The Wave Kinetic Theory for Quasilinear MMT Equation

Abstract

We study the one-dimensional quasilinear Majda--McLaughlin--Tabak (MMT) equation on a large torus $[0,L]$: \begin{align*} i \partial_t u +2π|\nabla|^σu +λ^{2}|\nabla|^β\left[ \left||\nabla|^βu\right|^{2} |\nabla|^βu\right]=0. \end{align*} Our focus is on the well-posedness of its dynamics and the emergence of kinetic behavior where the domain size $L$ tends to infinity and the nonlinearity $α=λ^2L^{-1}$ vanishes. In contrast to semilinear dispersive models, the quasilinear structure leads to unavoidable derivative loss, which prevents the construction of solutions via iteration of the Duhamel formula. Our results exhibit a dichotomy depending on the dispersion exponent $σ$. For $σ\in(1,2]$, we prove that, with high probability, solutions exist up to time scales $T_0 \sim α^{-\frac54+} \wedge α^{-\frac1{1-β}+}$, and that only trivial resonances occur, leading to a degenerate wave kinetic equation. For $σ\in(0,1)$, we prove the existence up to time scales $T_0 \sim α^{-1-}$ and show that the second-order statistics are well approximated by the wave kinetic equation. In both cases, the solutions remain smooth while exhibiting smallness in suitable $L^\infty$-based norms despite having large total energy. The proof proceeds in two main steps. First, we establish the propagation of randomness for a suitably truncated equation, which allows us to overcome the derivative loss and recover the kinetic description. Then, we perform deterministic high-order energy estimates and a bootstrap argument to extend the solution up to time $T_0$.