On the well-posedness of the initial value problem for the MMT model
Abstract
This work investigates the initial value problem (IVP) for the two-parameter family of dispersive wave equations known as the Majda-McLaughlin-Tabak (MMT) model, which arises in the weak turbulence theory of random waves. The MMT model can be viewed as a derivative nonlinear Schr\"odinger (dNLS) equation where both the nonlinearity and dispersion involve nonlocal fractional derivatives. The purpose of this study is twofold: first, to establish a sharp well-posedness theory for the MMT model; and second, to identify the critical threshold for the derivative in the nonlinearity relative to the dispersive order required to ensure well-posedness. As a by-product, we establish sharp well-posedness for non-local fractional dNLS equations; notably, our results resolve the regularity endpoint left open in https://www.aimsciences.org/article/doi/10.3934/dcdsb.2022039 .
Cite
@article{arxiv.2601.07771,
title = {On the well-posedness of the initial value problem for the MMT model},
author = {Mahendra Panthee and James Patterson and Yuzhao Wang},
journal= {arXiv preprint arXiv:2601.07771},
year = {2026}
}