Long time dynamics for complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds
Abstract
We study the energy-subcritical and energy-critical complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds. We establish the asymptotic behavior of solutions via a purely energy-based approach and subsequently prove a fast/slow dichotomy of the solutions. Moreover, for the slow solution, we show that the solution converges to an explicit asymptotic profile. In particular, we obtain a precise description of both the amplitude and the phase of the leading-order dynamics. As a by-product, the complex Ginzburg-Landau equation approximation yields global weak solutions to the energy-subcritical and energy-critical defocusing nonlinear Schrödinger equations on compact three-dimensional Riemannian manifolds. We also investigate the long time behavior of the solutions to the energy-subcritical and energy-critical complex Ginzburg-Landau equations on a class of complete noncompact three-dimensional Riemannian manifolds and prove decay of global solutions under suitable geometric assumptions.
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BibTeX
@misc{cheng2026long,
title = {{Long time dynamics for complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds}},
author = {Xing Cheng and Zeyu Lyu and Chenmin Sun and Yunrui Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11581},
eprint = {2610.11581},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11581},
}
RIS
TY - UNPB
TI - Long time dynamics for complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds
AU - Xing Cheng
AU - Zeyu Lyu
AU - Chenmin Sun
AU - Yunrui Zheng
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11581
UR - https://arxiv.org/abs/2610.11581
AB - We study the energy-subcritical and energy-critical complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds. We establish the asymptotic behavior of solutions via a purely energy-based approach and subsequently prove a fast/slow dichotomy of the solutions. Moreover, for the slow solution, we show that the solution converges to an explicit asymptotic profile. In particular, we obtain a precise description of both the amplitude and the phase of the leading-order dynamics. As a by-product, the complex Ginzburg-Landau equation approximation yields global weak solutions to the energy-subcritical and energy-critical defocusing nonlinear Schrödinger equations on compact three-dimensional Riemannian manifolds. We also investigate the long time behavior of the solutions to the energy-subcritical and energy-critical complex Ginzburg-Landau equations on a class of complete noncompact three-dimensional Riemannian manifolds and prove decay of global solutions under suitable geometric assumptions.
ER -