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.
Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation / Genovese, Giuseppe; Lucà, Renato; Valeri, Daniele. - In: SELECTA MATHEMATICA. - ISSN 1022-1824. - 22:3(2016), pp. 1663-1702. [10.1007/s00029-016-0225-2]
Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation
Valeri, Daniele
2016
Abstract
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.File | Dimensione | Formato | |
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