Exact stability conditions for the spatially coherent state

Establish the exact stability conditions for the spatially coherent stationary state in the two-dimensional swarmalator model with higher-order phase interactions, where all positions synchronize into one or two clusters separated by π and the phases remain distributed around two distinct values. Derive explicit conditions on J, K1, and K2 guaranteeing stability of this state.

Background

The spatially coherent state arises specifically due to repulsive higher-order phase interactions and attractive spatial coupling, leading to frozen spatial positions (one or two clusters separated by π) while phases show partial coherence around two values. The authors provide an approximate emergence condition (e.g., 0 < −K1/K2 < 1 with J, K1 > 0 and K2 < 0) but not the exact stability criterion.

A full stability analysis would determine when this state is robust and how higher-order interactions interact with pairwise couplings to produce spatial freezing along with partial phase organization.