A Variational Family of Traveling Vortex Dipoles: Transition, Uniqueness, and Stability
Abstract
We provide a unified description of a stable variational family of traveling vortex dipoles for the two-dimensional incompressible Euler equations, connecting the classical Lamb dipole to asymptotically radial dipoles in the large-impulse regime. We show that this family undergoes a sharp transition from the Lamb dipole, where the mass constraint is inactive, to non-explicit traveling waves for which the mass constraint becomes active, and determine the critical value exactly. We further prove that the variational solutions are unique up to translation both near the Lamb dipole and in the large-impulse regime. As a consequence, the corresponding individual traveling waves are orbitally stable. Our results provide a unified picture of these two distinct asymptotic regimes within a single variational family of traveling vortex dipoles.
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BibTeX
@misc{abe2026variational,
title = {{A Variational Family of Traveling Vortex Dipoles: Transition, Uniqueness, and Stability}},
author = {Ken Abe and In-Jee Jeong and Guolin Qin and Weicheng Zhan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17962},
eprint = {2609.17962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17962},
}
RIS
TY - UNPB
TI - A Variational Family of Traveling Vortex Dipoles: Transition, Uniqueness, and Stability
AU - Ken Abe
AU - In-Jee Jeong
AU - Guolin Qin
AU - Weicheng Zhan
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.17962
UR - https://arxiv.org/abs/2609.17962
AB - We provide a unified description of a stable variational family of traveling vortex dipoles for the two-dimensional incompressible Euler equations, connecting the classical Lamb dipole to asymptotically radial dipoles in the large-impulse regime. We show that this family undergoes a sharp transition from the Lamb dipole, where the mass constraint is inactive, to non-explicit traveling waves for which the mass constraint becomes active, and determine the critical value exactly. We further prove that the variational solutions are unique up to translation both near the Lamb dipole and in the large-impulse regime. As a consequence, the corresponding individual traveling waves are orbitally stable. Our results provide a unified picture of these two distinct asymptotic regimes within a single variational family of traveling vortex dipoles.
ER -