# The helical vortex filaments of Ginzburg-Landau system in ${\mathbb{R}}^{3}$

Lipeng Duan  Lipeng Duan,
School of Mathematics and Information Science, Guangzhou University,
Guangzhou 510006, P. R. China<EMAIL_ADDRESS>,  Qi Gao  Qi Gao,
Department of Mathematics, School of Science, Wuhan University of Technology,
Wuhan 430070, P. R. China<EMAIL_ADDRESS>and  Jun Yang§ Jun Yang,
School of Mathematics and Information Science, Guangzhou University,
Guangzhou 510006, P. R. China<EMAIL_ADDRESS>
###### Abstract.

We consider the following coupled Ginzburg-Landau system in ${\mathbb{R}}^{3}$

$\displaystyle\begin{cases}-\epsilon^{2}\Delta
w^{+}+\Big{[}A_{+}\big{(}|w^{+}|^{2}-{t^{+}}^{2}\big{)}+B\big{(}|w^{-}|^{2}-{t^{-}}^{2}\big{)}\Big{]}w^{+}=0,\\\\[8.53581pt]
-\epsilon^{2}\Delta
w^{-}+\Big{[}A_{-}\big{(}|w^{-}|^{2}-{t^{-}}^{2}\big{)}+B\big{(}|w^{+}|^{2}-{t^{+}}^{2}\big{)}\Big{]}w^{-}=0,\end{cases}$

where $w=(w^{+},w^{-})\in\mathbb{C}^{2}$ and the constant coefficients satisfy

$A_{+},A_{-}>0,\quad B^{2}<A_{+}A_{-},\quad
t^{\pm}>0,\quad{t^{+}}^{2}+{t^{-}}^{2}=1.$

If $B<0$, then for every $\epsilon$ small enough, we construct a family of
entire solutions $w_{\epsilon}(\tilde{z},t)\in\mathbb{C}^{2}$ in the
cylindrical coordinates $(\tilde{z},t)\in\mathbb{R}^{2}\times\mathbb{R}$ for
this system via the approach introduced by J. Dávila, M. del Pino, M. Medina
and R. Rodiac in arXiv:1901.02807. These solutions are $2\pi$-periodic in $t$
and have multiple interacting vortex helices. The main results are the
extensions of the phenomena of interacting helical vortex filaments for the
classical (single) Ginzburg-Landau equation in $\mathbb{R}^{3}$ which has been
studied in arXiv:1901.02807. Our results negatively answer the Gibbons
conjecture [10] for the Allen-Cahn equation in Ginzburg-Landau system version,
which is an extension of the question originally proposed by H. Brezis.

Keywords: Ginzburg-Landau model, Helical Vortex Filaments, Lyapunov-Schmidt
reduction

AMS Subject Classification: 35A01, 35B25, 35Q56.

Corresponding author: Jun Yang<EMAIL_ADDRESS>
##  1\. introduction

In this paper, we study the entire solutions of the following Ginzburg-Landau
system in $\mathbb{R}^{N}$, for complex vector-valued functions
$w=(w^{+},w^{-}):\mathbb{R}^{N}\rightarrow\mathbb{C}^{2}$:

$\displaystyle\begin{cases}-\epsilon^{2}\Delta_{{\mathbb{R}}^{N}}w^{+}+\Big{[}A_{+}\big{(}|w^{+}|^{2}-{t^{+}}^{2}\big{)}+B\big{(}|w^{-}|^{2}-{t^{-}}^{2}\big{)}\Big{]}w^{+}=0,\\\\[8.53581pt]
-\epsilon^{2}\Delta_{{\mathbb{R}}^{N}}w^{-}+\Big{[}A_{-}\big{(}|w^{-}|^{2}-{t^{-}}^{2}\big{)}+B\big{(}|w^{+}|^{2}-{t^{+}}^{2}\big{)}\Big{]}w^{-}=0,\end{cases}$
(1.1)

here $A_{\pm}>0$, $t^{\pm}$, $B$ and $\epsilon>0$ are parameters, the notation
$\Delta_{{\mathbb{R}}^{N}}$ is the Laplacian operator in $\mathbb{R}^{N}$.
Throughout the paper we make the following assumptions concerning the
constants appearing in (1.1):

$\displaystyle A_{+},A_{-}>0,\ B^{2}<A_{+}A_{-},$ (H1) $\displaystyle\
t^{\pm}>0,\ \text{and}~{}{t^{+}}^{2}+{t^{-}}^{2}=1.$ (1.2)

The corresponding energy functional to system (1.1) is

$\displaystyle{\mathbf{E}}(w)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla w|^{2}\
+\ $
$\displaystyle\frac{1}{4\epsilon^{2}}\int_{\mathbb{R}^{N}}\Big{[}\,A_{+}(|w^{+}|^{2}-{t^{+}}^{2})^{2}+A_{-}(|w^{-}|^{2}-{t^{-}}^{2})^{2}$
$\displaystyle\qquad\quad+2B(|w^{+}|^{2}-{t^{+}}^{2})(|w^{-}|^{2}-{t^{-}}^{2})\,\Big{]}.$

The hypothesis (H1) ensures that the potential term in the energy is positive
definite and attains its minimum when $|w^{\pm}|=t^{\pm}$. From the previous
work about the classical Ginzburg-Landau equation in [9], we would look for
solutions which satisfy

$\displaystyle\int_{\mathbb{R}^{N}}\Big{[}\,A_{+}(|w^{+}|^{2}-{t^{+}}^{2})^{2}+A_{-}(|w^{-}|^{2}-{t^{-}}^{2})^{2}+2B(|w^{+}|^{2}-{t^{+}}^{2})(|w^{-}|^{2}-{t^{-}}^{2})\,\Big{]}<+\infty.$

Therefore the function $w=(w^{+},w^{-})$ satisfies the superconducting
boundary condition

$\displaystyle|w|^{2}=|w^{+}|^{2}+|w^{-}|^{2}\rightarrow{t^{+}}^{2}+{t^{-}}^{2}=1,\quad\text{when}\quad|x|\rightarrow+\infty.$
(1.3)

When $N=2,3$ in (1.1), Ginzburg-Landau systems of this type have been
introduced in physical models of $p$-wave superconductors [26] and two-
component Bose-Einstein condensates (BEC) [3, 25]. When the coupled term $B$
vanishes, system (1.1) is decoupled and is precisely the classical (single-
component) Ginzburg-Landau equation

$\displaystyle\epsilon^{2}\Delta_{{\mathbb{R}}^{N}}u+(1-|u|^{2})u=0,\quad
u\in\mathbb{C},$ (1.4)

which arises in the theory of superconductivity and superfluids [20] without
external applied magnetic field. From the theory of superconductivity,
$|u|^{2}$ is proportional to the density of superconducting electrons, i.e,
$|u|^{2}\approx 1$ corresponds to the superconducting state and
$|u|^{2}\approx 0$ corresponds to the normal state. The zero set of $u$, which
the superconductivity is not present, is said to be vortex set or vorticity.
The solutions with vortex lines or vortex filaments to the system (1.1) are
the main objects in this paper.

###  1.1. The case $B=0$

For the classical Ginzburg-Landau equation in a smooth bounded domain
$\Omega\subset\mathbb{R}^{N}$,

$\displaystyle\begin{cases}\epsilon^{2}\Delta
u_{\epsilon}+(1-|u_{\epsilon}|^{2})u_{\epsilon}=0\quad\text{in}~{}\Omega\subset\mathbb{R}^{N},\quad
u_{\epsilon}\in\mathbb{C},\\\\[8.53581pt]
u_{\epsilon}=f\quad\text{on}\quad\partial\Omega,\end{cases}$ (1.5)

the solutions with vortices have been widely studied in the past years.

For $N=2$, F. Bethuel, H. Brezis, and F. Hélein [7] considered the asymptotic
behavior of $u_{\epsilon}$ with a boundary condition
$f:\partial\Omega\rightarrow\mathbb{S}^{1}$ of degree $k\geq 1$. They showed
that there exists a subsequence $\\{\epsilon_{j}\\}$ and exactly $k$ points
$b_{1},\cdots b_{k}$ in $\Omega$ such that

$\displaystyle u_{\epsilon_{j}}(x)\rightarrow{\bf
u}=e^{i\varphi(x)}\prod_{l=1}^{k}\frac{x-b_{l}}{|x-b_{l}|}\quad\text{in}~{}C_{loc}(\Omega\setminus\\{b_{1},\cdots
b_{k}\\}),\quad\text{as $\epsilon_{j}\rightarrow 0$},$ (1.6)

where $\varphi$ is a harmonic function satisfying
$\varphi|_{\partial\Omega}=f.$ Furthermore, they also showed that the
$k$-tuple $(b_{1},\cdots b_{k})$ globally minimizes the renormalized energy
functional  $\mathcal{W}(b_{1},\cdots b_{k})$ which is characterised by

$\displaystyle\mathcal{W}(b_{1},\cdots b_{k})=:\lim_{r\rightarrow
0}\int_{\Omega\setminus\cup_{l=1}^{k}B_{r}(b_{l})}\Big{[}|\nabla{\bf
u}|^{2}-k\pi|\ln r|\Big{]}.$ (1.7)

And in [7], an implicit expression of $\mathcal{W}(b_{1},\cdots b_{k})$ in
terms of Green’s function was given, see also [15, 36] for similar results.
One better approximation than ${\bf u}$ in (1.6) is

$e^{i\varphi(x)}\prod_{l=1}^{k}\omega\Big{(}\frac{x-b_{l}}{\epsilon}\Big{)},$
(1.8)

where $\omega$ is the standard degree $+1$ vortex solution of the following
Ginzburg-Landau equation in $\mathbb{R}^{2}$

$\Delta u+(1-|u|^{2})u=0,\quad u\in\mathbb{C}.$ (1.9)

In other words, equation (1.9) has a unique solution of the form

$\omega(\tilde{z})=e^{i\theta}U(r),\quad\tilde{z}=re^{i\theta},\quad
U\in\mathbb{R},$ (1.10)

where $U>0$ is a solution of

$\left\\{\begin{aligned}
&U^{\prime\prime}+\frac{U^{\prime}}{r}-\frac{U}{r^{2}}+(1-U^{2})U=0{\quad\hbox{in
}}(0,\infty),\\\ &U(0^{+})=0,\quad U(+\infty)=1,\end{aligned}\right.$

see [11, 21]. On the other hand, a natural question is to find solutions of
(1.5) with vortices at other critical points of the renormalized energy
functional $\mathcal{W}(b_{1},\cdots b_{k})$ (see [6, 17, 28, 29, 30, 33] and
the references therein for researchs on this problem). In [33], F. Pacard and
T. Rivière constructed solutions of (1.5) with vortices of combined degree
$\pm 1$ and showed that the vortex points converge to non-degenerate critical
point of corresponding renormalized energy. In [17], M. del Pino, M. Kowalczyk
and M. Musso considered the Ginzburg-Landau equation (1.5) with added zero
Neumann boundary condition and constructed solutions by using Lyapunov-Schmidt
reduction without non-degenerate assumption on the critical point of
renormalized energy.

