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Swarmalators are systems of agents which are both self-propelled particlesand oscillators. Each particle is endowed with a phase which modulates itsinteraction force with the other particles. In return, relative positionsmodulate phase synchronization between interacting particles. In the presentmodel, there is no force reciprocity: when a particle attracts another one, thelatter repels the former. This results in a pursuit behavior. In this paper, wederive a hydrodynamic model of this swarmalator system and show that it hasexplicit doubly-periodic travelling-wave solutions in two space dimensions.These special solutions enjoy non-trivial topology quantified by the index ofthe phase vector along a period in either dimension. Stability of thesesolutions is studied by investigating the conditions for hyperbolicity of themodel. Numerical solutions of both the particle and hydrodynamic models areshown. They confirm the consistency of the hydrodynamic model with the particleone for small times or large phase-noise but also reveal the emergence ofintriguing patterns in the case of small phase-noise.

Topological states and continuum model for swarmalators without force reciprocity