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

We study a class of periodic NLS-Dysthe (NLS-Dy) equations, also referred to as higher-order nonlinear Schrödinger equations (HNLS), of the form $i{\mathrm{\partial }}_{t}u+a{\mathrm{\partial }}_x^{2}u+b\mathrm{\mid }u{\mathrm{\mid }}^{2}u+i\epsilon (c{\mathrm{\partial }}_x^{3}u+d\mathrm{\mid }u{\mathrm{\mid }}^2{\mathrm{\partial }}_{x}u+e{u}^2{\mathrm{\partial }}_x\bar{u})=0$ with $x\in \mathbb{T}$, $a,b,c,d,e\in \mathbb{R}$, $abc\mathrm{\ne }0$, 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 $s>3\mathrm{/}2$ for arbitrary admissible coefficients, and establish an $H^1$ continuation criterion for $s\ge 2$. In the parameter regime $e\mathrm{\ne }0$, a lower-order conservation law provides uniform $H^1$ control and, together with the continuation criterion, yields global well-posedness for $s\ge 2$. This result does not require the non-resonance condition appearing in Takaoka’s periodic theory, while the condition $e\mathrm{\ne }0$ naturally arises in the modulation setting considered here. Finally, motivated by the long-time regime relevant for modulation, we establish uniform long-time control without imposing a smallness assumption on the initial datum. Using the conservation laws of cubic NLS and their invariance under the commuting complex mKdV flow, we obtain uniform $H^s$ bounds on time intervals of order ${\epsilon }^{-1}$ for integer $s\ge 2$. For non-integer real $s\ge 2$, the maximal $H^s$ solution exists on the same time scale, while uniform control is obtained at a lower regularity, with a loss of less than one derivative.