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

Abstract

This paper completes the program started in [14,15] aiming at providing a full rigorous justification of the wave kinetic theory for the nonlinear Schrödinger (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.

A note on cascade flux laws for the stochastically-driven nonlinear Schrödinger equation

“…The derivation is meant to hold for time-scales 1 t ≈ T kin , the so-called 'kinetic time', which is a characteristic time-scale for the leading order nonlinear effects. The WKE for the NLS was recently given a mathematically rigorous proof by Deng and Hani in [12][13][14][15][16] where it was shown to correctly predict the dynamics of the deterministic nonlinear Schrödinger with suitable small, random initial data (with no damping or driving) in a certain parameter regime for times t < δT kin , i.e. a small, fixed fraction of the kinetic time-scale (see also [9] for earlier progress) and then finally to arbitrary finite times in their recent work [16].…”

Global well-posedness and stability of the inhomogeneous kinetic wave equation near vacuum

“…Later, in a pioneering work, Deng and Hani [13] reached the kinetic timescale for the cubic (NLS), which provides the first full derivation of the homogeneous (KWE) for (NLS). The same authors addressed propagation of chaos and full range of scaling laws in [14,15]. Recently, they extended the derivation to longer times in [16].…”