Solve the nonstandard eigenvalue problem governing the melting of the 2D swarmalator async state

Determine the eigenvalues λ by solving the nonstandard eigenvalue equation that arises from linearizing around the static asynchronous disk state of the two-dimensional swarmalator model defined by Eqs. (2.15)–(2.16). Specifically, solve Equation (\eqref{hard_eval}) for λ to characterize the loss of stability of the asynchronous state and thereby identify the critical melting point K_m.

Background

The original 2D swarmalator model couples spatial motion and internal phase dynamics and exhibits a static asynchronous disk state that melts into an active phase wave at a critical coupling value K_m. Standard linear stability methods lead to an eigenvalue problem (Equation \eqref{hard_eval}) involving convolutions with Heaviside and delta functions, which is technically challenging.

Solving this eigenvalue equation would provide the spectrum governing stability of the async disk and enable analytic determination of K_m, a key theoretical quantity analogous to the Kuramoto critical coupling in classical synchronization.