In the high dimension $N\geq 3$, the locations of vortices to the classical
Ginzburg-Landau with suitable boundary conditions and energy levels do not
occur at points, but along a generalized minimal sub-manifold structure with
co-dimension $2$, which is naturally interpreted as “vortex sub-manifold”. In
$\mathbb{R}^{3}$, the structure of vortices should typically be the form of
curves which are called vortex filaments. In the work [16], M. del Pino and M.
Kowalczyk studied (1.5) in a cylinder $\Omega=B_{R}(0)\times(0,2\pi)$ in
$\mathbb{R}^{3}$ and formally gave an approximation of solution to (1.5) with
helical vortex filaments structure. Moreover, they conjectured in [16] that
(1.5) has a solution $u_{\epsilon{\bf g}}$ of the form

$\displaystyle u_{\epsilon{\bf g}}\approx
e^{i\varphi(\tilde{z},t)}\prod_{j=1}^{k}\omega\Big{(}\frac{\tilde{z}-g_{j}(t)}{\epsilon}\Big{)},\quad(\tilde{z},t)\in\mathbb{R}^{3},$
(1.11)

where $\varphi$ is a harmonic function satisfying the boundary condition and
${\bf g}=(g_{1},\cdots,g_{k})$ represents $k$ curves :
$t\rightarrow\big{(}g_{j}(t),t\big{)},j\leq k$. For simplicity, the authors
assumed that the $k$ curves are $2\pi$-periodic in $t$. The asymptotic
expansion of energy functional $E_{0\epsilon}(u_{\epsilon{\bf g}})$ of (1.5)
is

$I_{\epsilon}({\bf g}):=E_{0\epsilon}(u_{\epsilon{\bf g}})\approx 2\pi\times
k\pi|\log\epsilon|+\mathcal{I}_{\epsilon}({\bf g}),$

where

$E_{0\epsilon}(u)=\frac{1}{2}\int_{\Omega}|{\nabla}u|^{2}+\frac{1}{4\epsilon^{2}}\int_{\Omega}(1-|u|^{2})^{2},$

and

$\mathcal{I}_{\epsilon}(\mathbf{g}):=\pi\int_{0}^{2\pi}\Big{(}\,|\ln\epsilon|\frac{1}{2}\sum_{l=1}^{k}|g^{\prime}_{l}(t)|^{2}-\sum_{j\neq
l}\log|g_{j}(t)-g_{l}(t)|\,\Big{)}\,{\mathrm{d}}t.$

Therefore, the asymptotic expressions for equilibrium location in (1.11) are

$\displaystyle
g_{j}(t)\approx\frac{1}{\sqrt{|\ln\epsilon|}}\sqrt{k-1}e^{it}e^{2i(j-1)\pi/k},\
\ j=1,\cdots,k,$ (1.12)

and they can be obtained by solving the following Euler-Lagrange equations of
$\mathcal{I}_{\epsilon}(\mathbf{g})$

${\bf g}(t)=\frac{1}{\sqrt{|\ln\epsilon|}}\tilde{\bf g}(t),\quad\tilde{\bf
g}=(\tilde{g}_{1},\cdots,\tilde{g}_{k}),\quad-\tilde{g}_{l}^{\prime\prime}(t)=2\sum_{i\neq
l}\frac{\tilde{g}_{l}(t)-\tilde{g}_{i}(t)}{|\tilde{g}_{l}(t)-\tilde{g}_{i}(t)|^{2}}.$

Recently, J. Dávila, M. del Pino, M. Medina and R. Rodiac in [12] have
rigorously proved this conjecture in $\mathbb{R}^{3}$ by constructing a family
of entire solutions $u_{\epsilon}(\tilde{z},t)$ as $\epsilon\rightarrow 0$
with the form

$\displaystyle
u_{\epsilon}(\tilde{z},t)\approx\prod_{l=1}^{k}\omega\Big{(}\frac{\tilde{z}-g_{j}(t)}{\epsilon}\Big{)},\quad(\tilde{z},t)\in\mathbb{R}^{3}.$

Now we recall the main results provided in [12].

######  Theorem 1.1.

([12]). For every integer $k\geq 2$ and $\epsilon>0$ sufficiently small, there
exists a solution $u_{\epsilon}(\tilde{z},t)\in\mathbb{C}$ of (1.4) with
$N=3$, $2\pi$-periodic in the $t$-variable, with the following asymptotic
profile:

$u_{\epsilon}(\tilde{z},t)=\prod_{j=1}^{k}\omega\left(\frac{\tilde{z}-{g}^{\epsilon}_{j}(t)}{\epsilon}\right)+\varphi_{\epsilon}(\tilde{z},t),$

where ${g}_{j}^{\epsilon}(t)$ is $2\pi$-periodic with the asymptotic behavior
(1.12) and

$|\varphi_{\epsilon}(\tilde{z},t)|\,\leq\,\frac{C}{|\ln\epsilon|}.$

∎

It is worth pointing out that the solutions $u_{\epsilon}(\tilde{z},t)$
constructed in [12] satisfy

$\lim_{|\tilde{z}|\rightarrow+\infty}|u_{\epsilon}(\tilde{z},t)|=1\quad\text{uniformly
in}~{}t,$ (1.13)

and still depend on $t$. As we know that H. Brezis has postulated the
following Gibbons conjecture: whether a solution
$u_{\epsilon}(\tilde{z},t)\in\mathbb{C}$ of classical Ginzburg-Landau with the
uniform convergence condition (1.13) must necessarily be a function in
$\tilde{z}$. However, the results obtained in [12] actually negatively
answered the conjecture proposed by H. Brezis when $N=3$. The reader can refer
to [10], [19] for more information about the Gibbons conjecture.

###  1.2. The case $B\neq 0$

When $N=2$, with space rescaling, we can reduce the system (1.1) into the
following one:

$\displaystyle\begin{cases}-\Delta
w^{+}+\Big{[}A_{+}\big{(}|w^{+}|^{2}-{t^{+}}^{2}\big{)}+B\big{(}|w^{-}|^{2}-{t^{-}}^{2}\big{)}\Big{]}w^{+}=0,\\\\[8.53581pt]
-\Delta
w^{-}+\Big{[}A_{-}\big{(}|w^{-}|^{2}-{t^{-}}^{2}\big{)}+B\big{(}|w^{+}|^{2}-{t^{+}}^{2}\big{)}\Big{]}w^{-}=0,\end{cases}\quad\text{in}~{}\mathbb{R}^{2}.$
(1.14)

The two-component model (1.14) was studied by [1] and [2] in the “balanced”
case, $A_{+}=A_{-}$ and $t^{+}=t^{-}=1/\sqrt{2}$. S. Alama, L. Bronsard and P.
Mironescu in [1] and [2] proved the existence, uniqueness, monotonicity and
stability of radial solutions with symmetric vortices. Furthermore, under the
assumption (H1), S. Alama and Q. Gao obtained the existence, uniqueness,
asymptotic behaviors and quantization results of symmetric equivariant vortex
solution with degree pair $(n^{+},n^{-})$ in the form

$\displaystyle w_{n}^{+}(\ell,\theta)=W_{n}^{+}(\ell)e^{in^{+}\theta},\qquad
w_{n}^{-}(\ell,\theta)=W_{n}^{-}(\ell)e^{in^{-}\theta},$

to the system (1.14), where $(\ell,\theta)$ are the polar coordinates (see
[3]).

In order to give our results in a more precise way, we use the notation in the
rest of this paper to denote the radially symmetric solution with vortices of
degree pair $(n_{+},n_{-})=(+1,+1)$ as below:

$\displaystyle
w(\tilde{z})=\big{(}w^{+}(\tilde{z}),w^{-}(\tilde{z})\big{)}\quad\mbox{where}\quad
w^{\pm}(\tilde{z})=W^{\pm}(\ell)e^{i\theta},\quad~{}\tilde{z}=\ell
e^{i\theta}.$

Then we get the corresponding ODE system

$\displaystyle\begin{cases}-{W^{+}}^{\prime\prime}-\frac{1}{\ell}{W^{+}}^{\prime}+\frac{1}{\ell^{2}}{W^{+}}+\Big{[}\,A_{+}({W^{+}}^{2}-{t^{+}}^{2})+B({W^{-}}^{2}-{t^{-}}^{2})\,\Big{]}W^{+}=0,\vspace{3mm}\\\
-{W^{-}}^{\prime\prime}-\frac{1}{\ell}{W^{-}}^{\prime}+\frac{1}{\ell^{2}}{W^{-}}+\Big{[}\,A_{-}({W^{-}}^{2}-{t^{-}}^{2})+B({W^{+}}^{2}-{t^{+}}^{2})\,\Big{]}W^{-}=0,\end{cases}$
(1.15)

where $\ell\in[0,+\infty)$. The facts below about the properties of
$W=(W^{+},W^{-})$ are given in [3].

######  Lemma 1.2.

([3]). Suppose $W=(W^{+},W^{-})$ is a solution of the ODE system (1.15).Then
$W^{+}$ and $W^{-}$ satisfy the following:

$\displaystyle\begin{cases}&W^{\pm}\in C^{\infty}(0,\infty),\qquad
0<W^{\pm}<t^{\pm}~{}\text{for~{}all }~{}\ell>0,\\\\[5.69054pt]
&W^{\pm}\sim\ell~{}\text{as }~{}\ell\rightarrow 0,\qquad W^{\pm}\sim
t^{\pm}-\frac{c_{\pm}}{2\ell^{2}}~{}\text{as
}~{}\ell\rightarrow\infty,\\\\[5.69054pt]
&{W^{\pm}}^{\prime}>0~{}\text{when}~{}B<0,\qquad{W^{\pm}}^{\prime}\sim\frac{c_{\pm}}{\ell^{3}}~{}\text{as
}~{}\ell\rightarrow\infty,\end{cases}$ (1.16)

where

$c_{\pm}=\frac{A_{\mp}-B}{(A_{+}A_{-}-B^{2})t^{\pm}}.$

Moreover, $(W^{+},W^{-})$ is the unique solution of (1.15). ∎

The linear Ginzburg-Landau operator of (1.14) around the standard vortex
$w=(w^{+},w^{-})$ is given by

$L_{0}(\phi):=\big{(}L_{0}^{+},L_{0}^{-}\big{)}(\phi),$

with

$\displaystyle L_{0}^{\pm}(\phi)$
$\displaystyle=\Delta\phi^{\pm}+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-{W^{\pm}}^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-{W^{\mp}}^{2}\big{)}\Big{]}\phi^{\pm}$
$\displaystyle\quad-2A_{\pm}{\rm{Re}}\Big{(}w^{\pm}\overline{\phi^{\pm}}\Big{)}w^{\pm}\,-\,2B{\rm{Re}}\Big{(}w^{\mp}\overline{\phi^{\mp}}\Big{)}w^{\pm}.$
(1.17)

S. Alama and Q. Gao in [4] also studied the stability of $w(\tilde{z})$ in the
sense of spectrum of the linear operator $L_{0}(\phi)$ in a disk of
$\mathbb{R}^{2}$. Furthermore, in [18], L. Duan and J. Yang studied the
linearized operator $L_{0}$. Under an extra assumption $B<0$, they proved the
non-degeneracy result of degree pair $(+1,+1)$ vortex solution in a natural
Hilbert space $\mathscr{H}$ endowed with the norm $\|\cdot\|_{\mathscr{H}}$ in
the following form

$\displaystyle\|\phi\|_{\mathscr{H}}$
$\displaystyle=\int_{\mathbb{R}^{2}}(|{\nabla}\phi^{+}|^{2}+|{\nabla}\phi^{-}|^{2})+\int_{\mathbb{R}^{2}}\Big{[}A_{+}\big{(}{t^{+}}^{2}-{W^{+}}^{2}\big{)}-B\big{(}{t^{-}}^{2}-{W^{-}}^{2}\big{)}\Big{]}|\phi^{+}|^{2}$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}\Big{[}A_{+}\big{(}{t^{-}}^{2}-{W^{-}}^{2}\big{)}-B\big{(}{t^{+}}^{2}-{W^{+}}^{2}\big{)}\Big{]}|\phi^{-}|^{2}.$
(1.18)

For convenience of reader, we present the non-degeneracy result in the
following lemma.

