JournalsrlmVol. 30, No. 4pp. 865–880

A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

  • Marcel Guardia

    Universitat Politècnica de Catalunya, Barcelona, Spain and BGSMath Graduate School of Mathematics, Barcelona, Spain

  • Zaher Hani

    University of Michigan, Ann Arbor, USA

  • Emanuele Haus

    Università die Roma Tre, Italy

  • Alberto Maspero

    SISSA, Trieste, Italy

  • Michela Procesi

    Università die Roma Tre, Italy
A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation cover

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Abstract

We present the recent result in [29] concerning strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap solutions of the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the topology (0 < s < 1) and whose norm can grow by any given factor.

Cite this article

Marcel Guardia, Zaher Hani, Emanuele Haus, Alberto Maspero, Michela Procesi, A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 4, pp. 865–880

DOI 10.4171/RLM/873