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Long time stability for cubic nonlinear Schr\"odinger equations on non-rectangular flat tori

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arxiv 2402.04122 v2 pith:2UN22YVT submitted 2024-02-06 math.AP math.DS

Joackim Bernier , Nicolas Camps This is my paper · ORCID
classification math.APmath.DS
keywords equationscubicflatnonlinearodingerschrstabilitytori
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We consider nonlinear Schr\"odinger equations on flat tori satisfying a simple and explicit Diophantine non-degeneracy condition. Provided that the nonlinearity contains a cubic term, we prove the almost global existence and stability of most of the small solutions in high regularity Sobolev spaces. To this end, we develop a normal form approach designed to handle general resonant Hamiltonian partial differential equations for which it is possible to modulate the frequencies by using the initial data.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Long-time dynamics for the Kelvin-Helmholtz equations close to circular vortex sheets

    math.AP 2025-04 conditional novelty 8.0 of 10

    For almost all Weber numbers below 4(2+sqrt(3)), small-amplitude circular vortex sheets with surface tension stay O(ε)-close to the circular equilibrium for times of order ε^{-(N+1)} for every integer N.

  2. Infinite dimensional invariant tori for nonlinear Schr\"odinger equations

    math.AP 2024-12 accept novelty 8.0 of 10

    For 1D NLS without external parameters, the paper constructs non-resonant infinite-dimensional Kronecker tori (almost periodic, not quasi-periodic) arbitrarily close to Kuksin-Pöschel KAM tori.

  3. Almost Global Solutions of Kirchhoff Equation

    math.AP 2025-05 conditional novelty 7.0 of 10

    Almost all small solutions of the 1D Kirchhoff equation keep their Fourier actions almost frozen for ε^{-r} and sub-exponential times.

  4. On almost periodic solutions to NLS without external parameters

    math.AP 2025-05 conditional novelty 4.0 of 10

    The authors give an extended proof sketch showing that 1D NLS without external parameters supports plenty of non-resonant infinite-dimensional tori, hence almost periodic non-quasi-periodic solutions.

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