The graphic on the left is something fun I made in Mathematica. It's a surface in \(\mathbb{R}^3\) which is swept out in time by a solution to the vortex filament equation (VFE), given by $$\frac{\partial \gamma}{\partial t}=\gamma'\times\gamma'',$$ where \(\gamma(s,t)\) is a time-dependent curve in \(\mathbb{R}^3\) and the prime notation denotes differentiation with respect to the curve parameter \(s\). The VFE is a model for the centerline of a vortex in an incompressible ideal fluid.

The specific solution used in the image to the left arises from taking the 1-soliton solution to the 1-dimensional focusing nonlinear Schrödinger equation $$i\frac{\partial \psi}{\partial t}+\frac{\partial^2\psi}{\partial s^2}+\frac{1}{2}\psi|\psi|^2=0$$ and using the Hasimoto transformation $$ \psi(s,t)=\kappa(s,t)e^{i\int_{s_0}^s\tau(\sigma,t)\,d\sigma}$$ to find the curvature \(\kappa(s,t)\) and torsion \(\tau(s,t)\) of a solution to the VFE. We can then use those invariants to write down examples of solutions, and each cyan-colored curve on the surface to the left is precisely such a solution of the VFE with a fixed \(t\)-value.