Well-posedness of KdV with higher dispersion

It is shown that if the dispersion of the KdV equation is replaced by a higher order dispersion ∂xm, where m≥3m≥3 is an odd integer, then the critical Sobolev exponent for local well-posedness on the circle does not change. That is, the resulting equation is locally well-posed in Hs(T), s≥−1/2s≥−1/2.