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Invariant measures for mKdV and KdV in infinite volume

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arxiv 2401.04292 v1 pith:W2SELJ6N submitted 2024-01-09 math.AP

Justin Forlano , Rowan Killip , Monica Visan This is my paper · ORCID
classification math.AP
keywords dynamicsgibbsmeasuremkdvconstructequationinvariantline
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We construct dynamics for the defocusing real-valued (Miura) mKdV equation on the real line with initial data distributed according to Gibbs measure. We also prove that Gibbs measure is invariant under these dynamics. On the way, we provide a new proof of the invariance of the Gibbs measure under mKdV on the torus. Building on these results, we construct new measure-preserving dynamics for the KdV equation on the whole real line. Samples from this family of measures exhibit the same local regularity as white-noise, but completely different statistics!

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  1. Invariant Gibbs measures for the one-dimensional quintic nonlinear Schr\"odinger equation in infinite volume

    math.AP 2025-05 accept novelty 7.0 of 10

    The defocusing quintic NLS on the real line has an invariant infinite-volume Gibbs measure for p between 3 and 5.

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