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Modified scattering for the cubic Schr\"odinger equation on Diophantine waveguides

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arxiv 2404.16817 v3 pith:R2XAXM7N submitted 2024-04-25 math.AP

Nicolas Camps , Gigliola Staffilani This is my paper
classification math.AP
keywords diophantinecubicequationmodifiedodingerscatteringschrabsence
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We consider the cubic Schr\"odinger equation posed on a product space subject to a generic Diophantine condition. Our analysis shows that the small-amplitude solutions undergo modified scattering to an effective dynamics governed by interactions that induce no growth of high-order Sobolev norms. This is in sharp contrast with the infinite energy cascade scenario observed by Hani--Pausader--Tzvetkov--Visciglia in the absence of Diophantine conditions.

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  1. Non-equilibrium steady state for a three-mode energy cascade model

    math.PR 2025-05 conditional novelty 7.0 of 10

    For a three-mode reduced NLS model with stochastic forcing and damping at the boundary modes, the authors construct a unique invariant probability measure with polynomial convergence to the non-equilibrium steady state.

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