In this talk, we will explore the Schr\odinger map equation, a geometric PDE, by examining its evolution for polygonal curves in various geometric settings. This equation is a special case of the renowned Landau--Lifshitz equation for ferromagnetism, and in Euclidean space, it describes the evolution of a vortex filament in a real fluid, commonly known as the vortex filament equation. When solved numerically for polygonal initial data, the dynamics exhibit intriguing fluid-like behaviours such as axis switching and multifractality, phenomena often linked to turbulence. Moreover, the algebraic construction of these solutions not only supports the numerical evolution but also reveals a degree of randomness, frequently seen in natural processes. I will present recent findings, focusing particularly on helical vortices and curves in hyperbolic space, and demonstrate that this seemingly random behaviour arising from a differential equation appears to be a generic phenomenon.

[1]: S. Kumar, Pseudorandomness of the Schr\odinger map equation. arXiv:2311.01611.
[2]: S. Kumar, On the Schr\odinger map for regular helical polygons in the hyperbolic space. Nonlinearity 35(1) (2022), 84--109.
[3]: F. de la Hoz, S. Kumar and L. Vega, Vortex Filament Equation for a regular l-polygon in the hyperbolic plane. J. Nonlinear Sci. 32(9) (2022).