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.