Global Well-Posedness and Uniform Long-Time Control for a Class of Periodic NLS-Dysthe Equations

Abstract

We study a class of periodic NLS-Dysthe equations of the form $i{\partial }_{t}u+a{\partial }_x^{2}u+b|u{|}^{2}u+i\epsilon (c{\partial }_x^{3}u+d|u{|}^2{\partial }_{x}u+e{u}^2{\partial }_x\overline{u})=0$ with $x\in \mathbb{T}$, $a,b,c,d,e\in \mathbb{R}$, $abc\ne 0$, and $\epsilon >0$. For $0<\epsilon \ll 1$, equations of this type arise as higher-order modulation models for some nonlinear dispersive PDEs, including the Klein-Gordon equation and the Serre-Green-Naghdi equation modeling the evolution of nonlinear shallow water waves. We prove local well-posedness in $H^s(\mathbb{T})$ for $s>3/2$ for arbitrary admissible coefficients, and establish an $H^1$ continuation criterion for $s\ge 2$. Under the additional assumption $e\ne 0$, a lower-order conservation law provides uniform $H^1$ control and, together with the continuation criterion, yields global well-posedness for $s\ge 2$. Unlike the previous $H^1$ global theory based on Takaoka's periodic well-posedness estimates (2000), our result does not require any non-resonance condition on the coefficients. Our approach relies on the construction of modified higher-order energies, following the strategy developed for the generalized KdV equation by Planchon, Tzvetkov and Visciglia (2023) and for the generalized DNLS equation by Hayashi, Ozawa and Visciglia (2025). Motivated by higher-order modulation theory, we exploit the conservation laws of cubic NLS and their invariance under the commuting complex mKdV flow to establish Sobolev bounds uniform in the modulation parameter $\epsilon$ on time intervals of order ${\epsilon }^{-1}$. This corresponds to physical times of order ${\epsilon }^{-3}$, which is the natural time scale anticipated for the NLS-Dysthe approximation.