Hasimoto Transformation of General Flows Expressed in the Frenet Frame

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arXiv:2210.00381v1 [math.DG] 1 Oct 2022

Hasimoto Transformation of General Flows Expressed

in the Frenet Frame

Jacob S. Hofera, Scott A. Stronga

aDepartment of Applied Mathematics and Statistics, Colorado School of Mines, 1500

Illinois St., Golden, 80401-1887, CO, USA

Abstract

A one-dimensional space curve in R3is a useful nonlinear medium for model-

ing vortex ﬁlaments and biological soft-matter capable of supporting a variety

of wave motions. Hasimoto’s transformation deﬁnes a mapping between the

kinematic evolution of a space curve and nonlinear scalar equations evolving

its intrinsic curve geometry. This mapping is quite robust and able to trans-

form general vector ﬁelds expressed in the Frenet frame, resulting in a fully

nonlinear integro-diﬀerential evolution equation, whose coeﬃcient structure

is deﬁned by the coordinates of the ﬂow in the Frenet frame. In this paper,

we generalize the Hasimoto map to arbitrary ﬂows deﬁned on space curves,

which we test against several existing kinematic ﬂows. After this, we consider

the time dynamics of length and bending energy to see that binormal ﬂows

are generally length preserving, and bending energy is fragile and unlikely to

be conserved in the general case.

Keywords: Hasimoto transformation, vortex ﬁlament, Frenet-Serret, local

induction approximation, nonlinear Schr¨odinger equation

1. Introduction

The autonomous motions of slender ﬁlaments are essential to several ar-

eas of applied science, e.g., atmospheric, aerodynamic, oceanographic phe-

nomena, astrophysical plasmas, and superﬂuid and superconducting states

of matter [1]. Through the Frenet-Serret equations, the ﬁlament’s one-

Email addresses: jhofer@mines.edu (Jacob S. Hofer), sstrong@mines.edu (Scott

A. Strong)

Preprint submitted to Nonlinear Waves: Computation and Theory-XI October 4, 2022

dimensional centerline conﬁguration is prescribed by two scalar functions,

i.e., curvature and torsion. The curvature and torsion dynamics determine

the evolution of a ﬁlament, which can be found by evolving the Frenet-Serret

coordinate system itself. However, an auxiliary condition is required to close

the system of equations. Often the kinematics of the ﬁlament are known and

act as the constitutive relation connecting our geometric response to a phys-

ical context. Consequently, this framework seeks to reduce the problem of

understanding ﬁlament behavior through its curvature and torsion dynamics.

To aid in this, we have Hasimoto’s remarkable result [2], which complexiﬁes

the local curvilinear coordinates so that the kinematics can be married to

the Frenet-Serret diﬀerential equations resulting in a complex-valued partial

diﬀerential equation controlling the evolution of the centerline geometry.

Hasimoto’s transformation, initially established in the context of ﬂuid

mechanics where vortex lines served as an object for modeling and simula-

tion, maps a global prescription for the motion of the ﬁlament centerline to a

complex-valued scalar evolution of its curvature and torsion and continues to

see widespread use with 38% of its citations occurring in the last ten years.

This is partly due to the applicability of vortex ﬁlaments in the setting of

quantum liquids and Bose-Einstein condensation, the latter being a hotbed

of activity since their experimental inception 25 years ago [3, 4]. Here the

ﬁlaments act as the skeleton of quantum turbulence [5] but they need not be

exotic structures. Currently, it is hypothesized that they provide a natural

object for singularity formation in Navier-Stokes [6] and mediate the anoma-

lous dissipation conjectured by Onsager [7], with their helical wave motion

stitching together our picture of turbulent cascades [8].

While these techniques provide a dimensional reduction to ﬂuid problems,

i.e., ﬂuid ﬂow reduced to the evolution of a complex scalar representation of a

space curve, they are fundamentally geometric in character and when coupled

to the elastic mechanics pioneered by Euler and Bernoulli, similar analyses

yield possible geometric conﬁgurations of stiﬀ polymers structuring DNA

supercoiling, self-assembly of bacterial ﬁbers, actin ﬁlaments, and cell ﬂag-

ella [9]. In either ﬂuid or elastic contexts, the primary purpose of Hasimoto’s

transformation is to recast the behavior speciﬁed by a setting’s kinematics

to the nonlinear evolution of intrinsically geometric quantities where wave

motion is less opaque.

In the following, we generalize the Hasimoto transform to centerline evo-

lutions expressed in the Frenet-Serret basis with arclength and time depen-

dent coeﬃcients. We show that the transformations of known special cases,

e.g., local induction approximation [10, 11], generalized local induction equa-

tion [12, 13], and stiﬀ chain polymers [9], are special cases of our general scalar

evolution law. Additionally, we show speciﬁc conditions for which arclength

is conserved and that conservation of bending energy is fragile. The work is

organized into three sections. First, we outline the procedure necessary to

arrive at the main result, which is a nonlinear integro-diﬀerential equation

in evolutionary form that speciﬁes the curve implicitly through its curvature

and torsion dynamics. Next, we apply this result to known kinematic ﬂows

of space curves. After this, we consider the relationship between the kine-

matic ﬂows expressed in the Frenet-Serret basis and the global quantities of

arclength and bending energy, before we conclude.

2. Generalization of Hasimoto Transformation to Frenet Framed

Kinematics

We consider the Hasimoto transformation of a one-dimensional space

curve γ:R1+1 →R3, parameterized by arclength, s, and permitted to evolve

in time, t, where a background velocity ﬁeld deﬁnes the kinematic equation

γt=νT+µN+λB(1)

expressed in the local curvilinear basis deﬁned by the tangent, T, normal

N, and binormal, B, vectors whose coordinates λ,µ, and ν, are arbitrary

functions of arclength and time. This coordinate system implicitly deﬁnes

a space curve via its curvature, k, and torsion, τ, describing local rotations

about the binormal and tangent vectors, respectively. Hasimoto’s transfor-

mation begins by deﬁning a change of coordinates M= (N+iB)eiθ where

θ=Rsτds′, which is used in conjunction with a complex-valued wave func-

tion whose modulus deﬁnes curvature and derivative of phase deﬁnes torsion,

i.e., ψ=keiθ .

Through the Frenet-Serret equations, Ts=kN,Ns=−kT+τB, and

Bs=−τN, it is possible to derive the following expressions for the derivatives

of Mand Twith respect to arclength,

Ms=−ψT(2)

Ts=1

2(ψM+ψM),(3)

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摘要：

arXiv:2210.00381v1[math.DG]1Oct2022HasimotoTransformationofGeneralFlowsExpressedintheFrenetFrameJacobS.Hofera,ScottA.StrongaaDepartmentofAppliedMathematicsandStatistics,ColoradoSchoolofMines,1500IllinoisSt.,Golden,80401-1887,CO,USAAbstractAone-dimensionalspacecurveinR3isausefulnonlinearmediumformode...

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