A self propelled vortex dipole model on a surface of variable negative curvature

Abstract We investigate vortex dipoles on surfaces of variable negative curvature, focusing on a catenoid of arbitrary throat radius a as a concrete example. We construct the effective dynamical system by incorporating both inter-vortex interactions and the curvature-induced self-interaction arising from the geometry of the catenoid. The resulting Hamiltonian dynamics reveals that dipoles move along catenoid geodesics, consistent with recent works by Gustafsson (2022 J. Nonlinear Sci. 32 62) and Drivas et al (2024 Int. Math. Res. Not. 2024 10880–94)). We utilize the symplectic structure to construct a conserved momentum map J associated with the U (1) symmetry along the azimuthal direction. We explicitly demonstrate the conservation of both the Hamiltonian and J for an arbitrary throat radius a . We then apply the formalism to analyze direct and exchange scattering of classical vortices on the catenoid surface. For comparison, we show that chiral pairs with similar initial configurations exhibit collective rotational motion (with azimuthal drift) instead of scattering. Finally, we construct a finite-dipole dynamical system on the catenoid based on dipole–dipole interactions and demonstrate the emergence of self-propulsion terms to leading order in dipole size. This provides a concrete realization, on a curved minimal surface, of the intuitive statement that a finite dipole propels orthogonally to its axis, with a speed modulated by curvature.