Takashi Sakajo: Exact solution to a Liouville equation on a curved torus and quantized point vortex equilibria.

Vortragender: Takashi Sakajo (Department of Mathematics, Kyoto University)
Ansprechpartner: Prof. Dr. Axel Voigt
Link zur Videokonferenz: BigBlueButton

Title:
Exact solution to a Liouville equation on a curved torus and quantized point vortex equilibria.

Abstract:
In this talk, an analytic formula of solutions to a modified Liouville equation on a curved torus with major radius R and minor radius r is presented. These solutions are interpreted as a steady solution of the incompressible Euler equation on the toroidal surface with a smooth background vorticity. This is a generalization of the flows with smooth vorticity distributions owing to Stuart in the plane and Crowdy on the spherical surface. Hence, it is interesting to investigate how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α=R/r characterizing geometric features of torus, i.e., non-constant curvature and a handle structure. In addition, the analytic formula yields point vortex equilibria with strengths quantized by 2π in a background smooth Liouville-type vorticity field, which is a model of quantized vortex crystals appearing superfluids. We show that appropriate choices of the loxodromic function in the solution leads to stationary vortex patterns with point vortices of identical strengths. We also find solutions continuously dependent on a parameter, such that the point vortices remain fixed whereas the smooth background vorticity changes with the parameter. This talk is based on the paper “T. Sakajo, Proc. Roy. Soc. A vol. 475 (2019) (doi: 10.1098/rspa.2018.0666) (Open Access)”, and a joint work with Dr. V. S. Krishnamurthy at University of Vienna.