Nonlinear Schrödinger equation on a closed 3D elastica knot

Abstract

The present work introduces a new relation between closed elastica knots with the solution of the nonlinear Schrödinger equation (NLSE) through the Hasimoto transformation. 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 $\mathbf{r}(s,t)$ at a fixed time $t$, which is a solution of the curvature differential equation ${\partial }_s^2\kappa (s,t)=-{\kappa }^3/2+k_0^4{\tau }_0^2{\kappa }^{-3}+\lambda k_0^2\kappa /2$. 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 the constant of integration $\lambda$ is associated with the constrained variation. After briefly reviewing the Hasimoto transformation from a space curve $\mathbf{r}(s,t)$ to the nonlinear Schrödinger equation (NLSE) $-i{D}^{-1}{\partial }_t\psi ={\partial }_s^2\psi +\frac{1}{2}|\psi {|}^2\psi$, where the constant $D$ has units of fluid circulation (m${}^2$/sec), we show how the traveling-wave solution $\psi (s,t)=\Psi (s_t\equiv s-ct)\equiv \kappa (s_t)\exp [i\theta (s_t)]$ 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 NLSE traveling-wave solution on a closed elastica knot requires an extension of the classical elastica-knot parameter space.

31 pages, 15 figures