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KAM for Beating Solutions of the Quintic NLS

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Abstract

We consider the nonlinear Schrödinger equation of degree five on the circle \({\mathbb{T}= \mathbb{R} / 2\pi}\). We prove the existence of quasi-periodic solutions with four frequencies which bifurcate from “resonant” solutions [studied in Grébert and Thomann (Ann Inst Henri Poincaré Anal Non Linéaire 29(3):455–477, 2012)] of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Naples, Italy

    E. Haus

  2. Dipartimento di Matematica e Fisica, Università di Roma Tre, Rome, Italy

    M. Procesi

Authors
  1. E. Haus
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  2. M. Procesi
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Correspondence to E. Haus.

Additional information

Communicated by W. Schlag

This research was supported by the European Research Council under FP7, Project “Hamiltonian PDEs and small divisor problem: a dynamical systems approach” (HamPDEs). The first author was also supported by Programme STAR, financed by UniNA and Compagnia di San Paolo.

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Haus, E., Procesi, M. KAM for Beating Solutions of the Quintic NLS. Commun. Math. Phys. 354, 1101–1132 (2017). https://doi.org/10.1007/s00220-017-2925-7

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  • DOI: https://doi.org/10.1007/s00220-017-2925-7

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