On the Convexity of the KdV Hamiltonian

Abstract

Motivated by perturbation theory, we prove that the nonlinear part $ \mathit{H^*}$ of the KdV Hamiltonian $\mathit{H^kdv}$, when expressed in action variables $\mathit{I}=(\mathit{I_n})_{\mathit{n}\geqslant1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb{N})$ of $\ell^2(\mathbb{N})$ and is strictly concave near 0. As a consequence, the differential of $ \mathit{H^*}$ defines a local diffeomorphism near 0 of $\ell^2_\mathbb{C}(\mathbb{N})$. Furthermore, we prove that the Fourier-Lebesgue spaces $\mathcal{FL}^\mathit{s,p}$ with −1/2 $\leqslant \mathit {s} \leqslant$ 0 and 2 $\leqslant \mathit{p} < \infty$, admit global KdV-Birkhoff coordinates. In particular, it means that $\ell^2_+(\mathbb{N})$ is the space of action variables of the underlying phase space $\mathcal{FL}^{-1/2,4}$ and that the KdV equation is globally in time $\mathit{C}^0$-well-posed on $\mathcal{FL}^{-1/2,4}$.