######  Lemma 1.3.

([18]). Assume that (H1) and $B<0$ hold. Suppose that $L_{0}(\phi)=0$ for some
$\phi\in\mathscr{H}$. Then we have

$\phi=c_{1}\frac{\partial w}{\partial x_{1}}+c_{2}\frac{\partial w}{\partial
x_{2}},$

for certain constants $c_{1}$ and $c_{2}$. ∎

Note that we also get the non-degeneracy result for $w$ with the degree pair
$(-1,-1),(-1,1)$ or $(1,-1)$ in [18] under the same constraints. Furthermore,
in the Appendix, we show that the operator $L_{0}(\phi)$ does have one kernel
if $\phi$ enjoys some symmetry and decay assumptions, see Lemmas 8.1-8.2.

There are extensive researches on solutions with vortex structures to the
classical Ginzburg-Landau equation with suitable boundary conditions, see [8,
23, 27, 31, 34]. However, for the Ginzburg-Landau system, much less is known
about the vortices, especially for solutions with vortex filaments in the case
with higher dimension $N\geq 3$. The non-degeneracy arguments provided in [18]
for the vortex solutions with standard degree pair $(+1,+1)$ to the system
(1.14) make it possible to construct vortex solutions by using the singular
perturbation methods. Inspired by [12], we construct new solutions of (1.1)
with helical vortex filaments in $\mathbb{R}^{3}$. We now present the first
result as follows:

######  Theorem 1.4.

Assume that (H1) and $B<0$ hold. For any integer $k\geq 2$ and $\epsilon$
small enough, there exists a solution
$w_{\epsilon}(\tilde{z},t)=\big{(}w_{\epsilon}^{+}(\tilde{z},t),w_{\epsilon}^{-}(\tilde{z},t)\big{)}\in\mathbb{C}^{2}$
of (1.1) with $N=3$ in the form

$\displaystyle
w_{\epsilon}(\tilde{z},t)=\Bigg{(}(t^{+})^{1-k}\prod_{j=1}^{k}w^{+}\Big{(}\frac{\tilde{z}-g^{+}_{\epsilon,j}(t)}{\epsilon}\Big{)},\,(t^{-})^{1-k}\prod_{j=1}^{k}w^{-}\Big{(}\frac{\tilde{z}-g^{-}_{\epsilon,j}(t)}{\epsilon}\Big{)}\Bigg{)}+\phi_{\epsilon}(\tilde{z},t),$
(1.19)

where
$\phi_{\epsilon}(\tilde{z},t)=\big{(}\phi_{\epsilon}^{+}(\tilde{z},t),\phi_{\epsilon}^{-}(\tilde{z},t)\big{)}\in\mathbb{C}^{2}$
and all $g^{\pm}_{\epsilon,j}(t)$’s are $2\pi$ periodic in variable $t$ which
are given by

$\displaystyle
g^{\pm}_{\epsilon,j}(t)\approx\frac{1}{\sqrt{|\ln\epsilon|}}\sqrt{k-1}e^{it}e^{i\frac{2(j-1)\pi}{k}},\quad\text{for}~{}j=1,\cdots
k.$ (1.20)

Furthermore, we also have

$\displaystyle|\phi_{\epsilon}^{\pm}|\leq\frac{D}{|\ln\epsilon|},\quad\text{for
some constant $D>0$},$ (1.21)

and

$\displaystyle\lim_{|\tilde{z}|\rightarrow+\infty}|w_{\epsilon}(\tilde{z},t)|=\lim_{|\tilde{z}|\rightarrow+\infty}\sqrt{|w^{+}_{\epsilon}(\tilde{z},t)|^{2}+|w^{-}_{\epsilon}(\tilde{z},t)|^{2}}=1,\quad\text{uniformly
in}~{}t.$ (1.22)

∎

Note that, due to (1.22), the solutions given in Theorem 1.4 satisfies the
asymptotic property (1.3). Theorem 1.4 also gives a negative answer to an
analogue of Gibbons conjecture (or De Giorgi conjecture) for the Ginzburg-
Landau system.

In Theorem 1.4, we use the degree pair $(+1,+1)$ vortex solution $w$ as the
building block to construct solutions. Thus the solutions constructed in
Theorem 1.4 consist in $k$ vortex helices of degree pair $(+1,+1)$. Note that
in [18], the authors showed that $\hat{w}=(w^{+},\overline{w^{-}})$, the
degree pair $(+1,-1)$ vortex solution of (1.14), is non-degenerate. Taking
$\hat{w}$ as the building block and using the same method, under the same
conditions in Theorem 1.4 we can construct solution
$\hat{w}_{\epsilon}(\tilde{z},t)$ which consists in $k$ vortex helices of
degree pair $(1,-1)$ with the form

$\displaystyle\hat{w}_{\epsilon}(\tilde{z},t)\approx\Bigg{(}(t^{+})^{1-k}\prod_{j=1}^{k}w^{+}\Big{(}\frac{\tilde{z}-g^{+}_{\epsilon,j}(t)}{\epsilon}\Big{)},\,(t^{-})^{1-k}\prod_{j=1}^{k}\overline{w^{-}}\Big{(}\frac{\tilde{z}-g^{-}_{\epsilon,j}(t)}{\epsilon}\Big{)}\Bigg{)}.$
(1.23)

Analogous to the work in [12], we also consider solutions to (1.1) which
consist in $k\geq 4$ vortex helices of degree pair $(+1,+1)$ rotating around a
straight vortex filament of degree pair $(-1,-1)$. The result is as below.

######  Theorem 1.5.

Assume that (H1) and $B<0$ hold. For any integer $k\geq 4$, there exists
$\epsilon_{0}>0$ small such that for every $\epsilon<\epsilon_{0}$, there
exists a solution $w_{\epsilon}$ solving (1.1) with $N=3$. Furthermore, the
solution can be written in the form

$\displaystyle
w_{\epsilon}(\tilde{z},t)\approx\Bigg{(}(t^{+})^{-k}\,\overline{w^{+}}(\tilde{z})\prod_{j=1}^{k}w^{+}\Big{(}\frac{\tilde{z}-{\tilde{g}}^{+}_{\epsilon,j}(t)}{\epsilon}\Big{)},\,(t^{-})^{-k}\,\overline{w^{-}}(\tilde{z})\prod_{j=1}^{k}w^{-}\Big{(}\frac{\tilde{z}-{\tilde{g}}^{-}_{\epsilon,j}(t)}{\epsilon}\Big{)}\Bigg{)},$
(1.24)

where $\tilde{g}^{\pm}_{\epsilon,j}$ is $2\pi$ periodic in $t$ and satisfies
the asymptotic behavior

$\tilde{g}^{\pm}_{\epsilon,j}(t)\approx\frac{1}{\sqrt{|\ln\epsilon|}}\sqrt{k-3}e^{it}e^{i\frac{2(j-1)\pi}{k}},\quad\text{for}~{}j=1,\cdots,k.$

∎

Theorems 1.4 and 1.5 establish the existence of multiple helical vortex
filaments for Ginzburg-Landau system (1.1) in $\mathbb{R}^{3}$. In fact, the
existence of a solution with a multiple-helices vortex structure is an
interesting question. In classical fluid and Bose-Einstein condensates,
experiments and numerical methods show that the possible existence of
arbitrary number of interacting helical vortex filaments, see [5, 22]. The
reader can also refer to [13, 14, 24, 32, 37] for the constructions of
solutions with helical vortex structure, involving Schrödinger map equation,
Gross-Pitaevskii equation and Euler equation.

For convenience, we just provide the proof of Theorem 1.4 in the present paper
and Theorem 1.5 can be proved by an exactly similar way. Here is the outline
of procedure.

$\bullet$ Using the screw symmetry in (1.25), we reduce the original problem
(1.1) to a $2$-dimensional one, see (2.1). This step will be done in Section
2.1.

$\bullet$ Next, we will construct an approximation to the real solutions of
(2.1) which is denoted by $v_{d}=(v_{d}^{+},v_{d}^{-})$, see (2.9)-(2.10).

$\bullet$ Note that the coupling between the real part and imaginary part in
the linearized problem of Ginzburg-Landau system (1.14) around $v_{d}$ would
make the setting very complicated. We would use the idea in [17] to overcome
this obstacle. More precisely speaking, for a perturbation,

$\psi=(\psi^{+},\psi^{-})=(\psi^{+}_{1}+i\psi^{+}_{2},\psi^{-}_{1}+i\psi^{-}_{2})\in\mathbb{C}^{2}$

with symmetry (2.26), we take the solution in the form of (2.15). Then we
rewrite the original problem to a perturbation of the linearized problem (see
(2.22))

$\mathcal{L}_{\epsilon}\psi+{\mathcal{R}}+\mathcal{N}(\psi)=0,$

where
$\mathcal{L}_{\epsilon}=\Big{(}\mathcal{L}_{\epsilon}^{+},\mathcal{L}_{\epsilon}^{-}\Big{)}$
is the linearized operator,
${\mathcal{R}}=\Big{(}{\mathcal{R}}^{+},{\mathcal{R}}^{-}\Big{)}$ is the error
term, and
$\mathcal{N}(\psi)=\Big{(}\mathcal{N}^{+}(\psi),\mathcal{N}^{-}(\psi)\Big{)}$
is the nonlinear term of higher order. When far away from the vortices, the
linearized operator $\mathcal{L}_{\epsilon}\psi$ behaves as below

$\displaystyle\mathcal{L}_{\epsilon}\psi\approx\begin{cases}\Delta\psi^{+}-2iA_{\pm}|v_{d}^{+}|^{2}{\rm{Im}}(\psi^{+})-2iB|v_{d}^{-}|^{2}\rm{Im}(\psi^{-})\\\\[8.53581pt]
\Delta\psi^{-}-2iA_{\pm}|v_{d}^{+}|^{2}{\rm{Im}}(\psi^{-})-2iB|v_{d}^{-}|^{2}\rm{Im}(\psi^{+}).\end{cases}$

Although there are also coupling between the $``+"$ part and $``-"$ part, the
operator $\mathcal{L}_{\epsilon}\psi$ is good far away from vortices due to
the positive definite assumption (H1). On the other hand, when near the
vortices, from Lemma 2.2, Lemmas 8.1-8.2, we would deal with the problem by
setting up a projected nonlinear problem, see (4.1).

$\bullet$ Section 3 is devoted to estimating the error in a suitable weighted
norm.

$\bullet$ We study and solve the projected nonlinear problem (4.1) by using a
fixed point argument. This step is more or less standard and will be done in
Sections 4-6.

