We prove that the renormalized defocusing mKdV equation on the circle is locally in time C 0 -wellposed on the Fourier Lebesgue space F p for any 2 < p < ∞. The result implies that the defocusing mKdV equation itself is illposed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS hierarchy.A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing to analytically extend them to F p for any 2 p < ∞ and to deduce asymptotics for n → ±∞.

We prove that the nonlinear part H * of the KdV Hamiltonian H kdv , when expressed in action variables I = (In) n 1 , extends to a real analytic function on the positive quadrant

The defocusing NLS equation iu t = −u xx + 2|u| 2 u on the circle admits a global nonlinear

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 0 and illposed (in a strong sense) if s < 0.

The KdV-equation ut = −uxxx + 6uux on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of u are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms ||u||m, m 0, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order. We further obtain quantitative estimates of the nonlinear Fourier transformed in arbitrary weighted Sobolev spaces.