Linear Stability of the Lamb-Chaplygin Dipole
Abstract
We describe the linearized dynamics near the Lamb-Chaplygin dipole, a classical traveling solution of the two-dimensional Euler equations. Exploiting the Hamiltonian structure of the system together with its symmetries, we identify all possible sources of linear instability. For general perturbations in L1∩ Lp, p>2, growth can occur through two explicit mechanisms triggered by: {\rm (i)} a nonzero circulation on the core of the dipole, and {\rm (ii)} a nontrivial component along the generalized eigenvectors associated with the eigenvalue $0$. In particular, we completely classify the spectrum and the Jordan chains of the operator associated with the linear dynamics. Both mechanisms hint for a nonlinear dynamics that may drift along the symmetry-generated family of traveling dipoles without moving away from it.
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BibTeX
@misc{numero2026linear,
title = {{Linear Stability of the Lamb-Chaplygin Dipole}},
author = {Francesco Pio Numero and Paolo Ventura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.09775},
eprint = {2606.09775},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09775},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Linear Stability of the Lamb-Chaplygin Dipole
AU - Francesco Pio Numero
AU - Paolo Ventura
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2606.09775
UR - https://arxiv.org/abs/2606.09775
AB - We describe the linearized dynamics near the Lamb-Chaplygin dipole, a classical traveling solution of the two-dimensional Euler equations. Exploiting the Hamiltonian structure of the system together with its symmetries, we identify all possible sources of linear instability. For general perturbations in L^1∩ L^p, p>2, growth can occur through two explicit mechanisms triggered by: {\rm (i)} a nonzero circulation on the core of the dipole, and {\rm (ii)} a nontrivial component along the generalized eigenvectors associated with the eigenvalue $0$. In particular, we completely classify the spectrum and the Jordan chains of the operator associated with the linear dynamics. Both mechanisms hint for a nonlinear dynamics that may drift along the symmetry-generated family of traveling dipoles without moving away from it.
ER -