  We show how to apply ideas from the theory of rough paths to the analysis of
low-regularity solutions to non-linear dispersive equations. Our basic example
will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic
domain and with initial condition in $\FF L^{\alpha,p}$ spaces. We discuss
convergence of Galerkin approximations, a modified Euler scheme and the
presence of a random force of white-noise type in time.
