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

TL;DR

This paper establishes a sharp well-posedness theory for the MMT dispersive wave model, a nonlocal fractional derivative nonlinear Schrödinger equation, identifying critical thresholds for nonlinearity and dispersion to ensure solution existence and uniqueness.

Contribution

It provides the first sharp well-posedness results for the MMT model and related nonlocal fractional dNLS equations, resolving previous open regularity endpoint issues.

Findings

Sharp well-posedness thresholds identified

Results apply to a broad class of nonlocal fractional dNLS equations

Resolves open problem on regularity endpoint

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 .