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

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 ∫ℎ, ∈ℤ+. In each ∫ℎ2 the term with the highest regularity involves the Sobolev norm ˙() of the solution of the DNLS equation. We show that a functional measure on 2(), absolutely continuous w.r.t. the Gaussian measure with covariance (+(−))−1, is associated to each integral of motion ∫ℎ2, ≥1