Birkhoff coordinates for KdV on phase spaces of distributions

• T. Kappeler ^[1] ; C. Möhr ^[1] ; P. Topalov ^[2]
  1. [1] University of Zurich

     University of Zurich

     Zürich, Suiza

     
  2. [2] Northeastern University

     Northeastern University

     City of Boston, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 11, Nº. 1, 2005, págs. 37-98
• Idioma: inglés
• DOI: 10.1007/s00029-005-0009-6
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• Resumen

  • The purpose of this paper is to extend the construction of Birkhoff coordinates for the KdV equation from the phase space of square integrable 1-periodic functions with mean value zero to the phase space H−10(T) of mean value zero distributions from the Sobolev space H−1(T) endowed with the symplectic structure (∂/∂x)−1. More precisely, we construct a globally defined real-analytic symplectomorphism Ω:H−10(T)→h−1/2 where h−1/2 is a weighted Hilbert space of sequences (xn,yn)n⩾1 supplied with the canonical Poisson structure so that the KdV Hamiltonian for potentials in H10(T) is a function of the actions ((x2n+y2n)/2)n⩾1 alone.