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Invariant measures for the periodic derivative nonlinear Schrödinger equation

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Abstract

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in \(L^2\) and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of the Gaussian measure on \(L^2\) with covariance \((\mathbb {I}+(-\Delta )^k)^{-1}\) for any \(k\;\geqslant \;2\).

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Acknowledgements

This work was partially developed during the visit of the first two authors to the CIRM, Trento (through the program Research in Pairs) and to the Yau Mathematical Sciences Center of Tsinghua University, Beijing. Both these institutions are thankfully acknowledged. Furthermore, we are grateful to Lorenzo Carvelli for many stimulating discussions. The work of GG is supported through NCCR SwissMAP. The work of RL is supported through the ERC Grant 676675 FLIRT. The work of DV was supported through the NSFC “Research Fund for International Young Scientists” grant and through a Tshinghua University startup research grant when working in the Yau Mathematical Sciences Center.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, 8057, Zurich, Switzerland

    Giuseppe Genovese

  2. Department Mathematik und Informatik, Universität Basel Spiegelgasse 1, 4051, Basel, Switzerland

    Renato Lucà

  3. School of Mathematics and Statistics, University of Glasgow, G12 8QQ, Glasgow, UK

    Daniele Valeri

Authors
  1. Giuseppe Genovese
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  2. Renato Lucà
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  3. Daniele Valeri
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Correspondence to Renato Lucà.

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Communicated by Y. Giga.

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Genovese, G., Lucà, R. & Valeri, D. Invariant measures for the periodic derivative nonlinear Schrödinger equation. Math. Ann. 374, 1075–1138 (2019). https://doi.org/10.1007/s00208-018-1754-0

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  • DOI: https://doi.org/10.1007/s00208-018-1754-0

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