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Large KAM Tori for Quasi-linear Perturbations of KdV

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Abstract

In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question of whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.

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Acknowledgements

This project was motivated by questions raised by S. Kuksin and V. Zakharov. We would like to thank them for their input. We would also like to thank M. Procesi for very valuable feedback. Part of this work was written during the stay of M. Berti at FIM at ETHZ. We thank FIM for the kind hospitality and support. In addition, the research was partially supported by PRIN 2015KB9WPT005 (M.B.), by the Swiss National Science Foundation (T.K., R.M.), and by INDAM-GNFM (R.M.).

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Authors and Affiliations

  1. SISSA, Trieste, Italy

    Massimiliano Berti

  2. University of Zurich, Zurich, Switzerland

    Thomas Kappeler

  3. University of Milan, Via Saldini 50, Milan, 20133, Italy

    Riccardo Montalto

Authors
  1. Massimiliano Berti
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  2. Thomas Kappeler
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  3. Riccardo Montalto
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Correspondence to Massimiliano Berti.

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Communicated by N. Masmoudi.

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Berti, M., Kappeler, T. & Montalto, R. Large KAM Tori for Quasi-linear Perturbations of KdV. Arch Rational Mech Anal 239, 1395–1500 (2021). https://doi.org/10.1007/s00205-020-01596-2

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  • DOI: https://doi.org/10.1007/s00205-020-01596-2

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