Due to its geometric form and relationship with several physical phenomena, the Schrödinger map equation secures a unique place in the literature. For instance, in the three-dimensional Euclidean space, it is related to the vortex filament equation which describes the evolution of a vortex filament in an inviscid incompressible fluid, e.g., smoke rings, tornadoes, etc.
In this talk, we discuss the evolution of these equations when the initial data correspond to a filament curve with corners. Apart from describing the evolution with theoretical and numerical techniques, it will be shown that the path traced by a single point located on the polygonal curve follows a multifractal trajectory which can be compared with the graph of Riemann's non-differentiable function. We will also consider different initial data and geometric settings to claim that this multifractal behavior indeed appears as a generic phenomenon.
This work is in collaboration with Francisco de la Hoz (UPV/EHU) and Luis Vega (BCAM, UPV/EHU).
Reference:
[1] F. de la Hoz, S. Kumar and L. Vega. On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion . SIAM Journal on Applied Mathematics,, 80(2):1034{1056, 2020.
[2] F. de la Hoz, S. Kumar and L. Vega. Vortex Filament Equation for a regular l-polygon in the hyperbolic plane. Preprint: arXiv:2007.04944.
[3] S. Kumar. On the Schrödinger map for regular helical polygons in the hyperbolic space. Preprint: arXiv:2010.12045.