Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

Giuseppe Genovese ^[1] ; Renato Lucà ^[3] ; Daniele Valeri ^[2]
1. [1] University of Zurich
  
  University of Zurich
  
  Zürich, Suiza
2. [2] Tsinghua University
  
  Tsinghua University
  
  China
3. [3] CSIC
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Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 22, Nº. 3, 2016, págs. 1663-1702
Idioma: inglés
DOI: 10.1007/s00029-016-0225-2
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Resumen
- We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion hk , k ∈ Z+. In each h2k the term with the highest regularity involves the Sobolev norm H˙ k (T) of the solution of the DNLS equation. We show that a functional measure on L2(T), absolutely continuous w.r.t. the Gaussian measure with covariance (I + (−Δ)k )−1, is associated to each integral of motion h2k , k ≥ 1.