# The Hasimoto Transformation for a Finite Length Vortex Filament and its
Application

Masashi Aiki∗

###### Abstract

We consider two nonlinear equations, the Localized Induction Equation and the
cubic nonlinear Schrödinger Equation, and prove that the solvability of
certain initial-boundary value problems for each equation is equivalent
through the generalized Hasimoto transformation. As an application, we prove
the orbital stability of plane wave solutions of the nonlinear Schrödinger
equation based on stability estimates obtained for the Localized Induction
Equation by the author in a paper in preparation. As far as the author knows,
this is the first time that the analysis of the Localized Induction Equation,
along with the Hasimoto transformation, provided new insight for the nonlinear
Schrödinger equation.

∗Department of Mathematics
Faculty of Science and Technology, Tokyo University of Science
2641 Yamazaki, Noda, Chiba 278-8510, Japan
E-mail<EMAIL_ADDRESS>
ORCID: 0000-0002-3476-6894

##  1 Introduction and Problem Setting

A vortex filament is a space curve on which the vorticity of the fluid is
concentrated. Vortex filaments are used to model very thin vortex structures
such as vortices that trail off airplane wings or propellers. The model
equation we consider in this paper is the Localized Induction Equation (LIE)
given by

$\displaystyle\mbox{$x$}_{t}=\mbox{$x$}_{s}\times\mbox{$x$}_{ss}$

where $\mbox{$x$}(s,t)={}^{t}(x_{1}(s,t),x_{2}(s,t),x_{3}(s,t))$ is the
position vector of the vortex filament parametrized by its arc length $s$ at
time $t$, $\times$ is the exterior product in the three-dimensional Euclidean
space, and subscripts $s$ and $t$ are differentiations with the respective
variables.

The LIE, which is derived by applying the localized induction approximation to
the Biot–Savart integral, was first derived by Da Rios [13] in 1906 and was
re-derived twice independently by Murakami et al. [30] in 1937 and by Arms and
Hama [4] in 1965. Since then, many researches have been done on the LIE and
many results have been obtained. Nishiyama and Tani [32, 33] proved the unique
solvability of initial and initial-boundary value problems in Sobolev spaces.
The author [1] and the author and Iguchi [3] proved the unique solvability of
initial-boundary value problems in Sobolev spaces with different boundary
conditions. In the above papers, the equation for the tangent vector,
$\mbox{$v$}:=\mbox{$x$}_{s}$, given by

(1.1) $\displaystyle\mbox{$v$}_{t}=\mbox{$v$}\times\mbox{$v$}_{ss}$

is introduced in the analysis and plays an important role. Equation (1.1) is
sometimes called the Vortex Filament Equation (VFE).

Koiso [29] considered a geometrically generalized setting in which he
rigorously proved the equivalence of the solvability of the initial value
problem for the VFE and the cubic nonlinear Schrödinger equation. This
equivalence was first shown by Hasimoto [23] in which he studied the formation
of solitons on a vortex filament. He defined a transformation of variable
known as the Hasimoto transformation to transform the VFE into a nonlinear
Schrödinger equation. The Hasimoto transformation is a change of variable
given by

$q(s,t)=\kappa(s,t)\exp\left({\rm i}\int^{s}_{0}\tau(r,t)\,{\rm d}r\right),$

where ${\rm i}$ is the imaginary unit, $\kappa$ is the curvature, and $\tau$
is the torsion of the filament. Defined as such, it is well known that $q$
satisfies the nonlinear Schrödinger equation given by

(1.2) $\displaystyle{\rm i}q_{t}=q_{ss}+\frac{1}{2}\lvert q\rvert^{2}q.$

The original transformation proposed by Hasimoto uses the torsion of the
filament in its definition, which means that the transformation is undefined
at points where the curvature of the filament is zero. Koiso [29] constructed
a transformation, sometimes referred to as the generalized Hasimoto
transformation, and gave a mathematically rigorous proof of the equivalence of
the VFE and (1.2). More precisely, Koiso proved that the solvability of
initial value problems for the VFE and (1.2) are equivalent. More recently,
Banica and Vega [5, 6, 7] and Gutiérrez, Rivas, and Vega [22] utilized the
generalized Hasimoto transformation to construct and analyze a family of self-
similar solutions of the LIE which forms a corner in finite time. In Chang,
Shatah, and Uhlenbeck [12] and Nahmod, Shatah, Vega, and Zeng [28], they
considered the Schrödinger maps and employed a Hasimoto type transformation to
prove the correspondence between the solution of the Schrödinger maps and the
solution of a nonlinear Schrödinger type equation. The Schrödinger maps are a
generalization of the Heisenberg model for a ferromagnetic spin system given
by

$\displaystyle\mbox{$m$}_{t}=\mbox{$m$}\times\Delta\mbox{$m$},$

where $\Delta$ is the Laplacian in $\mathbf{R}^{n}$ and the unknown variable
$m$ takes values in $\mathbf{S}^{2}$. The Heisenberg model can be seen as a
multi-dimensional version of the VFE.

All of the results mentioned above which utilizes the Hasimoto transformation
consider either an infinitely long filament defined on the whole line or a
closed filament defined on the torus. As far as the author knows, the rigorous
justification and application of the Hasimoto transformation for problems
describing the motion of filaments with end-points have not been done. In this
paper, we justify the Hasimoto transformation and prove the equivalence of the
solvability of the following initial-boundary value problems.

(1.6)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$v$}_{t}=\mbox{$v$}\times\mbox{$v$}_{ss},&s\in
I_{L},\ t>0,\\\\[8.53581pt] \mbox{$v$}(s,0)=\mbox{$v$}_{0},&s\in I_{L},\
t>0,\\\\[8.53581pt] \mbox{$v$}(0,t)=\mbox{$e$}_{1},\
\mbox{$v$}(L,t)=\mbox{$b$},&t>0,\end{array}\right.$ (1.10)
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle{\rm
i}q_{t}=q_{ss}+\frac{1}{2}\lvert q\rvert^{2}q,&s\in I_{L},t>0,\\\\[8.53581pt]
q(s,0)=q_{0}(s),&s\in I_{L},\\\\[8.53581pt]
q_{s}(0,t)=q_{s}(L,t)=0,&t>0.\end{array}\right.$

Problem (1.6) is an initial-boundary value problem for the VFE which describes
the motion of a vortex filament on a slanted plane, considered in a previous
paper by the author [1]. Here, $L>0$ is the length of the initial filament,
$I_{L}\subset\mathbf{R}$ is the interval $(0,L)$,
$\mbox{$e$}_{1}={}^{t}(1,0,0)$, and $\mbox{$b$}\in\mathbf{R}^{3}$ is an
arbitrary constant vector with unit length. The boundary datum at $s=0$ was
chosen as $\mbox{$e$}_{1}$ without loss of generality, because the VFE is
invariant under rotation. Problem (1.10) is an initial-boundary value problem
for the focusing cubic nonlinear Schrödinger equation, where $q=q(s,t)$ is a
complex-valued function.

As an application of this equivalence, we will prove the orbital stability of
the plane wave solution $q_{R}$ of (1.10) given in the form

$\displaystyle q_{R}(t)=-\frac{1}{R}\exp\left\\{-\frac{{\rm
i}t}{2R^{2}}\right\\}$

for $R>L/\pi$ in the Sobolev space $H^{2}(I_{L})$. This will be done by
considering problem (1.6) with appropriate initial and boundary data,
utilizing energy estimates for the solution $v$ of (1.6) obtained by the
author in [1, 2], and transferring the estimates for $v$ into estimates for
solutions of (1.10).

As far as the author knows, the results of this paper is the first time the
generalized Hasimoto transformation is utilized to give new insight on the
nonlinear Schrödinger equation from known facts about the VFE. All of the
preceding works utilizing the generalized Hasimoto transformation did so to
analyze the solution of the VFE utilizing known facts about the nonlinear
Schrödinger equation.

The initial value problem for equation (1.2) on the torus, explicitly given by

(1.13) $\displaystyle\left\\{\begin{array}[]{ll}\displaystyle{\rm
i}u_{t}=u_{ss}+\frac{1}{2}\lvert
u\rvert^{2}u,&s\in\mathbf{T},t>0,\\\\[8.53581pt]
u(s,0)=u_{0}(s),&s\in\mathbf{T},\end{array}\right.$

where $\mathbf{T}=\mathbf{R}/[-L,L]$, is closely related to problem (1.10)
since the solvability of problem (1.10) can be reduced to the solvability of
problem (1.13) by reflection and periodic extension.

It is known by Zakharov and Shabat [38] that equation (1.2) is completely
integrable, and the solution to problem (1.13) posses infinitely many
conserved quantities. This in turn implies that solutions in the Sobolev space
$H^{m}$ for $m\in\mathbf{N}$ is bounded in $H^{m}$ for all time. The
solvability of problem (1.13) in Lebesgue or Sobolev spaces are known, for
example, by Bourgain [9]. Hence, it is natural to ask if particular types of
solutions are stable in Sobolev spaces. Namely, the stability of plane wave
solutions of equation (1.2) has been studied by many researchers.

Zhidkov [39] gives a detailed analysis of the plane wave solutions for the
initial value problem on the whole space $\mathbf{R}$. The stability of plane
wave solutions and periodic wave solutions for problem (1.13) was investigated
by Rowlands [35], Gallay and Hărăguş [18, 19], Faou, Gauckler, and Lubich
[14], and Wilson [37].

In particular, Gallay and Hărăguş [18, 19] considered problem (1.13) and
proved the orbital stability of periodic wave solutions in $H^{1}_{per}$,
i.e., the orbital stability in $H^{1}$ with the perturbation restricted to
periodic perturbations with the same period as the periodic wave solution. By
definition, periodic wave solutions include plane wave solutions and hence,
the stability results in [18, 19] are valid for plane wave solutions as well.

Faou, Gauckler, and Lubich [14] considered the initial value problem on the
torus with general dimension $d\geq 1$, and proved the long-time orbital
stability of plane wave solutions in $H^{r}$. By long-time they mean stability
up to time of order $O(\frac{1}{\varepsilon^{N}})$ where $\varepsilon>0$ is
the size of the perturbation in $H^{r}$ and $N\in\mathbf{N}$. The index $r>0$
must be chosen sufficiently large, depending on $N$ and the $L^{2}$ norm of
the perturbation.

Wilson [37] considered the stability problem of plane wave solutions for the
nonlinear Schrödinger equation with general power nonlinearities, and obtained
similar results as [14].

Other results on the initial–boundary value problems for the nonlinear
Schrödinger equation have been obtained by Holmer [24], Fokas and Its [15],
Fokas, Its, and Sung [17], Lenells and Fokas [25, 26], Bona, Sun, and Zhang
[8], and Fokas, Himonas, and Mantzavinos [16]. These results prove the well-
posedness of initial-boundary value problems under various boundary conditions
as well as obtain representation formulas for boundary values which represent
unknown boundary values of a solution by known boundary values, but do not
address the problem of stability of specific solutions.

In summary, we see that up until this paper, the stability of plane wave
solutions in higher order Sobolev spaces is only partially known.
Specifically, the regularity for which the stability is proved in [14, 37] is
not given explicitly, and the time-span for which the stability holds is not
global. On the other hand, the results in this paper give the time-global
orbital stability of plane wave solutions in Sobolev space $H^{2}$. This is
possible greatly due to the fact that the method of the proof given in this
paper is vastly different from the preceding works.

The methods utilized to prove the stability of solutions in [39, 35, 18, 19,
14, 37] can be broadly categorized into two types. Variational methods and
methods utilizing the Hamiltonian structure of the equation. The variational
approach was utilized in Cazenave and Lions [11] to prove orbital stability of
standing waves for the nonlinear Schrödinger equation. The approach utilizing
the Hamiltonian structure of the equation was introduced by Grillakis, Shatah,
and Strauss [20, 21] for a broad range of equations having a Hamiltonian
structure. These methods are widely adopted to approach stability problems for
a wide variety of dispersive equations. In contrast, the method employed in
this paper is tailor-made specifically for problem (1.6) and (1.10). Hence, we
are able to obtain more information for our specific problem, but the method
is not as widely applicable to other problems compared to traditional methods.

