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Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations

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Abstract

We study the energy cascade problematic for some nonlinear Schrödinger equations on \({\mathbb{T}^2}\) in terms of the growth of Sobolev norms. We define the notion of long-time strong instability and establish its connection to the existence of unbounded Sobolev orbits. This connection is then explored for a family of cubic Schrödinger nonlinearities that are equal or closely related to the standard polynomial one \({|u|^2u}\). Most notably, we prove the existence of unbounded Sobolev orbits for a family of Hamiltonian cubic nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first Birkhoff normal form).

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Zaher Hani

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  1. Zaher Hani
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Correspondence to Zaher Hani.

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Communicated by F. Lin

The author is supported by a Simons Postdoctoral Fellowship and NSF Grant DMS-1301647.

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Hani, Z. Long-time Instability and Unbounded Sobolev Orbits for Some Periodic Nonlinear Schrödinger Equations. Arch Rational Mech Anal 211, 929–964 (2014). https://doi.org/10.1007/s00205-013-0689-6

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  • DOI: https://doi.org/10.1007/s00205-013-0689-6

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