Communications in Mathematical Physics

, Volume 346, Issue 1, pp 191–236 | Cite as

On the Convexity of the KdV Hamiltonian

  • Thomas Kappeler
  • Alberto Maspero
  • Jan Molnar
  • Peter Topalov


Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), 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 \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), 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 \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Thomas Kappeler
    • 1
  • Alberto Maspero
    • 2
  • Jan Molnar
    • 1
  • Peter Topalov
    • 3
  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Laboratoire de Mathématiques Jean LerayUniversité de NantesNantes Cedex 03France
  3. 3.Department of MathematicsNortheastern UniversityBostonUSA

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