The contents of the rest of the paper are as follows. In Section 2, we
introduce basic notations and define compatibility conditions related to
problems (1.6) and (1.10). Then, we give a brief explanation of the Hasimoto
transformation and state our main theorems. In Section 3, we prove our main
theorems. We first prove that the solvability of problems (1.6) and (1.10) is
equivalent through the generalized Hasimoto transformation. We further prove
that plane wave solutions of problem (1.10) correspond to a particular type of
solution, which we call arc-shaped solutions, of problem (1.6).

Then, we prove stability estimates for arc-shaped solutions of problem (1.6)
and also prove that these stability estimates can be transferred to stability
estimates for plane wave solutions of problem (1.10) through the generalized
Hasimoto transformation. The stability estimates for arc-shaped solutions are
essentially derived from standard energy estimates for the perturbation, which
is much more simple than traditional methods.

##  2 Function Spaces, Notations, and Main Theorem

We introduce some function spaces that will be used throughout this paper, and
notations associated with the spaces. For a non-negative integer $m$ and
$1\leq p\leq\infty$, $W^{m,p}(I_{L})$ is the Sobolev space containing all
real-valued functions that have derivatives in the sense of distribution up to
order $m$ belonging to $L^{p}(I_{L})$. We set $H^{m}(I_{L}):=W^{m,2}(I_{L})$
as the Sobolev space equipped with the usual inner product, and set
$H^{1}_{0}(I_{L})$ as the closure, with respect to the $H^{1}$-norm, of the
set of smooth functions with compact support. The norm in $H^{m}(I_{L})$ is
denoted by $\|\cdot\|_{m}$ and we simply write $\|\cdot\|$ for
$\|\cdot\|_{0}$. Otherwise, for a Banach space $X$, the norm in $X$ is written
as $\|\cdot\|_{X}$. The inner product in $L^{2}(I_{L})$ is denoted by
$(\cdot,\cdot)$.

For $0<T\leq\infty$ and a Banach space $X$, $C^{m}([0,T];X)$ (
$C^{m}\big{(}[0,\infty);X\big{)}$ when $T=\infty$), denotes the space of
functions that are $m$ times continuously differentiable in $t$ with respect
to the norm of $X$. The space $L^{\infty}\big{(}0,\infty;X\big{)}$ denotes the
space of functions that are essentially bounded in $t$ with respect to the
norm of $X$

For any function space described above, we say that a vector valued function
belongs to the function space if each of its components does, and the same for
complex-valued functions if both the real and imaginary parts do.

Finally, vectors $\mbox{$e$}_{j}\in\mathbf{R}^{3}$ for $j=1,2,3$ denote the
standard basis of $\mathbf{R}^{3}$. In other words,
$\mbox{$e$}_{1}={}^{t}(1,0,0)$, $\mbox{$e$}_{2}={}^{t}(0,1,0)$, and
$\mbox{$e$}_{3}={}^{t}(0,0,1)$. Additionally, we denote the unit sphere in
$\mathbf{R}^{3}$ by $\mathbf{S}^{2}$.

We next introduce some definitions in order to state the main theorems of this
paper. First we define the compatibility conditions for both (1.6) and (1.10).

###  2.1 Compatibility Conditions for (1.6) and (1.10)

First we define the compatibility conditions needed in this paper for problem
(1.6).

######  Definition 2.1.

For $m=0\ \text{or}\ 1$, we say that $\mbox{$v$}_{0}\in H^{2m+1}(I_{L})$ and
$\mbox{$b$}\in\mathbf{S}^{2}$ satisfy the $m$-th order compatibility condition
for (1.6) if

$\displaystyle\mbox{$v$}_{0}(0)=\mbox{$e$}_{1},\quad\mbox{$v$}_{0}(L)=\mbox{$b$},$

when $m=0$, and

$\displaystyle\mbox{$v$}_{0}(0)\times\mbox{$v$}_{0ss}(0)=\mbox{$v$}_{0}(L)\times\mbox{$v$}_{0ss}(L)=\mbox{$0$}$

when $m=1$. We also say that $\mbox{$v$}_{0}$ and $b$ satisfy the
compatibility conditions for (1.6) up to order $1$ if $\mbox{$v$}_{0}$ and $b$
satisfy both the $0$-th order and the $1$-st order compatibility condition for
(1.6).

Next we define the corresponding compatibility condition for (1.10).

######  Definition 2.2.

For $q_{0}\in H^{2}(I_{L})$, we say that $q_{0}$ satisfies the $0$-th order
compatibility condition for (1.10) if

$\displaystyle q_{0s}(0)=q_{0s}(L)=0$

are satisfied.

###  2.2 The Hasimoto Transformation and the Main Theorems

To state our main theorem, we give a brief explanation of the Hasimoto
transformation. The Hasimoto transformation is a map that relates the solution
of equation (1.1) to the solution of equation (1.2) proposed by Hasimoto [23].
The original transformation proposed by Hasimoto isn’t always well-defined
because the transformation is defined using the torsion of the filament, which
is not defined at points where the curvature of the filament is zero. Later,
Koiso [29] proved that the solvability of the initial value problem on the
torus for (1.1) and (1.2) is equivalent. Koiso did so by constructing a
modified transformation, to which we refer to as the generalized Hasimoto
transformation, which maps solutions of equation (1.1) to solutions of
equation (1.2). This transformation is invertible, and hence, the solvability
is equivalent.

One of the aims of this paper is to prove that the generalized Hasimoto
transformation given by Koiso [29] can be further modified to prove the
equivalence of the solvability of problem (1.6) and problem (1.10). More
precisely, we prove the following.

######  Theorem 2.3.

For $q_{0}\in H^{2}(I_{L})$ satisfying the $0$-th order compatibility
condition for (1.10), there exists $\mbox{$v$}_{0}\in H^{3}(I_{L})$ and
$\mbox{$b$}\in\mathbf{S}^{2}$ satisfying $\lvert\mbox{$v$}_{0}\rvert\equiv 1$
and the compatibility conditions for (1.6) up to order $1$ such that the
following holds. The solution $\mbox{$v$}\in
C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ of problem (1.6) with initial
datum $\mbox{$v$}_{0}$ and boundary datum $b$ corresponds to the solution
$q\in C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ of problem (1.10) with initial
datum $q_{0}$ through the generalized Hasimoto transformation.

######  Theorem 2.4.

For $\mbox{$v$}_{0}\in H^{3}(I_{L})$ and $\mbox{$b$}\in\mathbf{S}^{2}$
satisfying $\lvert\mbox{$v$}_{0}\rvert\equiv 1$ and the compatibility
conditions for (1.6) up to order $1$, there exists $q_{0}\in H^{2}(I_{L})$
satisfying the $0$-th order compatibility condition for (1.10) such that the
following holds. The solution $q\in C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ of problem (1.10) with initial
datum $q_{0}$ corresponds to the solution $\mbox{$v$}\in
C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ of problem (1.6) with initial
datum $\mbox{$v$}_{0}$ and boundary datum $b$ through the inverse generalized
Hasimoto transformation.

Theorem 2.3 and 2.4 together states that the solvability of problem (1.6) and
problem (1.10) are equivalent in suitable Sobolev spaces. The two theorems
also imply that the compatibility conditions for (1.6) and (1.10) correspond
to each other through the generalized Hasimoto transformation. Recall from the
introduction that the solvability of problem (1.6) and problem (1.10) are
already known. Hence, the solvability of either problem in itself is not new,
but the fact that the solvability of the two problems are equivalent is new.

As an application of the above two theorems, we prove the following theorem.

######  Theorem 2.5.

The plane wave solution $q_{R}$ of problem (1.10) given by

$\displaystyle q_{R}(t)=-\frac{1}{R}\exp\left\\{-\frac{{\rm
i}t}{2R^{2}}\right\\}$

with $\displaystyle R>\frac{L}{\pi}$ is orbitally stable in $H^{2}(I_{L})$.
More specifically, for any $\varepsilon>0$, there exists a $\delta>0$ such
that for any $\phi_{0}\in H^{2}(I_{L})$ satisfying the $0$-th order
compatibility condition for (1.10) and $\|\phi_{0}\|_{2}\leq\delta$, the
solution $q\in C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ of problem (1.10) with initial
datum $q_{0}=q_{R}(0)+\phi_{0}$ satisfies

$\displaystyle\sup_{t\geq 0}\inf_{\theta\in\mathbf{R}}\|\exp({\rm
i}\theta)q(t)-q_{R}(t)\|_{2}<\varepsilon.$

From here on, we will refer to the maps $q_{0}\mapsto\mbox{$v$}_{0}$ and
$\mbox{$v$}\mapsto q$ implied in Theorem 2.3 as the generalized Hasimoto
transformation, and the maps $\mbox{$v$}_{0}\mapsto q_{0}$ and
$q\mapsto\mbox{$v$}$ implied in Theorem 2.4 as the inverse generalized
Hasimoto transformation.

##  3 Proof of Main Theorems

In this section, we prove Theorem 2.3, 2.4, and 2.5. The transformation
utilized in the proof of theorem 2.3 and 2.4 is mostly due to Koiso [29] with
some modifications to accommodate the presence of boundary conditions.

Although it is implied in Theorem 2.3 and 2.4, we explicitly state the
solvability of problems (1.6) and (1.10) in the form that we will take for
granted throughout this paper.

######  Theorem 3.1.

Let $\mbox{$v$}_{0}\in H^{3}(I_{L})$ and $\mbox{$b$}\in\mathbf{S}^{2}$ satisfy
$\lvert\mbox{$v$}_{0}\rvert\equiv 1$ and the compatibility condition for (1.6)
up to order $1$. Then, there exists a unique solution $\mbox{$v$}\in
C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ of problem (1.6). Furthermore,
$\lvert\mbox{$v$}(s,t)\rvert=1$ for all $s\in I_{L}$ and $t>0$. Additionally,
there exists $c_{\ast}>0$ such that $v$ satisfies

(3.1) $\displaystyle\sup_{t>0}\|\mbox{$v$}(t)\|_{3}\leq c_{\ast},$

where $c_{\ast}>0$ depends on $L$ and $\|\mbox{$v$}_{0}\|_{3}$ and is non-
decreasing with respect to $\|\mbox{$v$}_{0}\|_{3}$

######  Theorem 3.2.

Let $q_{0}\in H^{2}(I_{L})$ satisfy the $0$-th order compatibility condition
for (1.10). Then, there exists a unique solution $q\in
C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ of problem (1.10).

Theorem 3.1 is proved by the author in [1] and Theorem 3.2 is essentially due
to Bourgain [9, 10] since problem (1.10) can be reduced to the initial value
problem on the torus by reflection and periodic extension.

###  3.1 Proof of Theorem 2.3

Let $q_{0}\in H^{2}(I_{L})$ satisfy the $0$-th order compatibility condition
for problem (1.10) and define $q^{0}_{1}$ and $q^{0}_{2}$ by

$\displaystyle q_{0}(s)=q^{0}_{1}(s)+{\rm i}q^{0}_{2}(s).$

Furthermore, we define $\mbox{$v$}_{0},\mbox{$e$}^{0}$, and $\mbox{$w$}^{0}$
as the solution of the following system of ordinary differential equations.

