Streamwise-localised travelling-wave edge state in square-duct flow

We report the first streamwise-localised travelling-wave solution in square-duct flow that acts as an edge state in the full phase space, without any imposed spatial symmetries. Performing edge tracking and Newton iteration, we identify a steady travelling wave that possesses a codimension-one stable manifold, which (at least locally) forms the boundary between the basins of laminar and turbulent attractors. Parametric continuation identifies this solution as the lower branch of a saddle-node bifurcation pair. Perturbation analysis places both solutions on the laminar–turbulent boundary and uncovers a heteroclinic connection that links the two branches and is likewise confined to the basin boundary. This symmetry-free, localised edge state expands the catalogue of invariant solutions in wall-bounded shear flows and provides a geometric framework for understanding the transition dynamics in extended systems.