Nonlinear Schrödinger equation on a closed 3D elastica knot
An elastica knot is defined in terms of the Frenet-Serret curvature $κ(s,t)$ 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 $\partial^{2}_{s}κ(s,t) = -\;κ^{3}/2 + k_{0}^{4}τ_{0}^{2}\;κ^{-3} + λ\,k_{0}^{2}κ/2$ 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 $τ(s,t)$ satisfies the conservation law $κ^{2}(s,t)\,τ(s,t) \equiv k_{0}^{2}\,τ_{0}$, while $λ$ is a constant of integration. After briefly reviewing the Hasimoto transformation from a space curve ${\bf r}(s,t)$ to the nonlinear Schrödinger equation (NLSE) $-\,iD^{-1}\partial_{t}ψ= \partial^{2}_{s}ψ+ \frac{1}{2}\,|ψ|^{2}ψ$, where the constant $D$ has units of fluid circulation (m$^{2}$/sec), we show how the traveling-wave solution $ψ(s,t) = Ψ(s_{t} \equiv s - c\,t) \equiv κ(s_{t})\;\exp[iθ(s_{t})]$ is mapped onto the curvature equation for an elastica knot, with $θ^{\prime}(s_{t}) \equiv c/(2D) + k_{0}^{2}τ_{0}/κ^{2}(s_{t})$ and the elastica-knot constant $k_{0}^{2}λ= -\frac{1}{2}\,(c/D)^{2}$ 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.