  In form of a case study for the KdV and the KdV2 equations, we present a
novel approach of representing the frequencies of integrable PDEs which allows
to extend them analytically to spaces of low regularity and to study their
asymptotics. Applications include convexity properties of the Hamiltonians and
wellposedness results in spaces of low regularity. In particular, it is proved
that on $H^{s}$ the KdV2 equation is $C^{0}$-wellposed if $s\ge 0$ and illposed
(in a strong sense) if $s < 0$.
