Rigorous derivation of wave kinetic theory for the quasilinear MMT equation

Establish a rigorous derivation of the wave kinetic equation for the one-dimensional quasilinear Majda–McLaughlin–Tabak equation in the large-torus, weak-nonlinearity limit.

Background

The paper studies the one-dimensional quasilinear Majda–McLaughlin–Tabak equation with derivative nonlinearities and seeks to connect its microscopic wave interactions to a macroscopic wave kinetic equation. Unlike semilinear dispersive equations, the quasilinear structure causes derivative loss, preventing a direct iterative Duhamel or Feynman-tree construction of solutions.

The authors obtain long-time regularity and a first-order kinetic expansion under specific scalings and truncations, but the broader rigorous derivation of the kinetic description for the quasilinear MMT model remains unresolved.

References

Nevertheless, a rigorous derivation of the quasilinear MMT equation remains an open problem.

— The Wave Kinetic Theory for Quasilinear MMT Equation  (2608.22710 - Lü, 24 Aug 2026) in Introduction

However, an analogous rigorous kinetic derivation remains an active problem for continuum nonlinear wave equations.

— Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise  (2608.28459 - Liu et al., 28 Aug 2026) in Section 1, Introduction