Camassa-Holm type equations and vortexons in axisymmetric Poiseuille pipe flows

We present a study on the nonlinear dynamics of a disturbance to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. We show that the associated Navier-Stokes equations can be reduced to a set of coupled Camassa-Holm type equations. These support regular and singular inviscid travelling waves with wedge-type singularities, the so called peakons, which bifurcate from smooth solitary waves as their celerity increase. In physical space they correspond to localized/periodic toroidal vortices concentrated near the pipe boundaries (wall vortexon) or that wrap around the pipe axis (centre vortexon). The dynamics of a vortexon is also investigated by means of an accurate Fourier-based numerical scheme.