$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$v$}_{0s}=q^{0}_{1}\mbox{$e$}^{0}+q^{0}_{2}\mbox{$w$}^{0},&s\in
I_{L},\\\\[8.53581pt] \mbox{$e$}^{0}_{s}=-q^{0}_{1}\mbox{$v$}_{0},&s\in
I_{L},\\\\[8.53581pt] \mbox{$w$}^{0}_{s}=-q^{2}_{0}\mbox{$v$}_{0},&s\in
I_{L},\\\\[8.53581pt]
(\mbox{$v$}_{0},\mbox{$e$}^{0},\mbox{$w$}^{0})(0)=(\mbox{$e$}_{1},-\mbox{$e$}_{2},\mbox{$e$}_{3}).&\end{array}\right.$

By direct calculations, we see that the triplet
$\\{\mbox{$v$}_{0},\mbox{$e$}^{0},\mbox{$w$}^{0}\\}$ is an orthonormal basis
of $\mathbf{R}^{3}$ for all $s\in I_{L}$. Set $\mbox{$b$}:=\mbox{$v$}_{0}(L)$,
and we see that by definition, $\mbox{$v$}_{0}$ and $b$ satisfy the $0$-th
order compatibility condition of problem (1.6). We further calculate and see
that

$\displaystyle\mbox{$v$}_{0ss}=(-(q^{0}_{1})^{2}-(q^{0}_{2})^{2})\mbox{$v$}_{0}+q^{0}_{1s}\mbox{$e$}^{0}+q^{0}_{2s}\mbox{$w$}^{0},$

and hence,

$\displaystyle\mbox{$v$}_{0}\times\mbox{$v$}_{0ss}$
$\displaystyle=q^{0}_{1s}\mbox{$v$}_{0}\times\mbox{$e$}^{0}+q^{0}_{2s}\mbox{$v$}_{0}\times\mbox{$w$}^{0}$
$\displaystyle=-q^{0}_{1s}\mbox{$w$}^{0}+q^{0}_{2s}\mbox{$e$}^{0}.$

Here, we have used the fact that
$\\{\mbox{$v$}_{0},\mbox{$e$}^{0},\mbox{$w$}^{0}\\}$ is an orthonormal basis
of $\mathbf{R}^{3}$ and the orientation given at $s=0$ yields
$\mbox{$v$}_{0}\times\mbox{$e$}^{0}=-\mbox{$w$}^{0}$ and
$\mbox{$v$}_{0}\times\mbox{$w$}^{0}=\mbox{$e$}^{0}$. Since $q_{0}$ satisfies
the $0$-th order compatibility condition for (1.10), we see that

$\displaystyle\mbox{$v$}_{0}(0)\times\mbox{$v$}_{0ss}(0)=\mbox{$v$}_{0}(L)\times\mbox{$v$}_{0ss}(L)=\mbox{$0$}$

and $\mbox{$v$}_{0}$ and $b$ satisfies the $1$-st order compatibility
condition for (1.6).

Let, $\mbox{$v$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ be the solution of problem (1.6)
with initial datum $\mbox{$v$}_{0}$ and boundary datum $b$ just obtained. From
Theorem 3.1, we know that $\lvert\mbox{$v$}(s,t)\rvert=1$ for all $s\in I_{L}$
and $t>0$. We now define $e$ as the solution of

$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$e$}_{s}=-(\mbox{$v$}_{s}\cdot\mbox{$e$})\mbox{$v$},&s\in
I_{L},\\\\[8.53581pt] \mbox{$e$}(0)=\mbox{$e$}_{2},&\end{array}\right.$

and set $\mbox{$w$}=\mbox{$v$}\times\mbox{$e$}$. Here, $\cdot$ is the standard
inner product of $\mathbf{R}^{3}$. Note that $e$ and $w$ depend on $t>0$ as a
parameter through $v$. Again, we see that
$\\{\mbox{$v$},\mbox{$e$},\mbox{$w$}\\}$ is an orthonormal basis of
$\mathbf{R}^{3}$ for all $s\in I_{L}$ and $t>0$. Since
$\mbox{$v$}\cdot\mbox{$v$}_{s}\equiv 0$, we have the decomposition

(3.2) $\displaystyle\mbox{$v$}_{s}=\psi_{1}\mbox{$e$}+\psi_{2}\mbox{$w$}$

for some $\psi_{1}(s,t)$ and $\psi_{2}(s,t)$. Then we also see that

(3.3) $\displaystyle\mbox{$e$}_{s}$ $\displaystyle=-\psi_{1}\mbox{$v$},$ (3.4)
$\displaystyle\mbox{$w$}_{s}$ $\displaystyle=-\psi_{2}\mbox{$v$}.$

Taking the inner product of $e$ and $w$ with equation (3.2) we have

$\displaystyle\psi_{1}$ $\displaystyle=\mbox{$e$}\cdot\mbox{$v$}_{s},$
$\displaystyle\psi_{2}$ $\displaystyle=\mbox{$w$}\cdot\mbox{$v$}_{s},$

from which we deduce that $\psi_{1},\psi_{2}\in
C\big{(}[0,\infty);H^{2}(I_{L})\big{)}$ since $\mbox{$v$}\in
C\big{(}[0,\infty);H^{3}(I_{L})\big{)}$. From equation (3.2), we see that

$\displaystyle\lvert\psi_{1}\rvert^{2}+\lvert\psi_{2}\rvert^{2}=\lvert\mbox{$v$}_{s}\rvert^{2},$

which implies

$\displaystyle\|\psi_{1}(t)\|^{2}+\|\psi_{2}(t)\|^{2}=\|\mbox{$v$}_{s}(t)\|^{2}.$

Furthermore, we have

$\displaystyle\psi_{1s}\mbox{$e$}+\psi_{2s}\mbox{$w$}$
$\displaystyle=\mbox{$v$}_{ss}+(\psi_{1})^{2}\mbox{$v$}+(\psi_{2})^{2}\mbox{$v$},$
$\displaystyle\psi_{1ss}\mbox{$e$}+\psi_{2ss}\mbox{$w$}$
$\displaystyle=\mbox{$v$}_{sss}+3\psi_{1}\psi_{1s}\mbox{$v$}+3\psi_{2}\psi_{2s}\mbox{$v$}+\psi_{1}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}\mbox{$e$}$
$\displaystyle\hskip
28.45274pt+\psi_{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}\mbox{$w$},$

and after taking the inner product of the above equations with $e$ and $w$, we
derive

$\displaystyle\|\psi_{1s}(t)\|$ $\displaystyle\leq\|\mbox{$v$}_{ss}(t)\|,$
$\displaystyle\|\psi_{2s}(t)\|$ $\displaystyle\leq\|\mbox{$v$}_{ss}(t)\|,$
$\displaystyle\|\psi_{1ss}(t)\|$
$\displaystyle\leq\|\mbox{$v$}_{sss}(t)\|+C\|\mbox{$v$}_{s}(t)\|_{1}^{2}\|\mbox{$v$}_{s}(t)\|,$
$\displaystyle\|\psi_{2ss}(t)\|$
$\displaystyle\leq\|\mbox{$v$}_{sss}(t)\|+C\|\mbox{$v$}_{s}(t)\|_{1}^{2}\|\mbox{$v$}_{s}(t)\|,$

where $C>0$ is determined from the embedding $H^{1}(I_{L})\hookrightarrow
L^{\infty}(I_{L})$. Hence, we have

(3.5) $\displaystyle\|\psi_{1}(t)\|_{2}+\|\psi_{2}(t)\|_{2}\leq
C\big{(}\|\mbox{$v$}(t)\|_{3}+\|\mbox{$v$}(t)\|_{3}^{3}\big{)},$

for all $t>0$, where $C>0$ is independent of $t$ and $v$.

Since $\mbox{$v$}\cdot\mbox{$v$}_{t}\equiv 0$, a decomposition of the form
$\mbox{$v$}_{t}=p_{1}\mbox{$e$}+p_{2}\mbox{$w$}$ holds. From the equation
$(\mbox{$v$}_{t})_{s}=(\mbox{$v$}_{s})_{t}$ and
$(\mbox{$e$}_{t})_{s}=(\mbox{$e$}_{s})_{t}$, we deduce that $p_{1}=-\psi_{2s}$
and $p_{2}=\psi_{1s}$. Furthermore,

(3.6) $\displaystyle\mbox{$v$}_{t}$
$\displaystyle=-\psi_{2s}\mbox{$e$}+\psi_{1s}\mbox{$w$},$
$\displaystyle\mbox{$e$}_{t}$
$\displaystyle=\psi_{2s}\mbox{$v$}+\bigg{\\{}-\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}+\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}\big{\rvert}_{s=0}\bigg{\\}}\mbox{$w$},$
$\displaystyle\mbox{$w$}_{t}$
$\displaystyle=-\psi_{1s}\mbox{$v$}-\bigg{\\{}-\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}+\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}\big{\rvert}_{s=0}\bigg{\\}}\mbox{$e$},$

holds. Here, $\big{\rvert}_{s=0}$ denotes the trace at $s=0$. The above three
equations along with the fact that $\mbox{$v$}\in
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$,
$\psi_{1}=\mbox{$e$}\cdot\mbox{$v$}_{s}$, and
$\psi_{2}=\mbox{$w$}\cdot\mbox{$v$}_{s}$ implies that $\psi_{1},\psi_{2}\in
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$. This in turn allows us to
calculate as follows.

$\displaystyle\psi_{1t}$
$\displaystyle=-\psi_{2ss}+\bigg{\\{}-\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}+\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}\big{\rvert}_{s=0}\bigg{\\}}\psi_{2},$
$\displaystyle\psi_{2t}$
$\displaystyle=\psi_{1ss}-\bigg{\\{}-\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}+\frac{1}{2}\big{(}(\psi_{1})^{2}+(\psi_{2})^{2}\big{)}\big{\rvert}_{s=0}\bigg{\\}}\psi_{1}.$

Setting $\psi=\psi_{1}+{\rm i}\psi_{2}$, we see that

$\displaystyle{\rm
i}\psi_{t}-\psi_{ss}=\frac{1}{2}\lvert\psi\rvert^{2}\psi-\frac{1}{2}\lvert\psi(0,t)\rvert^{2}\psi,$

and after a gauge transform given by

(3.7) $\displaystyle q(s,t)=\psi(s,t)\exp\bigg{\\{}\frac{{\rm
i}}{2}\int^{t}_{0}\lvert\psi(0,\tau)\rvert^{2}\ d\tau\bigg{\\}},$

we see that $q$ satisfies

$\displaystyle{\rm i}q_{t}=q_{ss}+\frac{1}{2}\lvert q\rvert^{2}q.$

As $t$ tends to zero in equation (3.2), we see from the uniqueness of the
solution of ordinary differential equations that

$\displaystyle q(s,0)=q_{0}(s).$

Finally, we see that by taking the derivative of the boundary condition in
problem (1.6) with respect to $t$, we see that
$\mbox{$v$}_{t}(0,t)=\mbox{$v$}_{t}(L,t)=\mbox{$0$}$, and from equation (3.6),
we see that

$\displaystyle q_{s}(0,t)=q_{s}(L,t)=0.$

Also note that $q\in C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ from the definition of $q$ along
with the fact that $\psi_{1},\psi_{2}\in
C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$. Additionally, from equations
(3.5) and (3.7) we have

(3.8) $\displaystyle\|q(t)\|_{2}\leq
C\big{(}\|\mbox{$v$}(t)\|_{3}+\|\mbox{$v$}(t)\|^{3}_{3}\big{)},$

for all $t\geq 0$, where $C>0$ is independent of $t$ and $v$. Estimate (3.8)
will be utilized later. This finishes the proof of Theorem 2.3.
$\Box$

###  3.2 Proof of Theorem 2.4

Let $\mbox{$v$}_{0}\in H^{3}(I_{L})$ satisfying
$\lvert\mbox{$v$}_{0}\rvert\equiv 1$ and $\mbox{$b$}\in\mathbf{S}^{2}$ be
initial and boundary datum for problem (1.6) satisfying the compatibility
conditions up to order $1$. Define $\mbox{$e$}^{0}$ as the solution of

$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$e$}^{0}_{s}=-(\mbox{$v$}_{0s}\cdot\mbox{$e$}^{0})\mbox{$v$}_{0},&s\in
I_{L},\\\\[8.53581pt] \mbox{$e$}^{0}(0)=-\mbox{$e$}_{2},&\end{array}\right.$

and set $\mbox{$w$}^{0}=\mbox{$e$}^{0}\times\mbox{$v$}_{0}$. Similarly to
Section 3.1, the triplet $\\{\mbox{$v$}_{0},\mbox{$e$}^{0},\mbox{$w$}^{0}\\}$
is an orthonormal basis of $\mathbf{R}^{3}$ and the decomposition

$\displaystyle\mbox{$v$}_{0s}=q^{0}_{1}\mbox{$e$}^{0}+q^{0}_{2}\mbox{$w$}^{0}$

holds and direct calculation yields

$\displaystyle\mbox{$v$}_{0}\times\mbox{$v$}_{0ss}=q^{0}_{2s}\mbox{$e$}^{0}-q^{0}_{1s}\mbox{$w$}^{0}.$

Hence, setting $q_{0}=q^{0}_{1}+{\rm i}q^{0}_{2}$ we see that $q_{0}$
satisfies $q_{0s}(0,t)=q_{0s}(L,t)=0$ since $\mbox{$v$}_{0}$ and $b$ satisfy
the $1$-st order compatibility condition for (1.6). In other words, $q_{0}$
satisfies the $0$-th order compatibility condition for (1.10).

