Identify the mechanism determining the number of clusters in the splintered phase wave of the 2D model

Ascertain the mechanism that determines the number of spatial–phase clusters in the splintered phase wave state of the two-dimensional swarmalator model defined by Eqs. (2.15)–(2.16), including its dependence on interaction length scales and the coupling parameters J and K.

Background

When K is decreased below zero from the static phase wave regime, the annular phase wave breaks into multiple clusters of similar phase (the splintered phase wave). Empirically, fewer clusters are observed when spatial interaction functions operate at smaller length scales, and parameters J and K also influence clustering.

However, the mechanism—analytical or computational—that fixes the number of clusters has not been characterized. Resolving this would advance understanding of pattern selection and the role of coupling functions and scales in swarmalator pattern formation.