  In this paper, we prove the nonlinear orbital stability of vortex dipoles for
the quasi-geostrophic shallow-water (QGSW) equations. The vortex dipoles are
explicit travelling wave solutions to the QGSW equations, which are analogues
of the classical circular vortex of Lamb and Chaplygin for the steady planar
Euler equations. We establish a variational characterization of these vortex
poles, which provides a basis for the stability result.

Analysis of PDEs