A Self Propelled Vortex Dipole Model on Surfaces of Variable Negative Curvature

TL;DR

<3-5 sentence high-level summary> This paper develops a geometric framework for vortex dipoles on surfaces of variable negative curvature, using the catenoid as a canonical minimal surface. It derives a curvature-aware point-vortex Hamiltonian with mutual and self-interaction terms, identifies a conserved azimuthal momentum map J, and confirms that tightly bound dipoles follow catenoid geodesics classified by a single parameter Lambda. It further demonstrates direct and exchange scattering of dipoles, contrasts dipole behavior with co-rotating pairs, and constructs a finite-dipole dynamical system that yields curvature-modulated self-propulsion and orientation dynamics, validated numerically. Overall, the work provides a rigorous, symplectic description of vortex dynamics on curved surfaces and offers a concrete platform for exploring curvature-controlled transport in curved Bose-Einstein condensates and related systems.

Abstract

We investigate vortex dipoles on surfaces of variable negative curvature, focusing on a catenoid of arbitrary throat radius as a concrete example. We construct the effective dynamical system including mutual and geometric self-interaction terms and show that the resulting Hamiltonian dynamics makes dipoles follow catenoid geodesics, in agreement with recent works, Gustafsson (J. Nonlinear Sci. 32, 62, 2022) and by Drivas, Glukhovskiy and Khesin (Int. Math. Res. Not. 2024, 14, 10880-10894). We utilize the symplectic structure to find a conserved momentum map J related to the U(1) symmetry along the azimuthal direction. We verify the conservation of both the Hamiltonian and this momentum for arbitrary throat radius. We then demonstrate direct and exchange scattering of classical vortices on the catenoid, and we contrast this with the collective rotational motion (with azimuthal drift) that arises for chiral pairs. Finally, we build a finite-dipole dynamical system on the catenoid and show that the self-propulsion terms emerge to leading order in the dipole size. This provides a concrete realization, on a curved minimal surface, of the intuitive statement that a finite dipole propels orthogonal to the dipole axis, with a speed modulated by curvature.

Figures (8)

• Figure 1: Meridional geodesic evolution with initial conditions $(u_1,v_1,u_2,v_2)=(\epsilon,-2,-\epsilon,-2)$ with $\epsilon=0.05$ which yields $J=0$ (exact) and $\Lambda=0$ to leading order. Top row: time evolution and geodesic embedding. Bottom row: deviation of conserved quantities $\delta J$ and $\delta H$. In the $(u,v)$ trajectory plots, the green curve denotes the analytical geodesic solution, while the blue curve shows the corresponding vortex–dipole evolution. For clarity, the geodesic curve is intentionally terminated at an earlier value of the affine parameter so that both trajectories can be visualized distinctly.
• Figure 2: Critical (neck circle) geodesic evolution for with initial conditions $(u_1,v_1,u_2,v_2)=(0,\epsilon,0,-\epsilon)$ with $\epsilon=0.05$ which yields $\Lambda=1$ to leading order. Top row: time evolution and geodesic embedding. Bottom row: deviation of conserved quantities $\delta J$ and $\delta H$. In the $(u,v)$ trajectory plots, the green curve denotes the analytical geodesic solution, while the blue curve shows the corresponding vortex–dipole evolution. For clarity, the geodesic curve is intentionally terminated at an earlier value of the affine parameter so that both trajectories can be visualized distinctly.
• Figure 3: Trapped one-sided geodesic evolution for with initial conditions $(u_1,v_1,u_2,v_2)=(0,0.15+\epsilon,0,0.15-\epsilon)$ with $\epsilon=0.05$ which yields $\Lambda>1$ to leading order. Top row: time evolution and geodesic embedding. Bottom row: deviation of conserved quantities $\delta J$ and $\delta H$. In the $(u,v)$ trajectory plots, the green curve denotes the analytical geodesic solution, while the blue curve shows the corresponding vortex–dipole evolution. For clarity, the geodesic curve is intentionally terminated at an earlier value of the affine parameter so that both trajectories can be visualized distinctly.
• Figure 4: Direct scattering of two dipoles on a catenoid surface. The dipoles are initialized symmetrically about the diagonal with a small offset, with the first dipole located near the origin in the $(u,v)$ coordinate plane and the second approaching from the vicinity of $(1,1)$. A small parameter $\epsilon = 0.07, \delta = 0.03$ defines the initial configuration $\{(0,\epsilon),\,(\epsilon,0),\,(1-\delta,1),\,(1,1-\delta)\}$, with vortex strengths $\Gamma=\{-1,1,1,-1\}$. The top panels show the trajectories at the final integration time $t_f = 0.5$, both in the $(u,v)$ coordinate plane and mapped onto the catenoid surface. The lower panels display the temporal evolution of the conserved quantities $J$ and $H$, demonstrating their numerical conservation throughout the simulation.
• Figure 5: Exchange scattering of two dipoles on a catenoid surface. The dipoles are initialized symmetrically about the diagonal, with the first dipole located near the origin in the $(u,v)$ coordinate plane and the second approaching from the vicinity of $(1,1)$. A small parameter $\epsilon = \delta = 0.05$ defines the initial configuration $\{(0,\epsilon),\,(\epsilon,0),\,(1-\delta,1),\,(1,1-\delta)\}$, with vortex strengths $\Gamma=\{-1,1,1,-1\}$. The top panels show the trajectories at the final integration time $t_f = 0.5$, both in the $(u,v)$ coordinate plane and mapped onto the catenoid surface. The lower panels display the temporal evolution of the conserved quantities $J$ and $H$, demonstrating their numerical conservation throughout the simulation.