Let $q\in C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ be the solution of problem (1.10)
with initial datum $q_{0}$ just obtained and set $q(s,t)=q_{1}(s,t)+{\rm
i}q_{2}(s,t)$. We define $v$, $e$, and $w$ as the solution of the following
system of ordinary differential equations with respect to $s$.

(3.13)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$v$}_{s}=q_{1}\mbox{$e$}+q_{2}\mbox{$w$},&s\in
I_{L},\\\\[8.53581pt] \mbox{$e$}_{s}=-q_{1}\mbox{$v$},&s\in
I_{L},\\\\[8.53581pt] \mbox{$w$}_{s}=-q_{2}\mbox{$v$},&s\in
I_{L},\\\\[8.53581pt]
(\mbox{$v$},\mbox{$e$},\mbox{$w$})(0)=(\mbox{$e$}_{1},-\mbox{$e$}_{2},\mbox{$e$}_{3}).&\end{array}\right.$

Again, $\\{\mbox{$v$},\mbox{$e$},\mbox{$w$}\\}$ depends on $t>0$ as a
parameter and $\\{\mbox{$v$},\mbox{$e$},\mbox{$w$}\\}$ is an orthonormal basis
of $\mathbf{R}^{3}$ for all $s\in I_{L}$ and $t>0$. From (3.13), we deduce
that $\mbox{$v$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ and

$\displaystyle\|\mbox{$v$}_{s}(t)\|^{2}$
$\displaystyle=\|q_{1}(t)\|^{2}+\|q_{2}(t)\|^{2},$
$\displaystyle\|\mbox{$v$}_{ss}(t)\|$ $\displaystyle\leq
C\big{(}\|q_{1s}(t)\|+\|q_{2s}(t)\|+\|q_{1}(t)\|^{2}_{1}+\|q_{2}(t)\|^{2}_{1}\big{)},$
$\displaystyle\|\mbox{$v$}_{sss}(t)\|$ $\displaystyle\leq
C\big{(}\|q_{1ss}(t)\|+\|q_{2ss}(t)\|+\|q_{1}(t)\|^{3}_{2}+\|q_{2}(t)\|^{3}_{2}\big{)},$

where $C>0$ is independent of $t$ and $q$. Hence we have

$\displaystyle\lvert\mbox{$v$}(s,t)\rvert$ $\displaystyle=1\ \text{for all}\
s\in I_{L}\ \text{and}\ t\geq 0,$ $\displaystyle\|\mbox{$v$}_{s}(t)\|_{2}$
$\displaystyle\leq C\big{(}\|q(t)\|_{2}+\|q(t)\|^{3}_{2}\big{)}.$

Note that the definition of $\mbox{$v$}_{0}$, $\mbox{$e$}^{0}$, and
$\mbox{$w$}^{0}$ coincides with the limit as $t$ tends to zero in equations
(3.13).

Next, we see that

(3.14) $\displaystyle\mbox{$v$}_{t}$
$\displaystyle=q_{2s}\mbox{$e$}-q_{1s}\mbox{$w$},$
$\displaystyle\mbox{$e$}_{t}$
$\displaystyle=-q_{2s}\mbox{$v$}+(\frac{1}{2}\lvert
q\rvert^{2}-\frac{1}{2}\lvert q(0,t)\rvert^{2})\mbox{$w$},$
$\displaystyle\mbox{$w$}_{t}$
$\displaystyle=q_{1s}\mbox{$v$}-(\frac{1}{2}\lvert
q\rvert^{2}-\frac{1}{2}\lvert q(0,t)\rvert^{2})\mbox{$e$}.$

From equations (3.13), we see that

$\displaystyle\mbox{$v$}\times\mbox{$v$}_{ss}=q_{2s}\mbox{$e$}-q_{1s}\mbox{$w$},$

and hence, along with (3.14) we have

$\displaystyle\mbox{$v$}_{t}=\mbox{$v$}\times\mbox{$v$}_{ss}.$

The first equation in (3.13) with $t=0$ together with the uniqueness of the
solution of ordinary differential equations show that

$\displaystyle\mbox{$v$}(s,0)=\mbox{$v$}_{0}(s).$

Finally, taking the trace at $s=0$ and $s=L$ in equation (3.14) we see that
$\mbox{$v$}_{t}(0,t)=\mbox{$v$}_{t}(L,t)=\mbox{$0$}$ which in turn yields

$\displaystyle\mbox{$v$}(0,t)$
$\displaystyle=\mbox{$v$}_{0}(0)=\mbox{$e$}_{1},$
$\displaystyle\mbox{$v$}(L,t)$ $\displaystyle=\mbox{$v$}_{0}(L)=\mbox{$b$},$

and hence, $v$ satisfies the boundary condition of problem (1.6). This shows
that $v$ is the desired solution of problem (1.6). This finishes the proof of
Theorem 2.4. $\Box$

###  3.3 Proof of Theorem 2.5

We divide the proof of Theorem 2.5 into five steps.

We first establish that perturbations to the initial datum of problem (1.10)
correspond to perturbations to the initial datum of problem (1.6).
Furthermore, we derive estimates that show in what norms the perturbations
translate.

Secondly, we show that $q_{R}(0)$ corresponds to a particular type of initial
datum of problem (1.6). Solutions of problem (1.6) with this type of initial
datum will be referred to as an arc-shaped solution of problem (1.6).

Then, we prove stability estimates for arc-shaped solutions of problem (1.6),
which is related to the stability problem for plane wave solutions of problem
(1.10). The stability estimates for arc-shaped solutions are already utilized
by the author in [2], but we reiterate it here for completeness.

Finally, we combine the results of the previous steps to prove that the plane
wave solution $q_{R}$ is orbitally stable. More precisely, we show that the
stability estimates for problem (1.6) yield stability estimates for problem
(1.10) through the generalized Hasimoto transformation.

####  3.3.1 The Generalized Hasimoto Transformation of Perturbations

In this section, we show that perturbations to the initial datum of problem
(1.10) translate to perturbations to the initial datum of problem (1.6)
through the generalized Hasimoto transformation.

Let $q_{0}\in H^{2}(I_{L})$ and $\varphi_{0}\in H^{2}(I_{L})$ be such that
$q_{0}$ and $\varphi_{0}$ satisfy the $0$-th order compatibility condition for
problem (1.10). Note that $q_{0}+\varphi_{0}$ also satisfy the same
compatibility condition. Setting $\tilde{q}_{0}=q_{0}+\varphi_{0}$ and

$\displaystyle q_{0}=q^{0}_{1}+{\rm
i}q^{0}_{2}\qquad\text{and}\qquad\tilde{q}_{0}=\tilde{q}^{0}_{1}+{\rm
i}\tilde{q}^{0}_{2},$

define $\\{\mbox{$v$}_{0},\mbox{$e$}^{0},\mbox{$w$}^{0}\\}$ and
$\\{\tilde{\mbox{$v$}}_{0},\tilde{\mbox{$e$}}^{0},\tilde{\mbox{$w$}}^{0}\\}$
as the solution of

$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$v$}_{0s}=q^{0}_{1}\mbox{$e$}^{0}+q^{0}_{2}\mbox{$w$}^{0},&s\in
I_{L},\\\\[8.53581pt] \mbox{$e$}^{0}_{s}=-q^{0}_{1}\mbox{$v$}_{0},&s\in
I_{L},\\\\[8.53581pt] \mbox{$w$}^{0}_{s}=-q^{0}_{2}\mbox{$v$}_{0},&s\in
I_{L},\\\\[8.53581pt]
(\mbox{$v$}_{0},\mbox{$e$}^{0},\mbox{$w$}^{0})=(\mbox{$e$}_{1},-\mbox{$e$}_{2},\mbox{$e$}_{3}),&\end{array}\right.$
$\displaystyle\left\\{\begin{array}[]{ll}\tilde{\mbox{$v$}}_{0s}=q^{0}_{1}\tilde{\mbox{$e$}}^{0}+q^{0}_{2}\tilde{\mbox{$w$}}^{0},&s\in
I_{L},\\\\[8.53581pt]
\tilde{\mbox{$e$}}^{0}_{s}=-q^{0}_{1}\tilde{\mbox{$v$}}_{0},&s\in
I_{L},\\\\[8.53581pt]
\tilde{\mbox{$w$}}^{0}_{s}=-q^{0}_{2}\tilde{\mbox{$v$}}_{0},&s\in
I_{L},\\\\[8.53581pt]
(\tilde{\mbox{$v$}}_{0},\tilde{\mbox{$e$}}^{0},\tilde{\mbox{$w$}}^{0})=(\mbox{$e$}_{1},-\mbox{$e$}_{2},\mbox{$e$}_{3}),&\end{array}\right.$

respectively. Setting $\mbox{$V$}:=\mbox{$v$}_{0}-\tilde{\mbox{$v$}}_{0},\
\mbox{$E$}:=\mbox{$e$}^{0}-\tilde{\mbox{$e$}}^{0},$ and
$\mbox{$W$}:=\mbox{$w$}^{0}-\tilde{\mbox{$w$}}^{0}$, we see that

(3.19)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$V$}_{s}=(q^{0}_{1}-\tilde{q}^{0}_{1})\mbox{$e$}^{0}+\tilde{q}^{0}_{1}\mbox{$E$}+(q^{0}_{2}-\tilde{q}^{0}_{2})\mbox{$w$}^{0}+\tilde{q}^{0}_{2}\mbox{$W$},&s\in
I_{L},\\\\[8.53581pt]
\mbox{$E$}_{s}=(q^{0}_{1}-\tilde{q}^{0}_{1})\mbox{$v$}_{0}-\tilde{q}^{0}_{1}\mbox{$V$},&s\in
I_{L},\\\\[8.53581pt]
\mbox{$W$}_{s}=(\tilde{q}^{0}_{2}-q^{0}_{2})\mbox{$v$}_{0}-\tilde{q}^{0}_{2}\mbox{$V$},&s\in
I_{L},\\\\[8.53581pt]
(\mbox{$V$},\mbox{$E$},\mbox{$W$})(0)=(\mbox{$0$},\mbox{$0$},\mbox{$0$}),\end{array}\right.$

holds. Standard energy estimates yield

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}s}\bigg{\\{}\lvert\mbox{$V$}\rvert^{2}$
$\displaystyle+\lvert\mbox{$E$}\rvert^{2}+\lvert\mbox{$W$}\rvert^{2}\bigg{\\}}$
$\displaystyle\leq
C\big{(}1+\|\tilde{q}^{0}_{1}\|^{2}_{L^{\infty}(I_{L})}+\|\tilde{q}^{0}_{2}\|^{2}_{L^{\infty}(I_{L})}\big{)}\big{(}\lvert\mbox{$V$}\rvert^{2}+\lvert\mbox{$E$}\rvert^{2}+\lvert\mbox{$W$}\rvert^{2}\big{)}$
$\displaystyle\qquad+\big{(}\|\tilde{q}^{0}_{1}-q^{0}_{1}\|^{2}_{L^{\infty}(I_{L})}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|^{2}_{L^{\infty}(I_{L})}\big{)},$

and from Gronwall’s inequality, we have

(3.20)
$\displaystyle\lvert\mbox{$V$}\rvert^{2}+\lvert\mbox{$E$}\rvert^{2}+\lvert\mbox{$W$}\rvert^{2}\leq
Ce^{CL}\bigg{\\{}\|\tilde{q}^{0}_{1}-q^{0}_{1}\|^{2}_{L^{\infty}(I_{L})}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|^{2}_{L^{\infty}(I_{L})}\bigg{\\}},$

where $C>0$ depends on $\|q_{0}\|_{L^{\infty}(I_{L})}$ and
$\|\tilde{q}_{0}\|_{L^{\infty}(I_{L})}$. Hence, estimate (3.20) implies that
if $\tilde{q}_{0}-q_{0}$ is controlled in $L^{\infty}(I_{L})$, then we have a
control on $\mbox{$V$},\mbox{$E$},$ and $W$ in $L^{\infty}(I_{L})$, and as a
consequence, in $L^{2}(I_{L})$. Equations (3.19) together with inequality
(3.20) show that

