Stability of the 1d swarmalator model in the continuum limit

We study the 1d swarmalator model in the continuum limit. We examine the stability of its collective states which have compact support: synchrony, where the swarmalators lie in two sync dots (zero dimensional support), and the phase wave, where the swarmalators line up in a ring with uniformly spaced positions and phases (one dimensional support). The compact support imposes analytic difficulties that occur in many other swarmalator models and thus is blocking progress in the field. We show how to overcome this difficulty, deriving the two states' stability spectra exactly.