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 $L^1\cap L^p$, $p>2$, growth can occur only 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.