$\displaystyle\|\mbox{$V$}_{s}\|^{2}+\|\mbox{$E$}_{s}\|^{2}+\|\mbox{$W$}_{s}\|^{2}\leq
C\big{(}\|\tilde{q}^{0}_{1}-q^{0}_{1}\|^{2}_{L^{\infty}(I_{L})}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|^{2}_{L^{\infty}(I_{L})}\big{)},$
$\displaystyle\|\mbox{$V$}_{s}\|_{L^{\infty}(I_{L})}+\|\mbox{$E$}_{s}\|_{L^{\infty}(I_{L})}+\|\mbox{$W$}_{s}\|_{L^{\infty}(I_{L})}$
$\displaystyle\hskip 113.81102pt\leq
C\big{(}\|\tilde{q}^{0}_{1}-q^{0}_{1}\|_{L^{\infty}(I_{L})}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|_{L^{\infty}(I_{L})}\big{)},$

hold, where $C>0$ depends on $\|q_{0}\|_{L^{\infty}(I_{L})}$ and
$\|\tilde{q}_{0}\|_{L^{\infty}(I_{L})}$.

Taking the derivative with respect to $s$ of equations (3.19), we have

$\displaystyle\mbox{$V$}_{ss}$
$\displaystyle=(q^{0}_{1}-\tilde{q}^{0}_{1})_{s}\mbox{$e$}^{0}-q^{0}_{1}(q^{0}_{1}-\tilde{q}^{0}_{1})\mbox{$v$}_{0}+\tilde{q}^{0}_{1s}\mbox{$E$}+\tilde{q}^{0}_{1}\big{(}(q^{0}_{1}-\tilde{q}^{0}_{1})\mbox{$v$}_{0}-\tilde{q}^{0}_{1}\mbox{$V$}\big{)}$
$\displaystyle\qquad+(q^{0}_{2}-\tilde{q}^{0}_{2})_{s}\mbox{$w$}^{0}-q^{0}_{2}(q^{0}_{2}-\tilde{q}^{0}_{2})\mbox{$v$}_{0}+\tilde{q}^{0}_{2s}\mbox{$W$}+\tilde{q}^{0}_{2}\big{(}(\tilde{q}^{0}_{2}-q^{0}_{2})\mbox{$v$}_{0}-(\tilde{q}^{0}_{2})\mbox{$V$}\big{)}$
$\displaystyle=-\big{(}(q^{0}_{1}-\tilde{q}^{0}_{1})^{2}+(q^{0}_{2}-\tilde{q}^{0}_{2})^{2}\big{)}\mbox{$v$}_{0}+(q^{0}_{1}-\tilde{q}^{0}_{1})_{s}\mbox{$e$}^{0}+(q^{0}_{2}-\tilde{q}^{0}_{2})_{s}\mbox{$w$}^{0}$
$\displaystyle\qquad+\tilde{q}^{0}_{1s}\mbox{$E$}+\tilde{q}^{0}_{2s}\mbox{$W$}-\big{(}(\tilde{q}^{0}_{1})^{2}+(\tilde{q}^{0}_{2})^{2}\big{)}\mbox{$V$},$

from which we have the estimate

$\displaystyle\|\mbox{$V$}_{ss}\|\leq
C\big{(}\|\tilde{q}^{0}_{1}-q^{0}_{1}\|_{1}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|_{1}+\|\tilde{q}^{0}_{1}-q^{0}_{1}\|^{2}_{1}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|^{2}_{1}\big{)}.$

In a similar fashion, we can derive the estimate

$\displaystyle\|\mbox{$V$}_{sss}\|\leq
C\big{(}\|\tilde{q}^{0}_{1}-q^{0}_{1}\|_{2}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|_{2}+\|\tilde{q}^{0}_{1}-q^{0}_{1}\|^{3}_{2}+\|\tilde{q}^{0}_{2}-q^{0}_{2}\|^{3}_{2}\big{)}.$

In summary, we have the estimate

(3.21) $\displaystyle\|\tilde{\mbox{$v$}_{0}}-\mbox{$v$}_{0}\|_{3}\leq
C\big{(}\|\tilde{q}_{0}-q_{0}\|_{2}+\|\tilde{q}_{0}-q_{0}\|^{3}_{2}\big{)},$

where $C>0$ depends on $\|\tilde{q}_{0}\|_{2}+\|q_{0}\|_{2}$. We also see that
$C>0$ is non-decreasing with respect to $\|\tilde{q}_{0}\|_{2}+\|q_{0}\|_{2}$.
Inequality (3.21) shows that $H^{2}$-perturbations of $q_{0}$ correspond to
$H^{3}$-perturbations of $\mbox{$v$}_{0}$.

We summarize the conclusion of this section in the following proposition.

######  Proposition 3.3.

For $q_{0}\in H^{2}(I_{L})$ and $\varphi_{0}\in H^{2}(I_{L})$ which satisfy
the $0$-th order compatibility condition for (1.10), let $\mbox{$v$}_{0}$ and
$\tilde{\mbox{$v$}}_{0}$ be initial data constructed by the generalized
Hasimoto transformation from $q_{0}$ and $q_{0}+\varphi_{0}$, respectively.
Then, $\mbox{$v$}_{0},\tilde{\mbox{$v$}}_{0}\in H^{3}(I_{L})$ and satsify

$\displaystyle\|\tilde{\mbox{$v$}_{0}}-\mbox{$v$}_{0}\|_{3}\leq
C\big{(}\|\tilde{q}_{0}-q_{0}\|_{2}+\|\tilde{q}_{0}-q_{0}\|^{3}_{2}\big{)},$

where $C>0$ depends on $\|q_{0}\|_{2}+\|\varphi_{0}\|_{2}$, and is non-
decreasing with respect to $\|q_{0}\|_{2}+\|\varphi_{0}\|_{2}$.

####  3.3.2 The Inverse Generalized Hasimoto Transformation of the Plane Wave
Solution

We investigate the image of the plane wave solution by the inverse generalized
Hasimoto transformation. Recall that for $R>0$,

$\displaystyle q_{R}(t)=-\frac{1}{R}\exp\left\\{-\frac{{\rm
i}t}{2R^{2}}\right\\},$

and $q_{R}(0)=-\frac{1}{R}$. Following the arguments in Section 3.1, the
generalized Hasimoto transformation applied to the initial datum
$-\frac{1}{R}$ corresponds to defining $\mbox{$v$}_{0}$ and $\mbox{$e$}^{0}$
as the solution to

(3.25)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$v$}_{0s}=-\frac{1}{R}\mbox{$e$}^{0},&s\in
I_{L},\\\\[8.53581pt] \mbox{$e$}^{0}_{s}=\frac{1}{R}\mbox{$v$}_{0},&s\in
I_{L},\\\\[8.53581pt]
(\mbox{$v$}_{0},\mbox{$e$}^{0})(0)=(\mbox{$e$}_{1},-\mbox{$e$}_{2}),&\end{array}\right.$

and setting $\mbox{$w$}^{0}\equiv\mbox{$e$}_{3}$. System (3.25) can be solved
explicitly to obtain

$\displaystyle\mbox{$v$}_{0}(s)={}^{t}\big{(}\cos(s/R),-\sin(s/R),0\big{)}.$

The above $\mbox{$v$}_{0}$ corresponds to an arc-shaped filament, which is an
explicit solution of problem (1.6). Namely,

$\displaystyle\mbox{$v$}_{0}\times\mbox{$v$}_{0ss}\equiv\mbox{$0$}$

and hence, $\mbox{$v$}_{0}$ is a stationary solution of problem (1.6) with
$\mbox{$b$}={}^{t}(\cos(L/R),-\sin(L/R),0)$. We set

$\displaystyle\mbox{$v$}^{R}(s,t)=\mbox{$v$}^{R}(s):={}^{t}\big{(}\cos(s/R),-\sin(s/R),0\big{)}$

for $s\in I_{L}$ and $t>0$ and refer to $\mbox{$v$}^{R}$ as the arc-shaped
solution of problem (1.6). This shows that $q_{R}$ corresponds to
$\mbox{$v$}^{R}$ through the inverse generalized Hasimoto transformation.

####  3.3.3 Stability of Arc-Shaped Solutions of Problem (1.6)

We derive stability estimates for arc-shaped solutions of problem (1.6). These
estimates are already derived by the author in [2], but we reiterate the
statements and proofs for completeness.

For $R>0$, $\mbox{$v$}^{R}$ is the solution of problem (1.6) with

$\displaystyle\mbox{$v$}_{0}(s)=\mbox{$v$}^{R}(s)={}^{t}(\cos(s/R),-\sin(s/R),0),$
$\displaystyle\mbox{$b$}={}^{t}(\cos(L/R),-\sin(L,/R),0).$

Recall that $\mbox{$v$}^{R}$ is a stationary solution. We consider
perturbations of the arc-shaped solution $\mbox{$v$}^{R}$. Let,
$\mbox{$\varphi$}_{0}\in H^{3}(I_{L})$ be the initial perturbation satisfying
the following.

######  Assumption 3.4.

For the initial perturbation $\mbox{$\varphi$}_{0}$, we assume the following.

(A1)

$\lvert\mbox{$v$}_{0}(s)+\mbox{$\varphi$}_{0}(s)\rvert=1$ for all $s\in
I_{L}$.

(A2)

$\mbox{$v$}_{0}+\mbox{$\varphi$}_{0}$ and $b$ satisfy the compatibility
conditions for problem (1.6) up to order $1$.

Note that because $\mbox{$v$}_{0}$ and $b$ satisfy the $0$-th order
compatibility condition, assumption (A2) implies
$\mbox{$\varphi$}_{0}\rvert_{s=0}=\mbox{$\varphi$}_{0}\rvert_{s=L}=\mbox{$0$}$.
Let $\mbox{$v$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ be the solution of problem (1.6)
with initial datum $\mbox{$v$}_{0}+\mbox{$\varphi$}_{0}$ and boundary datum
$b$. For convenience, we refer to this solution $v$ as the perturbed arc-
shaped solution. Set $\mbox{$\varphi$}=\mbox{$v$}-\mbox{$v$}^{R}$. Then,
$\varphi$ satisfies

(3.29)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$\varphi$}_{t}=\mbox{$\varphi$}\times\mbox{$\varphi$}_{ss}+\mbox{$\varphi$}\times\mbox{$v$}^{R}_{ss}+\mbox{$v$}^{R}\times\mbox{$\varphi$}_{ss},&s\in
I_{L},t>0,\\\\[8.53581pt] \mbox{$\varphi$}(s,0)=\mbox{$\varphi$}_{0}(s),&s\in
I_{L},\\\\[8.53581pt]
\mbox{$\varphi$}(0,t)=\mbox{$\varphi$}(L,t)=\mbox{$0$}.&\end{array}\right.$

Note that $\mbox{$\varphi$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$. We first prove the following
lemma.

