Completeness of the bifurcation diagrams

Determine whether the bifurcation diagrams of stationary solutions to the continuum McKean–Vlasov equation u_t = ∇·(u(∇u − ∇V_per*u)) on the periodic torus with square and hexagonal lattice symmetry—constructed using the minimal Fourier-periodized attractive potentials—are complete, that is, ascertain if any additional equilibrium branches exist beyond those identified and whether all qualitatively distinct behaviors have been captured across parameter ranges near and away from the primary bifurcation.

Background

The paper constructs and analyzes bifurcation diagrams for a continuum limit of interacting particle systems with short-range repulsion and long-range attraction on square and hexagonal lattices. It identifies vertical branches, vacuum bubble and fissure formation, and their diffusive perturbations, together with numerical continuation and discrete-particle comparisons.

In the Discussion, the authors explicitly state that some questions remain open regarding the completeness of the presented bifurcation diagrams, suggesting potential additional branches or structures not captured in the current analysis. This invites a rigorous classification to confirm that no further branches exist under the considered symmetries and potentials.