$\bullet$ In order to finish the proof of Theorem 1.4, in Section 7 the
reduction process will be applied to force the vanishing of Lagrange
multiplier $c$ in (4.1). In other words, we need to adjust the parameter
$\tilde{d}$ such that

${\rm
Re}\int_{B_{\mathbb{R}_{\epsilon}}(0)}\Bigg{(}i\overline{w^{\pm}_{x_{1}}}(z)w^{\pm}(z)\Big{[}{\mathcal{L}}^{\pm}_{\epsilon}(\psi)+{\mathcal{R}}^{\pm}+\mathcal{N}^{\pm}(\psi)\Big{]}(z+{\tilde{d}})\Bigg{)}=0.$

The first two terms

${\rm
Re}\int_{B_{\mathbb{R}_{\epsilon}}(0)}\Bigg{(}i\overline{w^{\pm}_{x_{1}}}(z)w^{\pm}(z){\mathcal{L}}^{\pm}_{\epsilon}(\psi)(z+{\tilde{d}})\Bigg{)}\quad\text{and}\quad{\rm
Re}\int_{B_{\mathbb{R}_{\epsilon}}(0)}\Bigg{(}i\overline{w^{\pm}_{x_{1}}}(z)w^{\pm}(z){\mathcal{R}}^{\pm}(z+{\tilde{d}})\Bigg{)}$

in the above identity would be of size $O(\epsilon|\ln\epsilon|)$, while the
third term

${\rm
Re}\int_{B_{\mathbb{R}_{\epsilon}}(0)}\Bigg{(}i\overline{w^{\pm}_{x_{1}}}(z)w^{\pm}(z)\mathcal{N}^{\pm}(\psi)(z+{\tilde{d}})\Bigg{)}$

is of size $O(|\ln\epsilon|^{-2})$. Hence, much more delicate analysis will be
provided to capture fine estimates. This will be bone by adopting the ideas
and approaches of Fourier decomposition to the error and nonlinear terms
introduced in [12, 13] to handle this problem, see Sections 3.2 and 4.2. In
fact, a crucial linear theory which gives more precise estimates for the
perturbation term $\psi$ will be established, see Proposition 4.2 in Section
4.2.

Notation and Definitions: We end this introduction by giving some notation and
definitions. For any complex-valued vector function
$F=(F^{+},F^{-})\in\mathbb{C}^{2}$, we define its conjugation

$\overline{F}=:(\overline{F^{+}},\,\overline{F^{-}}).$

Next, we give the definition of screw-symmetry in the coordinates
$(r,\theta,t)$ with $(\tilde{z},t)=(re^{i\theta},t)$.

######  Definition 1.6.

We say that a function
$u=(u^{+},u^{-}):\mathbb{R}^{3}\rightarrow\mathbb{C}^{2}$ has screw-symmetry
if

$u(r,\theta+h,t+h)=u(r,\theta,t),\quad\text{i.e.,}\quad
u^{\pm}(r,\theta+h,t+h)=u^{\pm}(r,\theta,t),$ (1.25)

for any $h\in\mathbb{R}$. ∎

##  2\. formulation of the problem

###  2.1. Reduction to a 2-dimensional problem by using screw-symmetry

By using the screw-symmetry, we will transform the problem (1.1), which is of
$3$ dimensional in nature, to a $2$ dimensional one. For simplicity, we only
treat the case $k=2$ in Theorem 1.4 and the arguments for general $k>2$ can be
easily adapted.

Notice that the condition (1.25) in Definition 1.6 is equivalent to

$u(r,\theta,t+h)=u(r,\theta-h,t)\mbox{ for any }h\in\mathbb{R}.$

Then a function with screw-symmetry can be expressed as a function of two
variables, i.e. for any
$(r,\theta,t)\in\mathbb{R}^{+}\times\mathbb{R}\times\mathbb{R}$,

$u(r,\theta,t)=u(r,\theta-t,0)=:{\tilde{u}}(r,\theta-t).$

We will apply these facts to make a setting to reduce the dimension.

Here is the observation in [12]. We can write the standard degree pair
$(+1,+1)$ vortex solution for (1.14) in polar coordinates $(\ell,\theta)$,
i.e.,

$w(\tilde{z})=\big{(}w^{+}(\tilde{z}),w^{-}(\tilde{z})\big{)}=\Big{(}W^{+}(r)e^{i\theta},W^{-}(r)e^{i\theta}\Big{)},$

and consider a function $w_{d}$ in the form

$\displaystyle w_{d}(\tilde{z},t)$
$\displaystyle=\Big{(}w_{d}^{+}(\tilde{z},t),w_{d}^{-}(\tilde{z},t)\Big{)},$
(2.1)

where

$\displaystyle
w_{d}^{\pm}(\tilde{z},t)=({t^{\pm}})^{-1}\prod_{j=1}^{2}w^{\pm}\left(\frac{{{\rm{Re}}(\tilde{z})}-d\cos\big{(}t+(j-1)\pi\big{)}}{\epsilon},\,\frac{{{\rm{Im}}(\tilde{z})}-d\sin\big{(}t+(j-1)\pi\big{)}}{\epsilon}\right).$
(2.2)

It can be checked that

$w_{d}(r,\theta,t+h)=\Big{(}e^{2ih}w^{+}_{d}(r,\theta-h,t),e^{2ih}w^{-}_{d}(r,\theta-h,t)\Big{)}=e^{2ih}w_{d}(r,\theta-h,t),$

for any $h$ in $\mathbb{R}$. That is, $w_{d}$ does not have screw-symmetry but
$\tilde{w}_{d}(r,\theta,t):=e^{-2it}w_{d}(r,\theta,t)$ fulfills the symmetry.

The above arguments suggest that we should look for a solution $w_{\epsilon}$
of (1.1) that can be written as

$\displaystyle
w_{\epsilon}(r,\theta,t)=\Big{(}w_{\epsilon}^{+}(r,\theta,t),w_{\epsilon}^{-}(r,\theta,t)\Big{)}=\Big{(}e^{2it}\tilde{w}^{+}(r,\theta,t),e^{2it}\tilde{w}^{-}(r,\theta,t)\Big{)}$
(2.3)

with $\tilde{w}^{\pm}$ being screw-symmetric. Thus

$\displaystyle\tilde{w}^{\pm}(r,\theta,t)=u^{\pm}(r,\theta-t),$ (2.4)

where $u^{\pm}:\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{C}$ is
$2\pi$-periodic in the second variable. Denoting

$u^{\pm}(r,s)=u^{\pm}(r,\theta-t)\quad\mbox{with }s:=\theta-t,$

we can see that

$\partial_{r}w_{1}^{\pm}=e^{2it}\partial_{r}u^{\pm}(r,s),\quad\partial^{2}_{{r}{r}}w_{1}^{\pm}=e^{2it}\partial^{2}_{rr}u^{\pm}(r,s),\,\quad\partial_{\theta}w_{1}^{\pm}=e^{2it}\partial_{s}u^{\pm}(r,s),\quad\partial^{2}_{\theta\theta}w_{1}^{\pm}=e^{2it}\partial^{2}_{ss}u^{\pm}(r,s),$

$\partial_{t}w_{1}^{\pm}=[2iu^{\pm}-\partial_{s}u^{\pm}]e^{2it},\ \ \
\partial^{2}_{tt}w_{1}^{\pm}=[\partial^{2}_{ss}u^{\pm}-4i\partial_{s}u^{\pm}-4u^{\pm}]e^{2it}.$

Recalling the expression of the Laplacian operator in cylindrical coordinates
as below

$\partial^{2}_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{{r}^{2}}\partial^{2}_{\theta\theta}+\partial^{2}_{tt},$

we conclude that $w_{\epsilon}$ is a solution of (1.1) if and only if
$u=\big{(}u^{+},u^{-}\big{)}$ is a solution of the following equations

$\displaystyle\epsilon^{2}\Big{(}\partial^{2}_{rr}u^{\pm}+\frac{1}{r}\partial_{r}u^{\pm}$
$\displaystyle+\frac{1}{{r}^{2}}\partial^{2}_{ss}u^{\pm}+\partial^{2}_{ss}u^{\pm}-4i\partial_{s}u^{\pm}-4u^{\pm}\Big{)}$
(2.5)
$\displaystyle+\Big{[}A_{\pm}\big{(}{t}^{\pm}-|u^{\pm}|^{2}\big{)}+B\big{(}{t}^{\mp}-|u^{\mp}|^{2}\big{)}\Big{]}u^{\pm}=0\quad\text{
in }\mathbb{R}^{*}_{+}\times\mathbb{R}.$ (2.6)

We will work in the rescaled coordinates, that is, we define

$\displaystyle v^{\pm}(r,s):=u^{\pm}(\epsilon r,s),$ (2.7)

and search for a solution to the system of equations

$\displaystyle\partial^{2}_{rr}v^{\pm}+\frac{1}{r}\partial_{r}v^{\pm}+\frac{1}{r^{2}}\partial^{2}_{ss}v^{\pm}$
$\displaystyle+\epsilon^{2}(\partial^{2}_{ss}v^{\pm}-4i\partial_{s}v^{\pm}-4v^{\pm})$
$\displaystyle+\Big{[}A_{\pm}\big{(}{t}^{\pm}-|v^{\pm}|^{2}\big{)}+B\big{(}{t}^{\mp}-|v^{\mp}|^{2}\big{)}\Big{]}v^{\pm}=0\quad\text{
in }\mathbb{R}^{*}_{+}\times\mathbb{R}.$ (2.8)

From now on we will work in the plane $\mathbb{R}^{2}$ and use the notation
$z=x_{1}+ix_{2}=re^{is}$. We denote by $\Delta$ the Laplacian operator in
$2$-dimensional space, i.e.,

$\Delta=\partial^{2}_{x_{1}x_{1}}+\partial^{2}_{x_{2}x_{2}}=\partial^{2}_{rr}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial^{2}_{ss}.$

The approximation of a real solution to (2.1) can be chosen in the form

$\displaystyle v_{d}(z)=\big{(}v_{d}^{+}(z),v_{d}^{-}(z)\big{)}$ (2.9)

with

$\displaystyle
v_{d}^{\pm}=({t^{\pm}})^{-1}\prod_{j=1}^{2}w^{\pm}\Big{(}z-\tilde{d}e^{i(j-1)\pi}\Big{)},$
(2.10)

where $\tilde{d}=\frac{\hat{d}}{\epsilon|\ln\epsilon|^{\frac{1}{2}}}$ for some
$\hat{d}=O(1)$ to be determined later and $w=(w^{+},w^{-})$ is the standard
degree $(+1,+1)$ vortex solution of equation (1.14). For convenience, we use
the notation

$\displaystyle e_{j}=e^{i(j-1)\pi},\quad j=1,2.$ (2.11)

######  Remark 2.1.

Starting from (2.9) and following the transformations in (2.3), (2.4) and
(2.7), we claim that by some tedious but straightforward calculations, we can
get

$\displaystyle
e^{2it}\prod_{j=1}^{2}w^{\pm}\Big{(}\frac{z}{\epsilon}-\tilde{d}e^{i\frac{2(j-1)\pi}{2}}\Big{)}=\prod_{j=1}^{2}w^{\pm}\Big{(}\frac{\tilde{z}}{\epsilon}-\tilde{d}e^{it}e^{i\frac{2(j-1)\pi}{2}}\Big{)},$

with

$z=re^{i(\theta-t)},\quad\tilde{z}=re^{i\theta}.$

Equivalently, we get an approximation which has $2$ vortex helices with degree
pair $(+1,+1)$ to the system (1.1) with the form