######  Lemma 3.5.

For $\mbox{$\varphi$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$, set

$\displaystyle
E(\mbox{$\varphi$}(t)):=\|\mbox{$\varphi$}_{s}(t)\|^{2}-\frac{1}{R^{2}}\|\mbox{$\varphi$}(t)\|^{2}.$

Then, if $\varphi$ satisfies (3.29),
$E(\mbox{$\varphi$}(t))=E(\mbox{$\varphi$}_{0})$ for all $t>0$. In other
words, $E$ is a conserved quantity.

###### Proof.

From the first equation in (3.29), we have

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\|\mbox{$\varphi$}\|^{2}=2(\mbox{$\varphi$},\mbox{$\varphi$}_{t})=2(\mbox{$\varphi$},\mbox{$v$}^{R}\times\mbox{$\varphi$}_{ss})=-2(\mbox{$\varphi$},\mbox{$v$}^{R}_{s}\times\mbox{$\varphi$}_{s}),$

where integration by parts was used. Furthermore,

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\|\mbox{$\varphi$}_{s}\|^{2}=2(\mbox{$\varphi$}_{s},\mbox{$\varphi$}_{st})=-2(\mbox{$\varphi$}_{ss},\mbox{$\varphi$}_{t})$
$\displaystyle=-2(\mbox{$\varphi$}_{ss},\mbox{$\varphi$}\times\mbox{$v$}^{R}_{ss})$
$\displaystyle=2(\mbox{$\varphi$}_{s},\mbox{$\varphi$}\times\mbox{$v$}^{R}_{sss})$
$\displaystyle=-\frac{2}{R^{2}}(\mbox{$\varphi$},\mbox{$v$}^{R}_{s}\times\mbox{$\varphi$}_{s}),$

where we substituted
$\mbox{$v$}^{R}_{sss}=-\frac{1}{R^{2}}\mbox{$v$}^{R}_{s}$. Combining the two
equalities show that $\frac{{\rm d}}{{\rm d}t}E(\mbox{$\varphi$}(t))=0$, and
this proves the lemma. ∎

From Lemma 3.5, the following Corollary immediately follows.

######  Corollary 3.6.

For $\mbox{$\varphi$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ satisfying (3.29), if $R>L/\pi$,
there exists $C>0$ such that

$\displaystyle\|\mbox{$\varphi$}(t)\|_{1}\leq C\|\mbox{$\varphi$}_{0}\|_{1}$

holds for all $t>0$, where $C>0$ depends on $L$ and $R$.

###### Proof.

Since $\mbox{$\varphi$}\rvert_{s=0}=\mbox{$\varphi$}\rvert_{s=L}=\mbox{$0$}$,
the Poincaré inequality

$\displaystyle\|\mbox{$\varphi$}(t)\|\leq\frac{L}{\pi}\|\mbox{$\varphi$}_{s}(t)\|,$

is applicable and hence

$\displaystyle
E(\mbox{$\varphi$}(t))\geq\|\mbox{$\varphi$}_{s}(t)\|^{2}-\frac{1}{R^{2}}\left(\frac{L}{\pi}\right)^{2}\|\mbox{$\varphi$}_{s}(t)\|^{2}=c_{0}\|\mbox{$\varphi$}_{s}(t)\|^{2}$

holds for all $t>0$, where $c_{0}=1-\frac{L^{2}}{R^{2}\pi^{2}}\in(0,1)$. This
with Lemma 3.5 yields

$\displaystyle\|\mbox{$\varphi$}_{s}(t)\|^{2}\leq CE(\mbox{$\varphi$}_{0})\leq
C\|\mbox{$\varphi$}_{0}\|_{1},$

where $C>0$ depends on $L$ and $R$. Applying the Poincaré inequality once
more, we have

$\displaystyle\|\mbox{$\varphi$}(t)\|_{1}\leq C\|\mbox{$\varphi$}_{0}\|_{1}$

for all $t>0$.

∎

Next, we show higher-order estimates.

######  Lemma 3.7.

For $\mbox{$\varphi$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ satisfying (3.29), there exists
$C>0$ depending only on $L$ and $R$ such that if $R>L/\pi$, $\varphi$
satisfies

$\displaystyle\|\mbox{$\varphi$}_{ss}(t)\|_{1}\leq
C\big{(}\|\mbox{$\varphi$}_{0}\|_{3}+\|\mbox{$\varphi$}_{0}\|_{3}^{3}\big{)}$

for all $t>0$.

###### Proof.

We make use of conserved quantities for problem (1.6), which was also utilized
in [1]. Properties of solutions of problem (1.6) that we make use of here are
either well known, or proved in [1] and hence, we will use them without proof.
The perturbed arc-shaped solution $v$ satisfies
$\lvert\mbox{$v$}(s,t)\rvert=1$ for all $s\in I_{L}$ and $t>0$. Hence,

$\displaystyle
1=\lvert\mbox{$v$}(s,t)\rvert^{2}=\lvert\mbox{$v$}^{R}(s)+\mbox{$\varphi$}(s,t)\rvert^{2}$

for all $s\in I_{L}$ and $t>0$. Since $\lvert\mbox{$v$}^{R}(s)\rvert=1$, we
have

(3.30) $\displaystyle
2\mbox{$v$}^{R}(s)\cdot\mbox{$\varphi$}(s,t)=-\lvert\mbox{$\varphi$}(s,t)\rvert^{2}$

for all $s\in I_{L}$ and $t>0$. We also have the following conserved
quantities.

(3.31) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\bigg{\\{}\|\mbox{$v$}_{ss}\|^{2}-\frac{5}{4}\|\lvert\mbox{$v$}_{s}\rvert^{2}\|^{2}\bigg{\\}}=0,$
(3.32) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\bigg{\\{}\|\mbox{$v$}_{sss}\|^{2}-\frac{7}{2}\|\lvert\mbox{$v$}_{s}\rvert\lvert\mbox{$v$}_{ss}\rvert\|^{2}-14\|\mbox{$v$}_{s}\cdot\mbox{$v$}_{ss}\|^{2}+\frac{21}{8}\|\lvert\mbox{$v$}_{s}\rvert^{3}\|^{2}\bigg{\\}}=0.$

These quantities are well known, and the above equality can be verified by
direct calculation.

First we set

$\displaystyle
E_{1}(\mbox{$v$}(t)):=\|\mbox{$v$}_{ss}(t)\|^{2}-\frac{5}{4}\|\lvert\mbox{$v$}_{s}(t)\rvert^{2}\|^{2}.$

For $t\geq 0$ we substitute $\mbox{$v$}=\mbox{$v$}^{R}+\mbox{$\varphi$}$ and
decompose as follows.

$\displaystyle
E_{1}(\mbox{$v$}(t))=E_{1}(\mbox{$v$}^{R})+E_{1}(\mbox{$\varphi$}(t))+R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$}(t)),$

where $R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$}(t))$ is given by

$\displaystyle R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$}(t))$
$\displaystyle=2(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})-5(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}^{R}_{s})-5((\mbox{$v$}^{R}_{s}\cdot\mbox{$\varphi$}_{s})\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})$
$\displaystyle\qquad-\frac{5}{2}(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$\varphi$}_{s},\mbox{$\varphi$}_{s})-5(\lvert\mbox{$\varphi$}_{s}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})$

On the other hand, since $E_{1}$ is conserved, we have

$\displaystyle E_{1}(\mbox{$v$}(t))=E_{1}(\mbox{$v$}_{0})$
$\displaystyle=E_{1}(\mbox{$v$}^{R}+\mbox{$\varphi$}_{0})$
$\displaystyle=E_{1}(\mbox{$v$}^{R})+E_{1}(\mbox{$\varphi$}_{0})+R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$}_{0}),$

and combining the two equalities, we have

$\displaystyle
E_{1}(\mbox{$\varphi$}(t))=E_{1}(\mbox{$\varphi$}_{0})-R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$})+R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$}_{0}).$

We estimate $R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$})$ in detail and omit the
details for $R_{1}(\mbox{$v$}^{R},\mbox{$\varphi$}_{0})$ since they are the
same. The terms that are linear with respect to $\varphi$ can be estimated as
follows. First we have

$\displaystyle
2(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})=-\frac{2}{R^{2}}(\mbox{$v$}^{R},\mbox{$\varphi$}_{ss})$
$\displaystyle=\frac{2}{R^{2}}(\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})-\frac{2}{R^{2}}\big{[}\mbox{$v$}^{R}\cdot\mbox{$\varphi$}_{s}\big{]}^{L}_{s=0}$
$\displaystyle=-\frac{2}{R^{2}}(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$})-\frac{2}{R^{2}}\big{[}\mbox{$v$}^{R}\cdot\mbox{$\varphi$}_{s}\big{]}^{L}_{s=0}$
$\displaystyle=\frac{2}{R^{4}}(\mbox{$v$}^{R},\mbox{$\varphi$})-\frac{2}{R^{2}}\big{[}\mbox{$v$}^{R}\cdot\mbox{$\varphi$}_{s}\big{]}^{L}_{s=0}.$

Integrating equation (3.30) with respect to $s$ shows that
$(\mbox{$v$}^{R},\mbox{$\varphi$})=-\frac{1}{2}\|\mbox{$\varphi$}\|^{2}$.
Furthermore, differentiating equation (3.30) with respect to $s$ and taking
the trace $s=0,L$ yields
$\mbox{$v$}^{R}\cdot\mbox{$\varphi$}_{s}\rvert_{s=0,L}=0$. Hence we have

$\displaystyle
2(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})=-\frac{1}{R^{4}}\|\mbox{$\varphi$}\|^{2}.$

Similarly, we have

$\displaystyle-5(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})=-\frac{5}{R^{2}}(\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})=\frac{5}{2R^{4}}\|\mbox{$\varphi$}_{s}\|^{2},$

where $\lvert\mbox{$v$}^{R}_{s}\rvert\equiv\frac{1}{R}$ was substituted.
Taking into account the above two equalities, we have

$\displaystyle\lvert
2(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})-5(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})\rvert\leq
C\|\mbox{$\varphi$}\|_{1}^{2}\leq C\|\mbox{$\varphi$}_{0}\|_{1}^{2},$

where Corollary 3.6 was applied in the last inequality.

For the other terms, we have

$\displaystyle\big{\lvert}-5((\mbox{$v$}^{R}_{s}\cdot\mbox{$\varphi$}_{s})\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})-\frac{5}{2}(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$\varphi$}_{s},\mbox{$\varphi$}_{s})$
$\displaystyle-5(\lvert\mbox{$\varphi$}_{s}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})\big{\rvert}$
$\displaystyle\leq
C\big{(}\|\mbox{$\varphi$}_{s}\|^{2}+\|\mbox{$\varphi$}_{s}\|^{2}\|\mbox{$\varphi$}_{s}\|_{1}\big{)}$
$\displaystyle\leq
C\big{(}\|\mbox{$\varphi$}_{0}\|_{1}^{2}+\|\mbox{$\varphi$}_{0}\|_{1}^{3}\big{)}$

where $C>0$ depends only on $R$. Corollary 3.6 and the interpolation
inequality yield

$\displaystyle E_{1}(\mbox{$\varphi$}(t))$
$\displaystyle\geq\|\mbox{$\varphi$}_{ss}\|^{2}-C\|\mbox{$\varphi$}_{ss}\|\|\mbox{$\varphi$}_{s}\|^{3}-C\|\mbox{$\varphi$}_{s}\|^{2}$
$\displaystyle\geq
c_{1}\|\mbox{$\varphi$}_{ss}\|^{2}-C\|\mbox{$\varphi$}_{s}\|^{2}-C\|\mbox{$\varphi$}_{s}\|^{6}$
$\displaystyle\geq
c_{1}\|\mbox{$\varphi$}_{ss}\|^{2}-C\|\mbox{$\varphi$}_{0}\|_{1}^{2}-C\|\mbox{$\varphi$}_{0}\|_{1}^{6}$

where $c_{1}>0$ depends only on $L$ and $C>0$ depends on $L$ and $R$.
Combining all of the obtained estimates yields

(3.33) $\displaystyle\|\mbox{$\varphi$}_{ss}(t)\|^{2}\leq
C\big{(}\|\mbox{$\varphi$}_{0}\|_{2}^{2}+\|\mbox{$\varphi$}_{0}\|_{2}^{6}\big{)}$

for all $t>0$. Here, $C>0$ depends on $L$ and $R$.

