Suppression of the Benjamin-Feir instability by constant vorticity
Abstract
We study the effect of constant vorticity on the Benjamin-Feir instability of small-amplitude Stokes waves in infinite depth. Working with the full water-wave equations, we rigorously establish a transition between modulational instability and stability as the vorticity crosses a critical value. We consider right-propagating waves with constant vorticity γ, associated with the background shear current −γyex, and normalize the gravitational acceleration and carrier wavenumber to one. For each fixed γ<2/√3, we prove that sufficiently small-amplitude Stokes waves are Benjamin-Feir unstable. For each fixed γ>2/√3, these waves are spectrally stable with respect to longitudinal, long-wavelength perturbations. These results rigorously confirm, away from the critical value, the stability transition predicted by the formal nonlinear Schrödinger approximation of Thomas, Kharif, and Manna.
Related
- A Uniqueness theorem for the Vortex-Wave system with non-constant vorticity near the point vortex
- Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity
- Modulational stability of finite-depth Stokes waves with constant vorticity
- Unstable Stokes waves in constant vorticity flows of finite depth
- Stabilization of gravity-capillary water waves with constant vorticity
BibTeX
@misc{chen2026suppression,
title = {{Suppression of the Benjamin-Feir instability by constant vorticity}},
author = {Bowen Chen and Huy Q. Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39000},
eprint = {2610.39000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39000},
}
RIS
TY - UNPB
TI - Suppression of the Benjamin-Feir instability by constant vorticity
AU - Bowen Chen
AU - Huy Q. Nguyen
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.39000
UR - https://arxiv.org/abs/2610.39000
AB - We study the effect of constant vorticity on the Benjamin-Feir instability of small-amplitude Stokes waves in infinite depth. Working with the full water-wave equations, we rigorously establish a transition between modulational instability and stability as the vorticity crosses a critical value. We consider right-propagating waves with constant vorticity γ, associated with the background shear current −γye_x, and normalize the gravitational acceleration and carrier wavenumber to one. For each fixed γ<2/√3, we prove that sufficiently small-amplitude Stokes waves are Benjamin-Feir unstable. For each fixed γ>2/√3, these waves are spectrally stable with respect to longitudinal, long-wavelength perturbations. These results rigorously confirm, away from the critical value, the stability transition predicted by the formal nonlinear Schrödinger approximation of Thomas, Kharif, and Manna.
ER -