$\displaystyle\Bigg{(}({t^{+}})^{-1}\prod_{j=1}^{2}w^{+}\Big{(}\frac{\tilde{z}}{\epsilon}-\tilde{d}e^{it}e^{i\frac{2(j-1)\pi}{2}}\Big{)},\
({t^{-}})^{-1}\prod_{j=1}^{2}w^{-}\Big{(}\frac{\tilde{z}}{\epsilon}-\tilde{d}e^{it}e^{i\frac{2(j-1)\pi}{2}}\Big{)}\Bigg{)}.$
(2.12)

For the general integer $k>2$, the corresponding approximation to the solution
of system (2.1) would be

$\displaystyle\Bigg{(}(t^{+})^{1-k}\prod_{j=1}^{k}w^{+}\Big{(}z-\tilde{d}e^{i\frac{2(j-1)\pi}{k}}\Big{)},\
(t^{-})^{1-k}\prod_{j=1}^{k}w^{-}\Big{(}z-\tilde{d}e^{i\frac{2(j-1)\pi}{k}}\Big{)}\Bigg{)}.$
(2.13)

The asymptotic property (1.22) of our solutions holds from the construction
and Lemma 1.2. The functions in (2.12) and (2.13) will provide the profile of
the solution $w_{\epsilon}$ in Theorem 1.4. ∎

###  2.2. Additive-multiplicative perturbation

Let

$S(v):=\Big{(}S^{+}(v),S^{-}(v)\Big{)},\quad\text{for}~{}v=(v^{+},v^{-}),$

with

$S^{\pm}(v)=\Delta
v^{\pm}+\epsilon^{2}(\partial^{2}_{ss}v^{\pm}-4i\partial_{s}v^{\pm}-4v^{\pm})+\Big{[}A_{\pm}\big{(}{t}^{\pm}-|v^{\pm}|^{2}\big{)}+B\big{(}{t}^{\mp}-|v^{\mp}|^{2}\big{)}\Big{]}v^{\pm}.$

The operator $S$ can be decomposed as

$\displaystyle S=S_{0}+S_{1}\ ,$

where

$\displaystyle S_{0}(v)=\Big{(}S_{0}^{+}(v),S_{0}^{-}(v)\Big{)},\quad
S_{1}(v)=\Big{(}S_{1}^{+}(v),S_{1}^{-}(v)\Big{)},$

with

$\displaystyle S_{0}^{\pm}(v)=\Delta
v^{\pm}+\Big{[}A_{\pm}\big{(}{t}^{\pm}-|v^{\pm}|^{2}\big{)}+B\big{(}{t}^{\mp}-|v^{\mp}|^{2}\big{)}\Big{]}v^{\pm},$
$\displaystyle S_{1}^{\pm}(v)$
$\displaystyle=\epsilon^{2}(\partial^{2}_{ss}v^{\pm}-4i\partial_{s}v^{\pm}-4v^{\pm}).$

Therefore, the equation (2.1) can be written as

$\displaystyle S(v)=0.$ (2.14)

We first try to search for a solution of (2.14) with the form

$v=v_{d}+\phi,\quad\text{for some$~{}\phi=(\phi^{+},\phi^{-})$ small}.$

We see that

$\displaystyle S_{0}(v_{d}+\phi)$
$\displaystyle=S_{0}(v_{d})+{\mathcal{L}}_{0}(\phi)+{\mathcal{N}}_{0}(\phi),$
$\displaystyle S_{1}(v_{d}+\phi)$ $\displaystyle=S_{1}(v_{d})+S_{1}(\phi).$

In the above formulae, we have denoted the linear operator in the form

$\displaystyle{\mathcal{L}}_{0}(\phi)$
$\displaystyle=\Big{(}{\mathcal{L}}_{0}^{+},{\mathcal{L}}_{0}^{-}\Big{)}(\phi),$

where

$\displaystyle{\mathcal{L}}_{0}^{\pm}(\phi)$
$\displaystyle=\Delta\phi^{\pm}+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\phi^{\pm}$
$\displaystyle\quad\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{\phi^{\pm}}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{\phi^{\mp}}\Big{)}v_{d}^{\pm},$

and the nonlinear operator as the following

$\displaystyle{\mathcal{N}}_{0}(\phi)$
$\displaystyle=\Big{(}{\mathcal{N}}_{0}^{+},{\mathcal{N}}_{0}^{-}\Big{)}(\phi),$

where

$\displaystyle{\mathcal{N}}_{0}^{\pm}(\phi)$
$\displaystyle=-2A_{\pm}{\rm{Re}}(v_{d}^{\pm}\overline{\phi^{\mp}})\phi^{\pm}-A_{\pm}|\phi^{\pm}|^{2}(v_{d}^{\pm}+\phi^{\pm})$
$\displaystyle\quad\quad-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{\phi^{\mp}}\Big{)}\phi^{\pm}-B|\phi^{\mp}|^{2}(v_{d}^{\pm}+\phi^{\pm}).$

We note that, due to the strong coupling effect in the following terms

$-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{\phi^{\pm}}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{\phi^{\mp}}\Big{)}v_{d}^{\pm},$

it is very complicated to handle the linear operator
${\mathcal{L}}_{0}(\phi)$.

So, we follow the idea in the work of [17] to look for a solution of (2.14) in
the following form

$\displaystyle v=(v^{+},v^{-}),\qquad v^{\pm}=\eta
v_{d}^{\pm}(1+i\psi^{\pm})+(1-\eta)v_{d}^{\pm}e^{i\psi^{\pm}},$ (2.15)

where $v_{d}=(v_{d}^{+},v_{d}^{-})$ is the ansatz (2.9) and
$\psi=(\psi^{+},\psi^{-})=(\psi^{+}_{1}+i\psi^{+}_{2},\psi^{-}_{1}+i\psi^{-}_{2})$
is the unknown perturbation. The cut-off function $\eta$ is defined as

$\displaystyle\eta(z)=\eta_{1}(|z-\tilde{d}|)+\eta_{1}(|z+\tilde{d}|),\quad
z\in\mathbb{C}\cong\mathbb{R}^{2}\ ,$

and $\eta_{1}:\mathbb{R}\rightarrow[0,1]$ is a smooth cut-off function such
that

$\displaystyle\eta_{1}(t)=1\text{ for }t\leq 1\text{ and }\eta_{1}(t)=0\text{
for }t\geq 2.$ (2.16)

Our goal is to rewrite (2.14) in the form

$\displaystyle\mathcal{L}_{\epsilon}\psi+{\mathcal{R}}+\mathcal{N}(\psi)=0$
(2.17)

by identifying the linear operator
$\mathcal{L}_{\epsilon}=\Big{(}\mathcal{L}_{\epsilon}^{+},\mathcal{L}_{\epsilon}^{-}\Big{)}$,
the error term
${\mathcal{R}}=\Big{(}{\mathcal{R}}^{+},{\mathcal{R}}^{-}\Big{)}$ and the
nonlinear term
$\mathcal{N}(\psi)=\Big{(}\mathcal{N}^{+}(\psi),\mathcal{N}^{-}(\psi)\Big{)}$.
This will be done in the sequel and the result is given in (2.22).

Rewrite $v$ in (2.15) as

$\displaystyle v$
$\displaystyle=v_{d}+\phi=\big{(}v_{d}^{+},v_{d}^{-}\big{)}+\big{(}\phi^{+},\phi^{-}\big{)}$

with

$\displaystyle\phi^{\pm}$
$\displaystyle=iv_{d}^{\pm}\psi^{\pm}+\gamma(\psi^{\pm}),\quad\gamma(\psi^{\pm})=(1-\eta)v_{d}^{\pm}(e^{i\psi^{\pm}}-1-i\psi^{\pm}).$

Then

$\displaystyle S_{0}(v)$
$\displaystyle=S_{0}(v_{d})+{\mathcal{L}}_{0}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}+{\mathcal{L}}_{0}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}$
$\displaystyle\quad+{\mathcal{N}}_{0}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}+\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)},$
$\displaystyle S_{1}(v)$
$\displaystyle=S_{1}(V_{d})+S_{1}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}+S_{1}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}.$

Explicit expressions of some operators in the above will be derived one by
one.

$\bullet$ By some computations, we start from $S_{0}(v)$ and consider

$\displaystyle{\mathcal{L}}_{0}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}=\Big{(}{\mathcal{L}}_{0}^{+},{\mathcal{L}}_{0}^{-}\Big{)}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}$

where

$\displaystyle{\mathcal{L}}_{0}^{\pm}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}$
$\displaystyle=\Delta(iv_{d}^{\pm}\psi^{\pm})+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}(iv_{d}^{\pm}\psi^{\pm})$
$\displaystyle\qquad\qquad\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{(iv_{d}^{\pm}\psi^{\pm})}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{(iv_{d}^{\mp}\psi^{\mp})}\Big{)}v_{d}^{\pm}$
$\displaystyle=iv_{d}^{\pm}\Bigg{(}\frac{\Delta
v_{d}^{\pm}}{v_{d}^{\pm}}+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Bigg{)}\psi^{\pm}$
$\displaystyle\quad+iv_{d}^{\pm}\Bigg{(}\Delta\psi^{\pm}+\frac{\nabla
v_{d}^{\pm}\nabla\psi^{\pm}}{v_{d}^{\pm}}-2iA_{\pm}|v_{d}^{\pm}|^{2}{\rm{Im}}(\psi^{\pm})-2iB|v_{d}^{\mp}|^{2}\rm{Im}(\psi^{\mp})\Bigg{)}$
$\displaystyle=iv_{d}^{\pm}\Bigg{(}\frac{\Delta
v_{d}^{\pm}}{v_{d}^{\pm}}+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Bigg{)}\psi^{\pm}$
$\displaystyle\quad+iv_{d}^{\pm}\Bigg{(}\Delta\psi^{\pm}+\frac{\nabla
v_{d}^{\pm}\nabla\psi^{\pm}}{v_{d}^{\pm}}-2iA_{\pm}|v_{d}^{\pm}|^{2}{\rm{Im}}(\psi^{\pm})-2iB|v_{d}^{\mp}|^{2}{\rm{Im}}(\psi^{\mp})\Bigg{)}$
$\displaystyle=iv_{d}^{\pm}\Bigg{(}\frac{S_{0}^{\pm}(v_{d})}{v_{d}^{\pm}}\psi^{\pm}+\Delta\psi^{\pm}+\overbrace{{\color[rgb]{1,0,0}\frac{\nabla
v_{d}^{\pm}\nabla\psi^{\pm}}{v_{d}^{\pm}}}}^{\text{weak
effect}}-2iA_{\pm}|v_{d}^{\pm}|^{2}{\rm{Im}}(\psi^{\pm})-2iB|v_{d}^{\mp}|^{2}\rm{Im}(\psi^{\mp})\Bigg{)}$
$\displaystyle:=iv_{d}^{\pm}\Bigg{(}\frac{S_{0}^{\pm}(v_{d})}{v_{d}^{\pm}}\psi^{\pm}+\tilde{L}^{\pm}_{0}(\psi)\Bigg{)},$

with

$\displaystyle\tilde{L}^{\pm}_{0}(\psi)=\Delta\psi^{\pm}+\overbrace{{\color[rgb]{1,0,0}\frac{\nabla
v_{d}^{\pm}\nabla\psi^{\pm}}{v_{d}^{\pm}}}}^{\text{weak
effect}}-2iA_{\pm}|v_{d}^{\pm}|^{2}{\rm{Im}}(\psi^{\pm})-2iB|v_{d}^{\mp}|^{2}\rm{Im}(\psi^{\mp}).$
(2.18)

