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

Giuseppe Genovese, Renato Lucà, Daniele Valeri

• 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.