Year: 2026
Type: preprint
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*} with $σ\in(0,1)\cup (1,2]$ and $0<β<σ/4$. We consider Gaussian random initial data and work in the joint limit $L\to\infty$ with $α=λ^2L^{-1}\to0$. The quasilinear structure leads to unavoidable derivative loss, which prevents the direct construction of solutions via iteration of the Duhamel formula. For $σ\in(1,2]$, we prove that, with high probability, solutions exist up to time scales ${T} \simα^{-{\frac43}+} \wedge α^{-\frac1{σ/2-β}+}$, and that only trivial four-wave resonances occur, leading to a degenerate wave kinetic equation. For $σ\in(0,1)$, we prove the existence up to time scales $T\sim α^{-\frac43+}$, 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. The proof combines propagation of randomness for a truncated equation with deterministic energy estimates and a bootstrap argument. The main technical difficulties arise in the regime $σ\in(0,1)$. At low frequencies, the counting argument requires simultaneous control of the frequency gap and the associated resonant phase gap. In the energy estimate, nontrivial resonances generate quartic terms that prevent the energy estimates from closing the bootstrap on $[0,T]$. To overcome this obstruction, we develop a new normal-form $L^2$ energy estimates, using distinct frequency truncations for the initial data and the nonlinearity.
Source: arXiv (Cornell University)
Author: Lü, Huaxiang
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W7204171773