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Derivation of the wave kinetic equation: full range of scaling laws

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arxiv 2301.07063 v2 pith:SCBQF4N7 submitted 2023-01-17 math.AP

Yu Deng , Zaher Hani This is my paper · ORCID
classification math.AP
keywords diagramskineticequationexpansionfeynmanfullrobustwave
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This paper completes the program started in arXiv:2104.11204 and arXiv:2110.04565 aiming at providing a full rigorous justification of the wave kinetic theory for the nonlinear Schr\"odinger (NLS) equation. Here, we cover the full range of scaling laws for the NLS on an arbitrary periodic rectangular box, and derive the wave kinetic equation up to small multiples of the kinetic time. The proof is based on a diagrammatic expansion and a deep analysis of the resulting Feynman diagrams. The main novelties of this work are three-fold: (1) we present a robust way to identify arbitrarily large "bad" diagrams which obstruct the convergence of the Feynman diagram expansion, (2) we systematically uncover intricate cancellations among these large "bad" diagrams, and (3) we present a new robust algorithm to bound all remaining diagrams and prove convergence of the expansion. These ingredients are highly robust, and constitute a powerful new approach in the geneal mathematical study of Feynman diagrams.

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Cited by 6 Pith papers

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    For radial nonnegative data, a regularized 3-wave kinetic equation has a unique global strong solution that stays nonnegative and is time-analytic.

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