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arXiv · 2601.07771

On the well-posedness of the initial value problem for the MMT model

Mahendra Panthee·James Patterson·Yuzhao Wang
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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 Schrödinger (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 .

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Mahendra Panthee, James Patterson, Yuzhao Wang. 2026-02-09. On the well-posedness of the initial value problem for the MMT model. https://arxiv.org/abs/2601.07771

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