Similarly, if we set

$\displaystyle
E_{2}(\mbox{$v$}(t))=\|\mbox{$v$}_{sss}(t)\|^{2}-\frac{7}{2}\|\lvert\mbox{$v$}_{s}(t)\rvert\lvert\mbox{$v$}_{ss}(t)\rvert\|^{2}-14\|\mbox{$v$}_{s}(t)\cdot\mbox{$v$}_{ss}(t)\|^{2}+\frac{21}{8}\|\lvert\mbox{$v$}_{s}(t)\rvert^{3}\|^{2},$

we have

$\displaystyle E_{2}(\mbox{$v$}^{R}+\mbox{$\varphi$}_{0})$
$\displaystyle=E_{2}(\mbox{$v$}^{R})+E_{2}(\mbox{$\varphi$}_{0})+R_{2}(\mbox{$v$}^{R},\mbox{$\varphi$}_{0}),$
$\displaystyle E_{2}(\mbox{$v$}^{R}+\mbox{$\varphi$}(t))$
$\displaystyle=E_{2}(\mbox{$v$}^{R})+E_{2}(\mbox{$\varphi$}(t))+R_{2}(\mbox{$v$}^{R},\mbox{$\varphi$}(t)),$

and the conservation of $E_{2}$ implies that the above two are equal. Hence,

$\displaystyle
E_{2}(\mbox{$\varphi$}(t))=E_{2}(\mbox{$\varphi$}_{0})+R_{2}(\mbox{$v$}^{R},\mbox{$\varphi$}_{0})-R_{2}(\mbox{$v$}^{R},\mbox{$\varphi$}(t)).$

Here,

$\displaystyle R_{2}(\mbox{$v$}^{R},\mbox{$\varphi$}(t))$
$\displaystyle=2(\mbox{$v$}^{R}_{sss},\mbox{$\varphi$}_{sss})+2(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})+(\lvert\mbox{$v$}^{R}_{ss}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})$
$\displaystyle\qquad+8(\lvert\mbox{$v$}^{R}_{s}\rvert^{4}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})+N(\mbox{$v$}^{R},\mbox{$\varphi$}(t)),$

where $N(\mbox{$v$}^{R},\mbox{$\varphi$}(t))$ are terms that are nonlinear
with respect to $\varphi$. We estimate as follows. First we have

$\displaystyle
2(\mbox{$v$}^{R}_{sss},\mbox{$\varphi$}_{sss})=-\frac{2}{R^{2}}(\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{sss})=-\frac{2}{R^{2}}[\mbox{$v$}^{R}_{s}\cdot\mbox{$\varphi$}_{ss}]^{L}_{s=0}+\frac{2}{R^{2}}(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})$

Taking the exterior product of $\mbox{$v$}^{R}_{s}$ from the left with the
first equation in problem (3.29), we have

(3.34)
$\displaystyle\mbox{$v$}^{R}_{s}\times\mbox{$\varphi$}_{t}=\mbox{$v$}^{R}_{s}\times(\mbox{$\varphi$}\times\mbox{$\varphi$}_{ss})+\mbox{$v$}^{R}_{s}\times(\mbox{$\varphi$}\times\mbox{$v$}^{R}_{ss})+\mbox{$v$}^{R}_{s}\times(\mbox{$v$}^{R}\times\mbox{$\varphi$}_{ss}).$

Since $\mbox{$\varphi$}_{t}\rvert_{s=0,L}=\mbox{$0$}$, taking the trace at
$s=0,L$ in (3.34) yields

$\displaystyle(\mbox{$v$}^{R}_{s}\cdot\mbox{$\varphi$}_{ss})\mbox{$v$}^{R}\rvert_{s=0,L}=\mbox{$0$}.$

$\lvert\mbox{$v$}^{R}\rvert\equiv 1$ further implies that
$\mbox{$v$}^{R}_{s}\cdot\mbox{$\varphi$}_{ss}\rvert_{s=0,L}=0$. Hence, we have

$\displaystyle
2(\mbox{$v$}^{R}_{sss},\mbox{$\varphi$}_{sss})=\frac{2}{R^{2}}(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})=-\frac{1}{R^{6}}\|\mbox{$\varphi$}\|^{2}.$

Next we estimate

$\displaystyle
2(\lvert\mbox{$v$}^{R}_{s}\rvert^{2}\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})$
$\displaystyle=\frac{2}{R^{2}}(\mbox{$v$}^{R}_{ss},\mbox{$\varphi$}_{ss})=-\frac{1}{R^{6}}\|\mbox{$\varphi$}\|^{2},$
$\displaystyle(\lvert\mbox{$v$}^{R}_{ss}\rvert^{2}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})+8(\lvert\mbox{$v$}^{R}_{s}\rvert^{4}\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})$
$\displaystyle=\frac{9}{R^{4}}(\mbox{$v$}^{R}_{s},\mbox{$\varphi$}_{s})=-\frac{9}{2R^{4}}\|\mbox{$\varphi$}_{s}\|^{2},$

and finally,

$\displaystyle\lvert N(\mbox{$v$}^{R},\mbox{$\varphi$}(t))\rvert\leq
C\big{(}\|\mbox{$\varphi$}\|_{2}^{2}+\|\mbox{$\varphi$}\|_{2}^{5}\big{)}.$

Additionally,

$\displaystyle
E_{2}(\mbox{$\varphi$}(t))\geq\|\mbox{$\varphi$}_{sss}(t)\|^{2}-C\big{(}\|\mbox{$\varphi$}(t)\|_{2}^{2}+\|\mbox{$\varphi$}(t)\|_{2}^{6}\big{)}$

along with all the estimates previously obtained shows that

(3.35) $\displaystyle\|\mbox{$\varphi$}_{sss}(t)\|^{2}\leq
C\big{(}\|\mbox{$\varphi$}_{0}\|_{3}^{2}+\|\mbox{$\varphi$}_{0}\|_{3}^{6}\big{)}$

holds for all $t>0$, where $C>0$ depends on $L$ and $R$. Estimates (3.33) and
(3.35) combined proves Lemma 3.7. ∎

In summary, we have the following.

######  Proposition 3.8.

For $R>L/\pi$, the arc-shaped solution $\mbox{$v$}^{R}$ of problem (1.6) is
stable in the following sense. For $\mbox{$\varphi$}_{0}\in H^{3}(I_{L})$
satisfying Assumption 3.4, the perturbed arc-shaped solution $\mbox{$v$}\in
C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ satisfies

$\displaystyle\|\mbox{$v$}(t)-\mbox{$v$}^{R}\|_{3}\leq
C\big{(}\|\mbox{$\varphi$}_{0}\|_{3}+\|\mbox{$\varphi$}_{0}\|_{3}^{3}\big{)}$

for all $t>0$. Here, $C>0$ depends on $L$ and $R$.

####  3.3.4 Stability Estimates and the generalized Hasimoto transformation

In this section, we show that stability estimates for solutions of problem
(1.6) transfer to stability estimates for solutions of problem (1.10) through
the generalized Hasimoto transformation.

Let, $\mbox{$v$}_{1}$ and $\mbox{$v$}_{2}$ be solutions of problem (1.6)
belonging to $C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);$ $H^{1}(I_{L})\big{)}$ with possibly different initial
datum, but the same boundary datum. Additionally, suppose that there exists
$C_{0}>0$ independent of $\mbox{$v$}_{1}$ and $\mbox{$v$}_{2}$ and $M\geq 0$
such that

(3.36) $\displaystyle\|\mbox{$v$}_{1}(t)-\mbox{$v$}_{2}(t)\|_{3}\leq C_{0}M$

holds for all $t>0$. Define $\mbox{$e$}^{1}$ and $\mbox{$e$}^{2}$ as the
solution of

(3.39)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$e$}^{1}_{s}=-(\mbox{$v$}_{1s}\cdot\mbox{$e$}^{1})\mbox{$v$}_{1},&s\in
I_{L},\\\\[8.53581pt]
\mbox{$e$}^{1}\rvert_{s=0}=\mbox{$e$}_{2},&\end{array}\right.$

and

(3.42)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{$e$}^{2}_{s}=-(\mbox{$v$}_{2s}\cdot\mbox{$e$}^{2})\mbox{$v$}_{2},&s\in
I_{L},\\\\[8.53581pt]
\mbox{$e$}^{2}\rvert_{s=0}=\mbox{$e$}_{2},&\end{array}\right.$

respectively, and define $\mbox{$w$}^{1}$ and $\mbox{$w$}^{2}$ by

(3.43)
$\displaystyle\mbox{$w$}^{1}=\mbox{$v$}_{1}\times\mbox{$e$}^{1}\qquad\text{and}\qquad\mbox{$w$}^{2}=\mbox{$v$}_{2}\times\mbox{$e$}^{2}.$

Again, note that $\mbox{$e$}^{i}$ and $\mbox{$w$}^{i}$ depend on $t$ through
$\mbox{$v$}_{i}$. Similarly to Section 3.3.1, estimating
$\mbox{$e$}^{1}-\mbox{$e$}^{2}$ and $\mbox{$w$}^{1}-\mbox{$w$}^{2}$ using
(3.39), (3.42), and (3.43) yields

(3.44) $\displaystyle\|\mbox{$e$}^{1}(t)-\mbox{$e$}^{2}(t)\|_{3}$
$\displaystyle\leq CM,$ (3.45)
$\displaystyle\|\mbox{$w$}^{1}(t)-\mbox{$w$}^{2}(t)\|_{3}$ $\displaystyle\leq
CM,$

for $t>0$. Here, $C>0$ depends on $C_{0}$ and
$\|\mbox{$v$}_{1s}\|_{2}+\|\mbox{$v$}_{2s}\|_{2}$ and is non-decreasing with
respect to $C_{0}>0$ and $\|\mbox{$v$}_{1s}\|_{2}+\|\mbox{$v$}_{2s}\|_{2}$

Since $\lvert\mbox{$v$}_{i}\rvert^{2}\equiv 1$ $(i=1,2)$, we have the
decomposition

(3.46)
$\displaystyle\mbox{$v$}_{1s}=\psi^{1}_{1}\mbox{$e$}^{1}+\psi^{1}_{2}\mbox{$w$}^{1},$
(3.47)
$\displaystyle\mbox{$v$}_{2s}=\psi^{2}_{1}\mbox{$e$}^{2}+\psi^{2}_{2}\mbox{$w$}^{2},$

for some $\psi^{i}_{j}(s,t)$ $(i,j=1,2)$. Furthermore, from equations (3.46)
and (3.47), along with (3.39), (3.42), and (3.43), we have

(3.48) $\displaystyle\|\mbox{$v$}_{1s}(t)\|_{2}+\|\mbox{$v$}_{2s}(t)\|_{2}\leq
C\big{(}\sum_{i,j=1}^{2}\|\psi^{i}_{j}(t)\|_{2}+\sum_{i,j=1}^{2}\|\psi^{i}_{j}(t)\|_{2}^{3}\big{)},$

where $C>0$ is independent of $\psi^{i}_{j}\ (i,j=1,2)$. Substituting (3.46)
and (3.47) into the first equation of (3.39) and (3.42) yields

