  Near an arbitrary finite gap potential we construct real analytic, canonical
coordinates for the KdV equation on the torus having the following two main
properties: (1) up to a remainder term, which is smoothing to any given order,
the coordinate transformation is a pseudodifferential operator of order 0 with
principal part given by the Fourier transform and (2) the pullback of the KdV
Hamiltonian is in normal form up to order three and the corresponding
Hamiltonian vector field admits an expansion in terms of a paradifferential
operator. Such coordinates are a key ingredient for studying the stability of
finite gap solutions of the KdV equation under small, quasi-linear
perturbations.
