Nonlinear Schr\"{o}dinger Equation on a closed 3D Elastica Knot

An elastica knot is defined in terms of the Frenet-Serret curvature $\kappa(s,t)$ as a function of the arclength $s$ along the spatial curve ${\bf r}(s,t)$ at a fixed time $t$, which is a solution of the curvature differential equation $\partial^{2}_{s}\kappa(s,t) = -\;\kappa^{3}/2 + k_{0}^{4}\tau_{0}^{2}\;\kappa^{-3} + \lambda\,k_{0}^{2}\kappa/2$ that is obtained from a variational principle that minimizes the bending energy of the spatial curve under the constraint of a constant curve length. Here, the Frenet-Serret torsion $\tau(s,t)$ satisfies the conservation law $\kappa^{2}(s,t)\,\tau(s,t) \equiv k_{0}^{2}\,\tau_{0}$, while $\lambda$ is a constant of integration. After briefly reviewing the Hasimoto transformation from a space curve ${\bf r}(s,t)$ to the nonlinear Schr\"{o}dinger equation (NLSE) $-\,iD^{-1}\partial_{t}\psi = \partial^{2}_{s}\psi + \frac{1}{2}\,|\psi|^{2}\psi$, where the constant $D$ has units of fluid circulation (m$^{2}$/sec), we show how the traveling-wave solution $\psi(s,t) = \Psi(s_{t} \equiv s - c\,t) \equiv \kappa(s_{t})\;\exp[i\theta(s_{t})]$ is mapped onto the curvature equation for an elastica knot, with $\theta^{\prime}(s_{t}) \equiv c/(2D) + k_{0}^{2}\tau_{0}/\kappa^{2}(s_{t})$ and the elastica-knot constant $k_{0}^{2}\lambda = -\frac{1}{2}\,(c/D)^{2}$ expressed in terms of the traveling-wave NLSE parameters $(c,D)$. The constraint of a closed 3D elastica knot imposes spatial periodicity conditions that introduce a unique set of knot parameters for which the NLSE traveling wave can exist. The present work shows that the traveling-wave solution on a closed elastica knot requires an extension of the classical elastica-knot parameter space.