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
Article

Abstract

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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References

  1. 1.
  2. 2.
    Bikbaev R.F., Kuksin S.B.: On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. Krichever. Lett. Math. Phys. 28(2), 115–122 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bobenko A.I., Kuksin S.B.: Finite-gap periodic solutions of the KdV equation are nondegenerate. Phys. Lett. A 161(3), 274–276 (1991)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Djakov P., Mityagin B.: Instability zones of periodic 1-dimensional Schrödinger and Dirac operators. Russian Math. Surv. 61, 663–766 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Djakov P., Mityagin B.: Spectral gaps of Schrödinger operators with periodic singular potentials. Dyn. Partial Differ. Equ. 6(2), 95–165 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Grébert B., Kappeler T.: The defocusing NLS equation and its normal form. European Mathematical Society (EMS), Zürich (2014)CrossRefMATHGoogle Scholar
  7. 7.
    Henrici A., Kappeler T.: Nekhoroshev theorem for the periodic Toda lattice. Chaos 19(3), 033120 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Huang G., Kuksin S.B.: The KdV equation under periodic boundary conditions and its perturbations. Nonlinearity 27(9), R61–R88 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kappeler T., Mityagin B.: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator. SIAM J. Math. Anal. 33(1), 113–152 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kappeler T., Möhr C.: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials. J. Funct. Anal. 186(1), 62–91 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kappeler T., Möhr C., Topalov P.: Birkhoff coordinates for KdV on phase spaces of distributions. Selecta Math. (N.S.) 11(1), 37–98 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kappeler T., Pöschel J.: KdV & KAM. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  13. 13.
    Kappeler T., Serier F., Topalov P.: On the symplectic phase space of KdV. Proc. Am. Math. Soc. 136(5), 1691–1698 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kappeler T., Topalov P.: Riccati representation for elements in \({H^{-1}(\mathbb{T})}\) and its applications. Pliska Stud. Math. Bulgar. 15, 171–188 (2003)MathSciNetGoogle Scholar
  15. 15.
    Kappeler T., Topalov P.: Global wellposedness of KdV in \({H^{-1}(\mathbb{T},\mathbb{R})}\). Duke Math. J. 135(2), 327–360 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Korotyaev E.: Characterization of the spectrum of Schrödinger operators with periodic distributions. Int. Math. Res. Not. 2003(37), 2019–2031 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Korotyaev, E., Kuksin, S.B.: KdV hamiltonian as a function of actions. J. Dyn. Control Syst., pp. 1–22 (2015, online)Google Scholar
  18. 18.
    Krichever I.M.: Perturbation theory in periodic problems for two-dimensional integrable systems. Sov. Sci. Rev. C. Math. Phys. 9, 1–103 (1992)MATHGoogle Scholar
  19. 19.
    Molnar, J.C.: On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete Contin. Dyn. Syst. 36(6), 3339–3356 (2016)Google Scholar
  20. 20.
    Pöschel J.: Hill’s potentials in weighted Sobolev spaces and their spectral gaps. Math. Ann. 349(2), 433–458 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Savchuk A.M., Shkalikov A.A.: Sturm-Liouville operators with distribution potentials. Tr. Mosk. Math. Obs. 64, 159–212 (2003)MathSciNetMATHGoogle Scholar

Copyright information

© 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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