$\displaystyle(\psi^{1}_{1}-\psi^{2}_{1})\mbox{$v$}_{1}$
$\displaystyle=-(\mbox{$e$}^{1}-\mbox{$e$}^{2})_{s}+\psi^{2}_{1}(\mbox{$v$}_{2}-\mbox{$v$}_{1}),$
$\displaystyle(\psi^{1}_{2}-\psi^{2}_{2})\mbox{$v$}_{1}$
$\displaystyle=-(\mbox{$w$}^{1}-\mbox{$w$}^{2})_{s}+\psi^{2}_{2}(\mbox{$v$}_{2}-\mbox{$v$}_{1}),$

and taking the inner product of the above two equations with $\mbox{$v$}_{1}$
yields

(3.49) $\displaystyle(\psi^{1}_{1}-\psi^{2}_{1})$
$\displaystyle=-(\mbox{$e$}^{1}-\mbox{$e$}^{2})_{s}\cdot\mbox{$v$}_{1}+\psi^{2}_{1}\big{(}(\mbox{$v$}_{2}-\mbox{$v$}_{1})\cdot\mbox{$v$}_{1}\big{)},$
(3.50) $\displaystyle(\psi^{1}_{2}-\psi^{2}_{2})$
$\displaystyle=-(\mbox{$w$}^{1}-\mbox{$w$}^{2})_{s}\cdot\mbox{$v$}_{1}+\psi^{2}_{2}\big{(}(\mbox{$v$}_{2}-\mbox{$v$}_{1})\cdot\mbox{$v$}_{1}\big{)}.$

We see from direct estimates based on equations (3.49) and (3.50) that

$\displaystyle\|\psi^{1}_{1}(t)-\psi^{2}_{1}(t)\|_{2}\leq C_{1}M,$
$\displaystyle\|\psi^{1}_{2}(t)-\psi^{2}_{2}(t)\|_{2}\leq C_{1}M$

holds for all $t>0$, where $C_{1}>0$ depends on $C_{0}$ and
$\sum_{i,j=1}^{2}\|\psi^{i}_{j}(t)\|_{2}$ and is non-decreasing with respect
to $C_{0}>0$ and $\sum_{i,j=1}^{2}\|\psi^{i}_{j}(t)\|_{2}$. Here, (3.44),
(3.45), and (3.48) was also utilized.

Estimate (3.1) along with estimate (3.5) implies that there exists
$C_{\ast}>0$ such that $\sum_{i,j=1}^{2}\|\psi^{i}_{j}(t)\|_{2}\leq C_{\ast}$
holds for all $t>0$. Here, $C_{\ast}=C_{\ast}(K)$ depends on
$K=\|\mbox{$v$}_{1}(0)\|_{3}+\|\mbox{$v$}_{2}(0)\|_{3}$ and is non-decreasing
with respect to $K>0$. We briefly remark that estimate (3.1) was utilized in
[1] and is derived from the conserved quantities (3.31), (3.32), and

$\displaystyle\frac{{\rm d}}{{\rm d}t}\|\mbox{$v$}_{s}(t)\|=0,$

along with the fact that $\lvert\mbox{$v$}(s,t)\rvert=1$ for all $s\in I_{L}$
and $t>0$. Setting $\psi_{1}(s,t):=\psi^{1}_{1}(s,t)+{\rm i}\psi^{1}_{2}(s,t)$
and $\psi_{2}(s,t):=\psi^{2}_{1}(s,t)+{\rm i}\psi^{2}_{2}(s,t)$, we see that

$\displaystyle\|\psi_{1}(t)-\psi_{2}(t)\|_{2}\leq C_{2}M$

holds for all $t>0$, where $C_{2}>0$ depends on $C_{0}$ and $K$ and is non-
decreasing with respect to $C_{0}>0$ and $K>0$. Finally, setting

$\displaystyle q_{1}(s,t)$ $\displaystyle:=\psi_{1}(s,t)\exp\left\\{\frac{{\rm
i}}{2}\int^{t}_{0}\lvert\psi_{1}(0,\tau)\rvert^{2}\ d\tau\right\\},$
$\displaystyle q_{2}(s,t)$ $\displaystyle:=\psi_{2}(s,t)\exp\left\\{\frac{{\rm
i}}{2}\int^{t}_{0}\lvert\psi_{2}(0,\tau)\rvert^{2}\ d\tau\right\\},$

we have

$\displaystyle\|\psi_{1}(t)-\psi_{2}(t)\|_{2}$
$\displaystyle=\left\|\exp\left\\{\frac{{\rm
i}}{2}\int^{t}_{0}\big{(}\lvert\psi_{2}(0,\tau)\rvert^{2}-\lvert\psi_{1}(0,\tau)\rvert^{2}\big{)}\
d\tau\right\\}q_{1}(t)-q_{2}(t)\right\|_{2}$
$\displaystyle\geq\inf_{\theta\in\mathbf{R}}\|\exp\\{{\rm
i}\theta\\}q_{1}(t)-q_{2}(t)\|_{2}$

for all $t>0$. In summary, we have

######  Proposition 3.9.

Let $\mbox{$v$}_{1},\mbox{$v$}_{2}\in
C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ be solutions of problem (1.6) with
initial datum $\mbox{$v$}_{0,1},\mbox{$v$}_{0,2}\in H^{3}(I_{L})$,
respectively, and a common boundary datum $\mbox{$b$}\in\mathbf{S}^{2}$. Let
$q_{1},q_{2}\in C\big{(}[0,\infty);H^{2}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);L^{2}(I_{L})\big{)}$ be solutions of problem (1.10)
corresponding to $\mbox{$v$}_{1}$ and $\mbox{$v$}_{2}$ respectively, through
the generalized Hasimoto transformation. Furthermore, assume that there exists
$C_{0}>0$ independent of $\mbox{$v$}_{1}$ and $\mbox{$v$}_{2}$ and $M\geq 0$
such that

$\displaystyle\|\mbox{$v$}_{1}(t)-\mbox{$v$}_{2}(t)\|_{3}\leq C_{0}M$

holds for all $t>0$. Then, there exists $C_{\ast\ast}>0$ depending on $C_{0}$
and $\|\mbox{$v$}_{0,1}\|_{3}+\|\mbox{$v$}_{0,2}\|_{3}$ such that

$\displaystyle\inf_{\theta\in\mathbf{R}}\|\exp\\{{\rm
i}\theta\\}q_{1}(t)-q_{2}(t)\|_{2}\leq C_{\ast\ast}M$

holds for all $t>0$. Here, $C_{\ast\ast}>0$ is non-decreasing with respect to
$C_{0}>0$ and $\|\mbox{$v$}_{0,1}\|_{3}+\|\mbox{$v$}_{0,2}\|_{3}$.

Proposition 3.9 with $M\geq 0$ being small implies that Lyapunov stability
estimates for problem (1.6) translates to orbital stability estimates for
problem (1.10) through the generalized Hasimoto transformation.

####  3.3.5 Final Step of the Proof of Theorem 2.5

We combine all of the results of the preceding sections to finish the proof of
Theorem 2.5.

Let $q_{R}(t)=-\frac{1}{R}\exp\\{-\frac{{\rm i}t}{2R^{2}}\\}$ with $R>L/\pi$
be the plane wave solution of problem (1.10) and $\phi_{0}\in H^{2}(I_{L})$
satisfy the $0$-th order compatibility condition for (1.10). Let
$\mbox{$v$}^{R}$ be the arc-shaped solution of problem (1.6) which corresponds
to $q_{R}$ through the inverse generalized Hasimoto transformation as shown in
Section 3.3.2. Recall that $\mbox{$v$}^{R}=\mbox{$v$}^{R}(s)$ is a stationary
solution. Furthermore, let $\mbox{$v$}_{0}\in H^{3}(I_{L})$ be the initial
datum corresponding to $q_{R}(0)+\phi_{0}$ and set
$\mbox{$\varphi$}_{0}:=\mbox{$v$}_{0}-\mbox{$v$}^{R}$. Finally, let
$\mbox{$v$}\in C\big{(}[0,\infty);H^{3}(I_{L})\big{)}\cap
C^{1}\big{(}[0,\infty);H^{1}(I_{L})\big{)}$ be the solution of problem (1.6)
with initial datum $\mbox{$v$}_{0}$ and boundary datum
$\mbox{$b$}=\mbox{$v$}^{R}(L)$ and set
$\mbox{$\varphi$}:=\mbox{$v$}-\mbox{$v$}^{R}$. From the results of Section
3.3.1, we have

(3.51)
$\displaystyle\|\mbox{$\varphi$}_{0}\|_{3}=\|\mbox{$v$}_{0}-\mbox{$v$}^{R}\|_{3}\leq
C\big{(}\|\phi_{0}\|_{2}+\|\phi_{0}\|_{2}^{3}\big{)},$

where $C>0$ depends on $\|q_{R}(0)\|_{2}+\|\phi_{0}\|_{2}$ and is non-
decreasing with respect to $\|q_{R}(0)\|_{2}+\|\phi_{0}\|_{2}$. Note that
$q_{R}$ doesn’t depend on $s$, so that $C$ essentially depends on $R$ and
$\|\phi_{0}\|_{2}$, and is non-decreasing with respect to $\|\phi_{0}\|_{2}$.
From the above estimate and Proposition 3.8, there exists $C_{3}>0$, which
depends on $L$, $R$, and $\|\phi_{0}\|_{2}$, such that

$\displaystyle\|\mbox{$v$}(t)-\mbox{$v$}^{R}\|_{3}\leq
C_{3}\big{(}\|\phi_{0}\|_{2}+\|\phi_{0}\|^{9}_{2}\big{)}$

holds for all $t>0$. Proposition 3.9 further shows that there exists $C_{4}>0$
depending on $L$, $R$, $\|\phi_{0}\|_{2}$, and
$\|\mbox{$v$}^{R}\|_{3}+\|\mbox{$v$}_{0}\|_{3}$ such that

$\displaystyle\inf_{\theta\in\mathbf{R}}\|\exp\\{{\rm
i}\theta\\}q(t)-q_{R}(t)\|_{2}\leq
C_{4}\big{(}\|\phi_{0}\|_{2}+\|\phi_{0}\|^{9}_{2}\big{)}$

holds for all $t>0$, and $C_{4}>0$ is non-decreasing with respect to
$\|\phi_{0}\|_{2}$ and $\|\mbox{$v$}^{R}\|_{3}+\|\mbox{$v$}_{0}\|_{3}$. Again,
note that $\|\mbox{$v$}^{R}\|_{3}$ is essentially a constant depending on $R$.
From inequality (3.51) we see that

$\displaystyle\|\mbox{$v$}_{0}\|_{3}\leq
C(1+\|\phi_{0}\|_{2}+\|\phi_{0}\|_{2}^{3}),$

where $C>0$ depends on $R$ and $\|\phi_{0}\|_{2}$, and is non-decreasing with
respect to $\|\phi_{0}\|_{2}$. Hence, $C_{4}>0$ can be chosen uniformly with
respect to $\phi_{0}$ satisfying $\|\phi_{0}\|_{2}\leq 1$.

Finally, for an arbitrary $\varepsilon>0$, choose $0<\delta<1$ small enough
such that $C_{4}\big{(}\delta+\delta^{9}\big{)}<\varepsilon$ holds. Then, if
$\|\phi_{0}\|_{2}\leq\delta$,

$\displaystyle\inf_{\theta\in\mathbf{R}}\|\exp\\{{\rm
i}\theta\\}q(t)-q_{R}(t)\|_{2}<\varepsilon$

holds for all $t>0$, and this finishes the proof of Theorem 2.5.

$\Box$

## Acknowledgement

This work was supported in part by JSPS Grant-in-Aid for Early-Career
Scientists grant number 20K14348.

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