Thus we have

$\displaystyle S^{\pm}_{0}(v)$
$\displaystyle=S^{\pm}_{0}(v_{d})+{\mathcal{L}}^{\pm}_{0}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}+{\mathcal{L}}^{\pm}_{0}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}$
$\displaystyle\quad+{\mathcal{N}}^{\pm}_{0}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}+\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}$
$\displaystyle=iv_{d}^{\pm}\Bigg{(}-i\frac{S^{\pm}_{0}(v_{d})}{v_{d}^{\pm}}+\frac{S_{0}^{\pm}(v_{d})}{v_{d}^{\pm}}\psi^{\pm}+\tilde{L}^{\pm}_{0}(\psi)-i\frac{1}{v_{d}^{\pm}}{\mathcal{L}}^{\pm}_{0}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}$
$\displaystyle\quad-i\frac{1}{v_{d}^{\pm}}{\mathcal{N}}^{\pm}_{0}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}+\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}\Bigg{)}.$

Specially, when far away from the vortices,
$v=\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}$. This implies
that

$\displaystyle
S_{0}(v)=\Bigg{(}S_{0}^{+}\Big{(}\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}\Big{)},S_{0}^{-}\Big{(}\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}\Big{)}\Bigg{)},$
(2.19)

and

$\displaystyle
S_{0}^{\pm}\Big{(}\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}\Big{)}$
$\displaystyle=\Delta(v_{d}^{\pm}e^{i\psi^{\pm}})+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|(v_{d}^{\pm}e^{i\psi^{\pm}})|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|(v_{d}^{\mp}e^{i\psi^{\mp}})|^{2}\big{)}\Big{]}(v_{d}^{\pm}e^{i\psi^{\pm}})$
$\displaystyle=iv_{d}^{\pm}e^{i\psi^{\pm}}\Big{(}-i\frac{S_{0}^{\pm}({v_{d}})}{{v_{d}^{\pm}}}+\tilde{L}^{\pm}_{0}(\psi)+\tilde{N}^{\pm}_{0}(\psi)\Big{)},$

where

$\tilde{N}^{\pm}_{0}(\psi)=i(\nabla\psi^{\pm})^{2}+iA_{\pm}|v_{d}^{\pm}|^{2}(e^{-2\psi^{\pm}_{2}}-1+2\psi^{\pm}_{2})+iB|v_{d}^{\mp}|^{2}(e^{-2\psi^{\mp}_{2}}-1+2\psi^{\mp}_{2}).$

$\bullet$ We consider

$\displaystyle
S_{1}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}=\Bigg{(}S_{1}^{+}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)},S_{1}^{-}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}\Bigg{)},$

and it follows that

$\displaystyle
S_{1}^{\pm}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}\Big{)}=iv_{d}^{\pm}\left(\frac{S^{\pm}_{1}(v_{d})}{v_{d}^{\pm}}\psi^{\pm}+\tilde{L}_{1}^{\pm}(\psi)\right)$

with

$\displaystyle\tilde{L}_{1}^{\pm}(\psi)=\epsilon^{2}\Bigl{(}\partial_{ss}^{2}\psi^{\pm}+\frac{2\partial_{s}v_{d}^{\pm}}{v_{d}^{\pm}}\partial_{s}\psi^{\pm}-4i\partial_{s}\psi^{\pm}\Bigr{)}.$
(2.20)

Specially, we have that, when far away from the vortices

$\displaystyle
S_{1}(v)=\Bigg{(}S_{1}^{+}\Big{(}\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}\Big{)},S_{1}^{-}\Big{(}\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}\Big{)}\Bigg{)},$
(2.21)

with

$\displaystyle
S_{1}^{\pm}\Big{(}\big{(}v_{d}^{+}e^{i\psi^{+}},v_{d}^{-}e^{i\psi^{-}}\big{)}\Big{)}=iv^{\pm}_{d}e^{i\psi^{\pm}}\left[-i\frac{S_{1}^{\pm}(v_{d})}{v^{\pm}_{d}}+\tilde{L}_{1}^{\pm}(\psi)+\epsilon^{2}i(\partial_{s}\psi^{\pm})^{2}\right].$

In order to draw a conclusion, a new cut-off function will be introduced as
the following

$\tilde{\eta}(z)=\eta_{1}(|z-\tilde{d}|-1)+\eta_{1}(|z+\tilde{d}|-1)$

with $\eta_{1}$ defined in (2.16). Then the problem (2.14) can be rewritten as

$\displaystyle 0$
$\displaystyle=\tilde{\eta}i{v_{d}^{\pm}}\Bigg{[}-i\frac{S_{0}^{\pm}({v_{d}})}{{v_{d}^{\pm}}}+\tilde{L}_{0}^{\pm}(\psi)+\frac{S_{0}^{\pm}({v_{d}})}{{v_{d}^{\pm}}}\psi^{\pm}-i\frac{S_{1}^{\pm}({v_{d}})}{{v_{d}^{\pm}}}+\tilde{L}_{1}^{\pm}(\psi)+\frac{S_{1}^{\pm}({v_{d}})}{{v_{d}^{\pm}}}\psi^{\pm}$
$\displaystyle\qquad-\frac{i}{{v_{d}^{\pm}}}{\mathcal{L}}_{0}^{\pm}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}-\frac{i}{{v_{d}^{\pm}}}{\mathcal{N}}_{0}^{\pm}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}+\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}$
$\displaystyle\qquad-\frac{i}{{v_{d}^{\pm}}}S_{1}^{\pm}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}\Bigg{]}$
$\displaystyle\quad+(1-\tilde{\eta})i{v_{d}^{\pm}}e^{i\psi^{\pm}}\left[-i\frac{S_{0}^{\pm}({v_{d}})}{{v_{d}^{\pm}}}+\tilde{L}_{0}^{\pm}(\psi)+\tilde{N}_{0}^{\pm}(\psi)-i\frac{S_{1}^{\pm}({v_{d}^{\pm}})}{{v_{d}^{\pm}}}+\tilde{L}_{1}(\psi^{\pm})+\epsilon^{2}i(\partial_{s}\psi^{\pm})^{2}\right].$

By the definitions of $\mathcal{L},\mathcal{R}$ and $\mathcal{N}$, we can have
a simpler representation of the above equation for (2.14):

$\displaystyle\mathcal{L}_{\epsilon}\psi+{\mathcal{R}}+\mathcal{N}(\psi)=0,\quad
i.e.\quad\mathcal{L}_{\epsilon}^{\pm}\psi+{\mathcal{R}}^{\pm}+\mathcal{N}^{\pm}(\psi)=0,$
(2.22)

where

$\displaystyle\mathcal{L}_{\epsilon}^{\pm}(\psi)$
$\displaystyle:=(\tilde{L}_{0}^{\pm}+\tilde{L}_{1}^{\pm})(\psi)+\tilde{\eta}\frac{S^{\pm}(v_{d})}{v_{d}^{\pm}}\psi^{\pm},\qquad\quad{\mathcal{R}}^{\pm}:=-i\frac{S(v_{d}^{\pm})}{v_{d}^{\pm}},$
$\displaystyle\mathcal{N}^{\pm}(\psi)$
$\displaystyle:=\tilde{\eta}\Bigl{(}\frac{1}{\tilde{\eta}+(1-\tilde{\eta})e^{i\psi^{\pm}}}-1\Bigr{)}\frac{S^{\pm}(v_{d})}{v_{d}^{\pm}}\psi^{\pm}$
$\displaystyle\qquad-\frac{i}{v_{d}^{\pm}}\frac{\tilde{\eta}}{\tilde{\eta}+(1-\tilde{\eta})e^{i\psi^{\pm}}}\Bigg{[}{\mathcal{L}}_{0}^{\pm}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}+S_{1}^{\pm}\Big{(}\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}$
$\displaystyle\qquad\qquad+{\mathcal{N}}_{0}^{\pm}\Big{(}\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)}+\big{(}\gamma(\psi^{+}),\gamma(\psi^{-})\big{)}\Big{)}\Bigg{]}$
$\displaystyle\qquad+\frac{(1-\tilde{\eta})e^{i\psi^{\pm}}}{\tilde{\eta}+(1-\tilde{\eta})e^{i\psi^{\pm}}}\Bigl{[}\tilde{N}_{0}^{\pm}(\psi)+\epsilon^{2}i(\partial_{s}\psi^{\pm})^{2}\Bigr{]}.$

From (2.18) and (2.20), we know that

$\displaystyle\mathcal{L}_{\epsilon}^{\pm}(\psi)$
$\displaystyle=\Delta\psi^{\pm}+{\frac{\nabla
v_{d}^{\pm}\nabla\psi^{\pm}}{v_{d}^{\pm}}}-2iA_{\pm}|v_{d}^{\pm}|^{2}{\rm{Im}}(\psi^{\pm})-2iB|v_{d}^{\mp}|^{2}\rm{Im}(\psi^{\mp})$
$\displaystyle\quad+\epsilon^{2}\Bigl{(}\partial_{ss}^{2}\psi^{\pm}+\frac{2\partial_{s}v_{d}^{\pm}}{v_{d}^{\pm}}\partial_{s}\psi^{\pm}-4i\partial_{s}\psi^{\pm}\Bigr{)}+\tilde{\eta}\frac{S^{\pm}(v_{d})}{v_{d}^{\pm}}\psi^{\pm}.$

Note that when $|z\pm\tilde{d}|\geq 3$ the nonlinear terms take the form

$\displaystyle\mathcal{N}^{\pm}(\psi)$
$\displaystyle=\tilde{N}_{0}^{\pm}(\psi)+\epsilon^{2}i(\partial_{s}\psi^{\pm})^{2}$
$\displaystyle=i(\nabla\psi^{\pm})^{2}+iA_{\pm}|v_{d}^{\pm}|^{2}(e^{-2\psi^{\pm}_{2}}-1+2\psi^{\pm}_{2})+iB|v_{d}^{\mp}|^{2}(e^{-2\psi^{\mp}_{2}}-1+2\psi^{\mp}_{2})+\epsilon^{2}i(\partial_{s}\psi^{\pm})^{2}.$

###  2.3. Another form of the equation near each vortex

Note that the expressions of the terms $S_{0}(v)$ and $S_{1}(v)$ in the outer
region (far from the vortices) are given in (2.19) and (2.21). In this part,
we will write the operator $\mathcal{L}_{\epsilon}\psi$ in another form when
it is near the vortices so that we can know it better.

