Number of parameter constraints for higher-order analogies

Determine whether the physically admissible solutions associated with higher-order mathematical problems having increasingly singular functions \(\xi(z)\) at infinity continue to require only a single constraint between the physical parameters.

Background

The paper identifies a hierarchy of mathematical free-boundary problems in which the analytic function ξ(z)\xi(z) has progressively higher-order singularities at infinity. For Problem 3, the analogous capillary-bubble solutions require one scalar constraint among the parameters, producing a two-dimensional admissible parameter surface.

The authors expect analogous constraints for higher-order problems associated with steadily rotating configurations of polygonal satellite vortices, but explicitly leave unresolved whether the number of constraints remains one as the order of the singularity increases.

References

Whether it remains a single constraint as the order of the singularity of $\xi(z)$ at infinity increases remains to be determined.

— Extension of a mathematical analogy between vortical flows without surface tension and irrotational capillary flows  (2610.10146 - Kunz et al., 7 Oct 2026) in Section 6, Discussion