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 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. Here, the Frenet-Serret torsion τ(s,t) satisfies the conservation law κ2(s,t) τ(s,t) ≡ k02 τ0, while the constant of integration λ is associated with the constrained variation. 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 NLSE traveling-wave solution on a closed elastica knot requires an extension of the classical elastica-knot parameter space.
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BibTeX
@misc{brizard2026nonlinear,
title = {{Nonlinear Schrödinger equation on a closed 3D elastica knot}},
author = {Alain J. Brizard},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.21750},
eprint = {2607.21750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21750},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonlinear Schrödinger equation on a closed 3D elastica knot
AU - Alain J. Brizard
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2607.21750
UR - https://arxiv.org/abs/2607.21750
AB - 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 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 ∂^2_sκ(s,t) = − κ^3/2 + k_0^4τ_0^2 κ^−3 + λ k_0^2κ/2. Here, the Frenet-Serret torsion τ(s,t) satisfies the conservation law κ^2(s,t) τ(s,t) ≡ k_0^2 τ_0, while the constant of integration λ is associated with the constrained variation. After briefly reviewing the Hasimoto transformation from a space curve bf r(s,t) to the nonlinear Schrödinger equation (NLSE) − iD^−1∂_tψ= ∂^2_sψ+ 1/2 |ψ|^2ψ, 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 ≡ s − c t) ≡ κ(s_t) exp[iθ(s_t)] is mapped onto the curvature equation for an elastica knot, with θ^′(s_t) ≡ c/(2D) + k_0^2τ_0/κ^2(s_t) and the elastica-knot constant k_0^2λ= −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.
ER -