Vanishing geodesic distance for the Riemannian metric with geodesic   equation the KdV-equation
  The Virasoro-Bott group endowed with the right-invariant $L^2$-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance. 
Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor
math.AP math.DG