In order to do so, we first introduce some notation. We denote

${z}_{j}=z-\tilde{d}e^{i\frac{2(j-1)\pi}{2}}\qquad\mbox{for }j=1,2,$

and then let

$\Phi_{j}(z)=\big{(}\Phi_{j}^{+}(z),\Phi_{j}^{-}(z)\big{)},\qquad\Phi_{j}^{\pm}(z)=iw^{\pm}({z}_{j})\psi^{\pm}(z).$

Furthermore, when near the vortex center, we also assume

$\displaystyle\Phi(z)=\big{(}\Phi^{+}(z),\Phi^{-}(z)\big{)},\qquad\Phi^{\pm}(z)=iv_{d}^{\pm}(z)\psi^{\pm}(z),$
$\displaystyle\Omega_{j}(z)=\big{(}\Omega^{+}_{j}(z),\Omega^{-}_{j}(z)\big{)},\qquad\Omega_{j}^{\pm}(z)=\frac{v_{d}^{\pm}(z)}{w^{\pm}({z}_{j})},$
$\displaystyle\mathbb{E}:=S(v_{d}),\quad\qquad\mathbb{E}^{\pm}=S^{\pm}(v_{d}).$

By some calculations, the following identity holds

$\displaystyle iw^{\pm}(z_{j})\mathcal{L}_{\epsilon}^{\pm}\psi(z)$
$\displaystyle=\frac{L^{\pm}_{d}(\Phi)}{\Omega_{j}^{\pm}}(z)+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}(z)$
$\displaystyle=\frac{L^{\pm}_{d}\Big{(}\big{(}\Phi^{+}_{j}\Omega^{+}_{j},\
\Phi^{-}_{j}\Omega^{-}_{j}\big{)}\Big{)}}{\Omega_{j}^{\pm}}(z)+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}(z)\
,$

where

$\displaystyle L^{\pm}_{d}(\Phi)$
$\displaystyle=\Delta\Phi^{\pm}+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Phi^{\pm}$
$\displaystyle\quad\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{\Phi^{\pm}}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{\Phi^{\mp}}\Big{)}v_{d}^{\pm}+\epsilon^{2}(\partial^{2}_{ss}\Phi^{\pm}-4i\partial_{s}\Phi^{\pm}-4\Phi^{\pm}).$

Recall that

$\Phi=\big{(}iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-}\big{)},$

and define

$\displaystyle
L_{d,j}(\Phi)=\Big{(}L^{+}_{d,j},L^{-}_{d,j}\Big{)}(\Phi),\quad\text{with}\quad
L^{\pm}_{d,j}(\Phi):=iw^{\pm}(z_{j})\mathcal{L}_{\epsilon}^{\pm}(\psi).$
(2.23)

In order to describe the properties of ${\mathcal{L}}_{\epsilon}^{\pm}(\psi)$
near the vortex, we give the following lemma.

######  Lemma 2.2.

When near the vortices, we can see that the linear operator
$L_{d,j}(\Phi)=\Big{(}L^{+}_{d,j},L^{-}_{d,j}\Big{)}(\Phi)$ defined in (2.23)
is a small perturbation of $L^{0}(\Phi)$, where $L^{0}$ is the linearized
Ginzburg-Landau operator around $\Big{(}w^{+}(z_{j}),w^{-}(z_{j})\Big{)}$,
i.e,

$L^{0}(\phi):=\big{(}{L^{0}_{+}}(\phi),{L^{0}_{-}}(\phi)\big{)}\quad\mbox{with
}\phi=(\phi^{+},\phi^{-}),$

and

$\displaystyle{L^{0}_{\pm}}(\phi)$
$\displaystyle=\Delta\phi^{\pm}\,+\,\Big{[}\,A_{\pm}\big{(}{t}^{\pm}-|{W^{\pm}(|z_{j}|)}|^{2}\Big{)}+B\Big{(}{t^{\mp}}^{2}-|{W^{\mp}(|z_{j}|)}|^{2}\big{)}\,\Big{]}\phi^{\pm}$
$\displaystyle\quad\,-\,2A_{\pm}{\rm{Re}}\Big{(}w^{\pm}\overline{\phi^{\pm}}\Big{)}w^{\pm}(z_{j})\,-\,2B{\rm{Re}}\Big{(}w^{\mp}\overline{\phi^{\mp}}\Big{)}w^{\pm}(z_{j}).$
(2.24)

###### Proof.

From the definition, we know that

$\displaystyle L^{\pm}_{d,j}(\Phi)$
$\displaystyle=\frac{L^{\pm}_{d}\Big{(}\big{(}\Phi^{+}_{j}\Omega^{+}_{j},\Phi^{-}_{j}\Omega^{-}_{j}\big{)}\Big{)}}{\Omega_{j}^{\pm}}+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}$
$\displaystyle=\frac{1}{\Omega_{j}^{\pm}}\Bigg{\\{}\Delta(\Phi^{\pm}_{j}\Omega^{\pm}_{j})+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}(\Phi^{\pm}_{j}\Omega^{\pm}_{j})$
$\displaystyle\qquad\quad\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{(\Phi^{\pm}_{j}\Omega^{\pm}_{j})}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{(\Phi^{\mp}_{j}\Omega^{\mp}_{j})}\Big{)}v_{d}^{\pm}$
$\displaystyle\qquad\quad+\epsilon^{2}\Big{[}\partial^{2}_{ss}(\Phi^{\pm}_{j}\Omega^{\pm}_{j})-4i\partial_{s}(\Phi^{\pm}_{j}\Omega^{\pm}_{j})-4(\Phi^{\pm}_{j}\Omega^{\pm}_{j})\Big{]}\Bigg{\\}}\
+\ (\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}$
$\displaystyle=\frac{1}{\Omega_{j}^{\pm}}\Big{(}\Phi^{\pm}_{j}\Delta\Omega^{\pm}_{j}+2\nabla\Phi^{\pm}_{j}\nabla\Omega^{\pm}_{j}+\Omega^{\pm}_{j}\Delta\Phi^{\pm}_{j}\Big{)}+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}$
$\displaystyle\quad+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Phi^{\pm}_{j}$
$\displaystyle\quad+\frac{1}{\Omega_{j}^{\pm}}\Big{[}\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{(\Phi^{\pm}_{j}\Omega^{\pm}_{j})}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{(\Phi^{\mp}_{j}\Omega^{\mp}_{j})}\Big{)}v_{d}^{\pm}\Big{]}$
$\displaystyle\quad+\epsilon^{2}\frac{1}{\Omega_{j}^{\pm}}\Big{[}\Omega^{\pm}_{j}\partial^{2}_{ss}\Phi^{\pm}_{j}+\Phi^{\pm}_{j}\partial^{2}_{ss}\Omega^{\pm}_{j}+2\partial_{s}\Omega^{\pm}_{j}\partial_{s}\Phi^{\pm}_{j}-4i(\partial_{s}\Omega^{\pm}_{j}\Phi^{\pm}_{j}+\partial_{s}\Phi^{\pm}_{j}\Omega^{\pm}_{j})-4(\Phi^{\pm}_{j}\Omega^{\pm}_{j})\Big{]}$
$\displaystyle:={\mathfrak{A}}_{1}+{\mathfrak{A}}_{2}+{\mathfrak{A}}_{3}+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm},$

with

$\displaystyle{\mathfrak{A}}_{1}$
$\displaystyle=\Delta\Phi^{\pm}_{j}+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Phi^{\pm}_{j}$
$\displaystyle\quad\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{(\Phi^{\pm}_{j}\Omega^{\pm}_{j})}\Big{)}w^{\pm}(z_{j})\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{(\Phi^{\mp}_{j}\Omega^{\mp}_{j})}\Big{)}w^{\pm}(z_{j}),$
${\mathfrak{A}}_{2}=\epsilon^{2}\Big{[}\partial^{2}_{ss}\Phi^{\pm}_{j}-4i\partial_{s}\Phi^{\pm}_{j}-4\Phi^{\pm}_{j}\Big{]}+\frac{1}{\Omega_{j}^{\pm}}\Big{[}2\nabla\Phi^{\pm}_{j}\nabla\Omega^{\pm}_{j}+2\epsilon^{2}\partial_{s}\Omega^{\pm}_{j}\partial_{s}\Phi^{\pm}_{j}\Big{]},$
${\mathfrak{A}}_{3}=\frac{1}{\Omega_{j}^{\pm}}\Big{[}\Delta\Omega^{\pm}_{j}+\epsilon^{2}\partial^{2}_{ss}\Omega^{\pm}_{j}-4i\epsilon^{2}\partial_{s}\Omega^{\pm}_{j}\Big{]}\Phi^{\pm}_{j}.$

For ${\mathfrak{A}}_{1}$, we have

$\displaystyle{\mathfrak{A}}_{1}$
$\displaystyle={L^{0}_{\pm}}(\Phi_{j})+\Big{[}A_{\pm}\big{(}|w^{\mp}(z_{j})|^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}|w^{\mp}(z_{j})|^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Phi^{\pm}_{j}$
$\displaystyle\quad+2A_{\pm}{\rm{Re}}\Big{(}w^{\pm}(z_{j})\overline{\Phi^{\pm}_{j}}\Big{)}w^{\pm}(z_{j})+2B{\rm{Re}}\Big{(}w^{\mp}(z_{j})\overline{\Phi^{\mp}_{j}}\Big{)}w^{\pm}(z_{j})$
$\displaystyle\quad\,-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{(\Phi^{\pm}_{j}\Omega^{\pm}_{j})}\Big{)}w^{\pm}(z_{j})-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{(\Phi^{\mp}_{j}\Omega^{\mp}_{j})}\Big{)}w^{\pm}(z_{j})$
$\displaystyle={L^{0}_{\pm}}(\Phi_{j})+\Big{[}A_{\pm}\big{(}|w^{\pm}(z_{j})|^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}|w^{\mp}(z_{j})|^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Phi^{\pm}_{j}$
$\displaystyle\quad+2A_{\pm}\big{(}1-|\Omega^{\pm}_{j}|^{2}\big{)}{\rm{Re}}\Big{(}w^{\pm}(z_{j})\overline{\Phi^{\pm}_{j}}\Big{)}w^{\pm}(z_{j})+2B\big{(}1-|\Omega^{\mp}_{j}|^{2}\big{)}{\rm{Re}}\Big{(}w^{\mp}(z_{j})\overline{\Phi^{\mp}_{j}}\Big{)}w^{\pm}(z_{j}).$

Since

$\displaystyle\mathbb{E}^{\pm}$ $\displaystyle=S^{\pm}(v_{d})$
$\displaystyle=\Delta(w^{\pm}(z_{j})\Omega^{\pm}_{j})+\Big{[}A_{\pm}\big{(}1-|w^{\pm}(z_{j})|^{2}|\Omega^{\pm}_{j}|^{2}\big{)}+B\big{(}1-|w^{\mp}(z_{j})|^{2}|\Omega^{\mp}_{j}|^{2}\big{)}\Big{]}w^{\pm}(z_{j})\Omega^{\pm}_{j}$
$\displaystyle\quad+\epsilon^{2}\Big{[}\partial^{2}_{ss}(w^{\pm}(z_{j})\Omega^{\pm}_{j})-4i\partial_{s}(w^{\pm}(z_{j})\Omega^{\pm}_{j})-4w^{\pm}(z_{j})\Omega^{\pm}_{j}\Big{]}$
$\displaystyle=w^{\pm}(z_{j})\Big{[}\Delta\Omega^{\pm}_{j}+\epsilon^{2}\partial^{2}_{ss}\Omega^{\pm}_{j}-4i\epsilon^{2}\partial_{s}\Omega^{\pm}_{j}\Big{]}-4\epsilon^{2}w^{\pm}(z_{j})\Omega^{\pm}_{j}$
$\displaystyle\quad+\Big{[}A_{\pm}|w^{\pm}(z_{j})|^{2}\big{(}1-|\Omega^{\pm}_{j}|^{2}\big{)}+B|w^{\mp}(z_{j})|^{2}\big{(}1-|\Omega^{\mp}_{j}|^{2}\big{)}\Big{]}w^{\pm}(z_{j})\Omega^{\pm}_{j}$
$\displaystyle\quad+2\nabla
w^{\pm}(z_{j})\nabla\Omega^{\pm}_{j}+2\epsilon^{2}\partial_{s}w^{\pm}(z_{j})\partial_{s}\Omega^{\pm}_{j}+\epsilon^{2}\Big{[}\partial_{ss}w^{\pm}(z_{j})-4i\partial_{s}w^{\pm}(z_{j})\Big{]}\Omega^{\pm}_{j},$

we therefore conclude that

$\displaystyle\Delta\Omega^{\pm}_{j}+\epsilon^{2}\partial^{2}_{ss}\Omega^{\pm}_{j}-4i\epsilon^{2}\partial_{s}\Omega^{\pm}_{j}$
$\displaystyle=\frac{\mathbb{E}^{\pm}}{w^{\pm}(z_{j})}+4\epsilon^{2}\Omega^{\pm}_{j}-\frac{2\nabla
w^{\pm}(z_{j})\nabla\Omega^{\pm}_{j}+2\epsilon^{2}\partial_{s}w^{\pm}(z_{j})\partial_{s}\Omega^{\pm}_{j}}{w^{\pm}(z_{j})}$
$\displaystyle\quad-\Big{[}A_{\pm}|w^{\pm}(z_{j})|^{2}\big{(}1-|\Omega^{\pm}_{j}|^{2}\big{)}+B|w^{\mp}(z_{j})|^{2}\big{(}1-|\Omega^{\mp}_{j}|^{2}\big{)}\Big{]}\Omega^{\pm}_{j}$
$\displaystyle\quad-\epsilon^{2}\frac{\Omega^{\pm}_{j}\Big{[}\partial_{ss}w^{\pm}(z_{j})-4i\partial_{s}w^{\pm}(z_{j})\Big{]}}{w^{\pm}(z_{j})}.$

