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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 \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).

Keywords

Action Variable Local Diffeomorphism Dirichlet Eigenvalue Complex Neighborhood Geometric Multiplicity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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)ADSMathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grébert B., Kappeler T.: The defocusing NLS equation and its normal form. European Mathematical Society (EMS), Zürich (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Henrici A., Kappeler T.: Nekhoroshev theorem for the periodic Toda lattice. Chaos 19(3), 033120 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huang G., Kuksin S.B.: The KdV equation under periodic boundary conditions and its perturbations. Nonlinearity 27(9), R61–R88 (2014)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kappeler T., Pöschel J.: KdV & KAM. Springer, Berlin (2003)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Korotyaev E.: Characterization of the spectrum of Schrödinger operators with periodic distributions. Int. Math. Res. Not. 2003(37), 2019–2031 (2003)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Savchuk A.M., Shkalikov A.A.: Sturm-Liouville operators with distribution potentials. Tr. Mosk. Math. Obs. 64, 159–212 (2003)MathSciNetzbMATHGoogle 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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