Classify vortex street patterns for Λ=2 on the cylinder

Determine a complete classification of periodic (cylindrical) vortex equilibria for two species of vortices with strengths −1 and 2 (Λ=2), governed by the periodic equilibrium equations (e.g., ∑_{j≠i} Q_j cot(z_i − z_j) = 0 for static or k + ∑_{j≠i} Q_j cot(z_i − z_j) = 0 for translating configurations), including whether non-terminating sequences exist and the full set of terminating families.

Background

For Λ=1, all static and translating vortex street patterns on a cylinder (period π) are generated by chains of Darboux transformations using trigonometric seed eigenfunctions of the free Schrödinger operator, and these families are identified with trigonometric soliton τ-functions of the KdV hierarchy.

For Λ=2, the relevant Lax operator is third order and lacks trigonometric eigenfunctions analogous to the Λ=1 case, preventing construction of non-terminating sequences via intertwining methods. The authors attempted a trigonometric analog of the bilinear construction and found only short terminating sequences. They explicitly state that a complete classification of Λ=2 vortex patterns on the cylinder remains unresolved.