One-dimensional wave kinetic theory
Katja D. Vassilev
TL;DR
The paper establishes a rigorous kinetic-limit description for the one-dimensional MMT model with nonlinearity strength α and dispersion ω(k)=|k|^σ, addressing the longstanding gap in 1D wave turbulence theory. It develops a detailed Feynman-diagram framework using ternary trees, couples, and molecule graphs to track fourth-order correlations, introduces a Stage 1 splicing to cancel irregular chains, and implements a Stage 2 counting algorithm to bound decorations while controlling the remainder via a linear operator ℒ. Under the scaling α=L^{−γ}, the authors derive kinetic-time-scale results up to T∼L^{−ε}α^{−5/4} (or T∼L^{−ε}T_{ ext{kin}}^{5/8}) and highlight that for σ>1 the collision kernel is trivial, leading to no nontrivial second-moment dynamics up to T_{ ext{kin}}. The work thus clarifies the nature and limits of kinetic theories in 1D, showing that in certain regimes the second-moment dynamics are governed by a trivial collision operator within the considered timescales, and lays groundwork for further refinement toward longer timescales. The methodology combines probabilistic bounds, hypercontractivity, and intricate combinatorial graph-analysis to overcome 1D counting obstacles and provides a template for future advances in 1D wave turbulence.
Abstract
Although wave kinetic equations have been rigorously derived in dimension $d \geq 2$, both the physical and mathematical theory of wave turbulence in dimension $d = 1$ is less understood. Here, we look at the one-dimensional MMT (Majda, McLaughlin, and Tabak) model on a large interval of length $L$ with nonlinearity of size $α$, restricting to the case where there are no derivatives in the nonlinearity. The dispersion relation here is $|k|^σ$ for $0 < σ\leq 2$ and $σ\neq 1$, and when $σ= 2$, the MMT model specializes to the cubic nonlinear Schrödinger (NLS) equation. In the range of $1 < σ\leq 2$, the proposed collision kernel in the kinetic equation is trivial, begging the question of what is the appropriate kinetic theory in that setting. In this paper we study the kinetic limit $L \to \infty$ and $α\to 0$ under various scaling laws $α\sim L^{-γ}$ and exhibit the wave kinetic equation up to timescales $T \sim L^{-ε}α^{-\frac{5}{4}}$ (or $T \sim L^{-ε} T_{\mathrm{kin}}^{\frac{5}{8}}$). In the case of a trivial collision kernel, our result implies there can be no nontrivial dynamics of the second moment up to timescales $T_{\mathrm{kin}}$.