2026/07/23 by Alain J. Brizard
#math-ph #math.MP #physics.plasm-ph
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 ∂2sκ(s,t) = - κ3/2 + k04τ02 κ-3 + λ k02κ/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 κ2(s,t) τ(s,t) ≡ k02 τ0, 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∂tψ= ∂2sψ+ (1)/(2) |ψ|2ψ, where the constant D has units of fluid circulation (m2/sec), we show how the traveling-wave solution ψ(s,t) = Ψ(st ≡ s - c t) ≡ κ(st) exp[iθ(st)] is mapped onto the curvature equation for an elastica knot, with θ′(st) ≡ c/(2D) + k02τ0/κ2(st) and the elastica-knot constant k02λ= -(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.