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KdV Hamiltonian as a Function of Actions

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Abstract

We prove that the non-linear part of the Hamiltonian of the KdV equation on the circle, written as a function of the actions, defines a continuous convex function on the ℓ 2 space and derive for it lower and upper bounds in terms of some functions of the ℓ 2-norm. The proof is based on a new representation of the Hamiltonian in terms of the quasimomentum, obtained via the conformal mapping theory.

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Notes

  1. About the behavior of K(I) of the finite-dimensional subspaces \(\tilde \ell ^{N}\), defined below in Eq. 1.11, we know more. See in [20] and below in Introduction.

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Acknowledgments

A first draft of this paper was written at the CMLS, Ecole Polytechnique, France, during E.K.’s visit there in April–July, 2010. E.K. is grateful to the centre for their hospitality. This work was supported by the RSF grant No 15-11-30007, and by l’ Agence Nacionale de la Recherche, grant ANR-10-BLAN 0102.

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Authors and Affiliations

  1. Mathematical Physics Department, Faculty of Physics, Ulianovskaya 2, St. Petersburg State University, St. Petersburg, 198904, Russia

    Evgeny L. Korotyaev

  2. Université Paris-Diderot (Paris 7), UFR de Mathématiques, Batiment Sophie Germain, 5 rue Thomas Mann, 75205, Paris Cedex 13, France

    Sergei Kuksin

Authors
  1. Evgeny L. Korotyaev
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  2. Sergei Kuksin
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Correspondence to Sergei Kuksin.

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Korotyaev, E.L., Kuksin, S. KdV Hamiltonian as a Function of Actions. J Dyn Control Syst 22, 661–682 (2016). https://doi.org/10.1007/s10883-015-9289-0

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  • DOI: https://doi.org/10.1007/s10883-015-9289-0

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