Confinement of Fractional Vortices and Free Composites in a Three-Component Ginzburg–Landau System with Josephson Coupling

We study topological vortex solutions in a three-component Ginzburg–Landau (GL) system with Josephson intercomponent coupling γ . Fractional vortices are attached to domain walls whose tension, in the rigid-amplitude approximation, satisfies the BPS bound σ DW = 8 v 2 2 γ ; the Josephson component σ J = σ DW / 2 is verified to within a few percent via BPS equipartition and grid refinement. Finite-size scaling over L = 8 to 512 confirms E J ∝ L for isolated fractional vortices (confinement) and E J = 0 for full vortices (free). A two-dimensional scan over both quark separation R sep and system size L reveals E J comp ∝ R sep 2 / L for the three-vortex composite, consistent with vanishing Josephson energy in the thermodynamic limit—the composite carries no Josephson confinement. Per-pair energy decomposition shows that the composite distributes Josephson energy equally among the three intercomponent pairs. At γ = 0 the tension ratio T 3 : T 2 : T 1 = 3 : 2 : 1 is a counting identity; finite γ introduces corrections of order 7% at γ = 0.5 .