  In this paper we prove new qualitative features of solutions of KdV on the
circle. The first result says that the Fourier coefficients of a solution of
KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first
order with strongly oscillating phase factors defined in terms of the KdV
frequencies. The second result provides estimates for the approximation of such
a solution by trigonometric polynomials of sufficiently large degree.
