  Swarmalators are systems of agents which are both self-propelled particles
and oscillators. Each particle is endowed with a phase which modulates its
interaction force with the other particles. In return, relative positions
modulate phase synchronization between interacting particles. In the present
model, there is no force reciprocity: when a particle attracts another one, the
latter repels the former. This results in a pursuit behavior. In this paper, we
derive a hydrodynamic model of this swarmalator system and show that it has
explicit doubly-periodic travelling-wave solutions in two space dimensions.
These special solutions enjoy non-trivial topology quantified by the index of
the phase vector along a period in either dimension. Stability of these
solutions is studied by investigating the conditions for hyperbolicity of the
model. Numerical solutions of both the particle and hydrodynamic models are
shown. They confirm the consistency of the hydrodynamic model with the particle
one for small times or large phase-noise but also reveal the emergence of
intriguing patterns in the case of small phase-noise.
