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 (NLS-Dy) equations, also referred to as higher-order nonlinear Schrödinger equations (HNLS), of the form i∂t u+a∂x2u+b|u|2u+iε(c∂x3u+d|u|2∂xu+eu2∂xu̅)=0 with $ x\in\mathbb T$, a,b,c,d,e∈ℝ, abc≠0, and ε>0. For 0<ε≪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 Hs(𝕋) for s>3/2 for arbitrary admissible coefficients, and establish an H1 continuation criterion for s≥2. In the parameter regime e≠0, a lower-order conservation law provides uniform H1 control and, together with the continuation criterion, yields global well-posedness for s≥2. This result does not require the non-resonance condition appearing in Takaoka's periodic theory, while the condition e≠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 Hs bounds on time intervals of order ε−1 for integer s≥2. For non-integer real s≥2, the maximal Hs solution exists on the same time scale, while uniform control is obtained at a lower regularity, with a loss of less than one derivative.
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BibTeX
@misc{belloni2026global,
title = {{Global Well-Posedness and Uniform Long-Time Control for a Class of Periodic NLS-Dysthe Equations}},
author = {Andrea Belloni and Filippo Giuliani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14786},
eprint = {2610.14786},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14786},
}
RIS
TY - UNPB
TI - Global Well-Posedness and Uniform Long-Time Control for a Class of Periodic NLS-Dysthe Equations
AU - Andrea Belloni
AU - Filippo Giuliani
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14786
UR - https://arxiv.org/abs/2610.14786
AB - 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∂_t u+a∂_x^2u+b|u|^2u+iε(c∂_x^3u+d|u|^2∂_xu+eu^2∂_xu̅)=0 with $ x\in\mathbb T$, a,b,c,d,e∈ℝ, abc≠0, and ε>0. For 0<ε≪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(𝕋) for s>3/2 for arbitrary admissible coefficients, and establish an H^1 continuation criterion for s≥2. In the parameter regime e≠0, a lower-order conservation law provides uniform H^1 control and, together with the continuation criterion, yields global well-posedness for s≥2. This result does not require the non-resonance condition appearing in Takaoka's periodic theory, while the condition e≠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 ε^−1 for integer s≥2. For non-integer real s≥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.
ER -