Exact stability region of the thick phase wave in two-dimensional swarmalators

Determine the exact stability region of the thick phase wave stationary state in the two-dimensional swarmalator model with periodic boundary conditions and higher-order phase interactions (with spatial coupling J and phase couplings K1 and K2). Derive precise conditions on J, K1, and K2 under which the thick phase wave is linearly stable.

Background

The thick phase wave is a stationary state where the swarmalator phases correlate with one spatial coordinate (either x or y), producing nonzero order parameter S1+ (or T1+) while the other order parameters remain near zero. It is believed to bifurcate from the asynchronous state and eventually transition to the thin phase wave state, but simulations show it can also appear where the thin phase wave does not, complicating the theoretical picture.

The authors provide numerical evidence for the thick phase wave in various parameter regimes, yet an exact analytical characterization of its stability region in terms of J, K1, and K2 is lacking. Establishing the precise stability conditions would clarify where this state exists and how higher-order interactions shape its emergence.