV}^{-1}$ over the mass range $m_{\rm a}=3.9-4.7\,\mu{\rm eV}$ using this
data. This limit relies on PSR J2144$-$3933 not being an extremely aligned
rotator, as strongly supported by simple arguments based on the observed pulse
profile width. We discuss the possibilities of improving this limit using
future observations with MeerKAT and also SKA1-mid and the possibility of using
other objects. Finally, to evade modelling uncertainties in axion radio
signals, we also carry out a generic ``any periodic-signal search" in the data,
finding no evidence for an axion signal.
