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Conservation of Resonant Periodic Solutions for the One-Dimensional Nonlinear Schrödinger Equation

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Abstract

We consider the one-dimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough, say greater than a certain threshold value depending on the number of resonant modes. If the frequencies of the latter are close enough to such a threshold, then they can not be too distant from each other. Hence we can interpret such solutions as perturbations of wave packets with large wave number.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tre, 00146, Roma, Italy

    Guido Gentile

  2. SISSA, 34014, Trieste, Italy

    Michela Procesi

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  1. Guido Gentile
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  2. Michela Procesi
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Communicated by G. Gallavotti

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Gentile, G., Procesi, M. Conservation of Resonant Periodic Solutions for the One-Dimensional Nonlinear Schrödinger Equation. Commun. Math. Phys. 262, 533–553 (2006). https://doi.org/10.1007/s00220-005-1409-3

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