Then we can get

$\displaystyle{\mathfrak{A}}_{3}$
$\displaystyle=\frac{1}{\Omega_{j}^{\pm}}\Big{[}\Delta\Omega^{\pm}_{j}+\epsilon^{2}\partial^{2}_{ss}\Omega^{\pm}_{j}-4i\epsilon^{2}\partial_{s}\Omega^{\pm}_{j}\Big{]}\Phi^{\pm}_{j}$
$\displaystyle=\Bigg{\\{}\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}+4\epsilon^{2}-\frac{2\nabla
w^{\pm}(z_{j})\nabla\Omega^{\pm}_{j}+2\epsilon^{2}\partial_{s}w^{\pm}(z_{j})\partial_{s}\Omega^{\pm}_{j}}{v_{d}^{\pm}}$
$\displaystyle\qquad-\epsilon^{2}\frac{\Big{[}\partial_{ss}w^{\pm}(z_{j})-4i\partial_{s}w^{\pm}(z_{j})\Big{]}}{w^{\pm}(z_{j})}\Bigg{\\}}\Phi_{j}^{\pm}$
$\displaystyle\quad-\Big{[}A_{\pm}|w^{\pm}(z_{j})|^{2}\big{(}1-|\Omega^{\pm}_{j}|^{2}\big{)}+B|w^{\mp}(z_{j})|^{2}\big{(}1-|\Omega^{\mp}_{j}|^{2}\big{)}\Big{]}\Phi^{\pm}_{j}\
.$

Combining all the above calculations, we obtain

$\displaystyle L^{\pm}_{d,j}(\Phi_{j})$
$\displaystyle={\mathfrak{A}}_{1}+{\mathfrak{A}}_{2}+{\mathfrak{A}}_{3}+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}$
$\displaystyle={L^{0}_{\pm}}(\Phi_{j})+2A_{\pm}\big{(}1-|\Omega^{\pm}_{j}|^{2}\big{)}{\rm{Re}}\Big{(}w^{\pm}(z_{j})\overline{\Phi^{\pm}_{j}}\Big{)}w^{\pm}(z_{j})+2B\big{(}1-|\Omega^{\mp}_{j}|^{2}\big{)}{\rm{Re}}\Big{(}w^{\mp}(z_{j})\overline{\Phi^{\mp}_{j}}\Big{)}w^{\pm}(z_{j})$
$\displaystyle\quad+\epsilon^{2}\Big{[}\partial^{2}_{ss}\Phi^{\pm}_{j}-4i\partial_{s}\Phi^{\pm}_{j}-4\Phi^{\pm}_{j}\Big{]}+\frac{1}{\Omega_{j}^{\pm}}\Big{[}2\nabla\Phi^{\pm}_{j}\nabla\Omega^{\pm}_{j}+2\epsilon^{2}\partial_{s}\Omega^{\pm}_{j}\partial_{s}\Phi^{\pm}_{j}\Big{]}$
$\displaystyle\quad+\Bigg{\\{}4\epsilon^{2}-\frac{2\nabla
w^{\pm}(z_{j})\nabla\Omega^{\pm}_{j}+2\epsilon^{2}\partial_{s}w^{\pm}(z_{j})\partial_{s}\Omega^{\pm}_{j}}{v_{d}^{\pm}}-\epsilon^{2}\frac{\Big{[}\partial_{ss}w^{\pm}(z_{j})-4i\partial_{s}w^{\pm}(z_{j})\Big{]}}{w^{\pm}(z_{j})}\Bigg{\\}}\Phi_{j}^{\pm}$
$\displaystyle\quad+\eta_{1}\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}\
.$ (2.25)

We point out that for $|{z}_{j}|<3,$ from Lemma (1.2), there holds

$|\Omega^{\pm}_{j}({z}_{j})|=1+O_{\epsilon}(\epsilon^{2}|\ln\epsilon|),\ \ \
\nabla\Omega^{\pm}_{j}({z}_{j})=O_{\epsilon}(\epsilon\sqrt{|\ln\epsilon|}),\ \
\ |\Delta\Omega^{\pm}_{j}({z}_{j})|=O_{\epsilon}(\epsilon^{2}|\ln\epsilon|).$

Therefore, the linear operator $L^{\pm}_{d,j}$ is a small perturbation of
$L^{0}_{\pm}$ when near the vortices.

∎

###  2.4. Symmetry assumptions on the perturbation

At the last of this part, we point out that we would make some symmetry
assumptions for the perturbation in our problem. In fact, from the expression
of $v_{d}=(v_{d}^{+},v_{d}^{-})$ in (2.9), we know that, under the coordinates
$z=x_{1}+ix_{2}$, there hold

$\displaystyle
v_{d}^{\pm}(-x_{1},x_{2})=\overline{v_{d}}^{\pm}(x_{1},x_{2}),\quad
v_{d}^{\pm}(x_{1},-x_{2})=\overline{v_{d}}^{\pm}(x_{1},x_{2}).$

Note that these symmetries are compatible with the solution operator $S(v)$:

$\mbox{if }S^{\pm}(v)=0\ \mbox{ and }\
u(z)=(u^{+},u^{-})(z):=\big{(}{\overline{v}^{+}},{\overline{v}^{-}}\big{)}(x_{1},-x_{2}),\quad\mbox{then
}S(u)=0,$

and the same property holds for
$u(z)=(u^{+},u^{-})(z):=\Big{(}v^{+}(-x_{1},x_{2}),v^{-}(-x_{1},x_{2})\Big{)}.$
Then we naturally assume that the perturbation $\psi$ satisfies

$\displaystyle\psi(x_{1},-x_{2})=-\overline{\psi^{\pm}}(x_{1},x_{2}),\quad\psi(-x_{1},x_{2})=-\overline{\psi^{\pm}}(x_{1},x_{2}).$
(2.26)

##  3\. Error estimate and Fourier decomposition

###  3.1. Error estimate

Recall the approximation $v_{d}$ of $S(v)=0$ in (2.9)-(2.10). Suppose

$z-e_{j}=\ell_{j}e^{i\theta_{j}}\quad\text{for}~{}j=1,2,$

where the $e_{j}$’s are given in (2.11). We can rewrite $v_{d}$ in the new
coordinates $(\ell_{j},\theta_{j})$ as

$\displaystyle v_{d}=(v_{d}^{+},v_{d}^{-}),\quad
v_{d}^{\pm}(z)=({t^{\pm}})^{-1}W^{\pm}(\ell_{1})W^{\pm}(\ell_{2})e^{i(\theta_{1}+\theta_{2})}.$
(3.1)

In this section, we would give an accurate estimation of $S(v_{d})$. Let

$S(v_{d})=\big{(}S^{+}(v_{d}),S^{-}(v_{d})\big{)}:=i(v_{d}^{+}{\mathcal{R}}^{+},v_{d}^{-}{\mathcal{R}}^{-}),$

and

$S^{\pm}(v_{d})=S^{\pm}_{0}(v_{d})+S^{\pm}_{1}(v_{d}):=iv_{d}^{\pm}{\mathcal{R}}_{0}^{\pm}+iv_{d}^{\pm}{\mathcal{R}}_{1}^{\pm},$

with

$S^{\pm}_{0}(v_{d})=iv_{d}^{\pm}{\mathcal{R}}_{0}^{\pm}.$

To estimate the size of error
${\mathcal{R}}=({\mathcal{R}}^{+},{\mathcal{R}}^{-})$, we introduce the norm
$\|\cdot\|_{**}$. For given $0<\alpha,\sigma<1$,
${\mathcal{H}}=({\mathcal{H}}^{+},{\mathcal{H}}^{-})$, we define the norm

$\displaystyle\|{\mathcal{H}}\|_{**}=\|{\mathcal{H}}^{+}\|_{**,1}\ +\
\|{\mathcal{H}}^{-}\|_{**,1},$ (3.2)

with

$\displaystyle\|{\mathcal{H}}^{\pm}\|_{**,1}$
$\displaystyle:=\sum_{j=1}^{2}\|v_{d}^{\pm}{\mathcal{H}}^{\pm}\|_{C^{\alpha}(\ell_{j}<3)}\
+\
\sup_{\ell_{1}>2,\ell_{2}>2}\Bigg{[}\frac{|{\rm{Re}}({\mathcal{H}}^{\pm})|}{\ell_{1}^{-2}+\ell_{2}^{-2}+\epsilon^{2}}+\frac{|{\rm{Im}}({\mathcal{H}}^{\pm})|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}+\epsilon^{2-\sigma}}\Bigg{]}$
$\displaystyle\quad\ +\
\sup_{2<\ell_{1}<2\mathbb{R}_{\epsilon},\,2<\ell_{2}<2\mathbb{R}_{\epsilon}}\frac{[{\rm{Re}}({\mathcal{H}}^{\pm})]_{\alpha,B_{|z|/2}(z)}}{\ell_{1}^{-2-\alpha}+\ell_{2}^{-2-\alpha}}\
+\
\sup_{2<\ell_{1}<2\mathbb{R}_{\epsilon},\,2<\ell_{2}<2\mathbb{R}_{\epsilon}}\frac{[{\rm{Im}}({\mathcal{H}}^{\pm})]_{\alpha,B_{1}(z)}}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}},$
(3.3)

where
$\mathbb{R}_{\epsilon}=\frac{\alpha_{0}}{\epsilon|\ln\epsilon|^{\frac{1}{2}}}$
for some $\alpha_{0}$ not large such that
$\mathbb{R}_{\epsilon}\leq\frac{1}{2}{\tilde{d}}$, and
$\|\cdot\|_{C^{\alpha}(D)},[\cdot]_{\alpha,D},\|\cdot\|_{C^{k,\alpha}(D)}$ are
defined as

$\displaystyle\|f\|_{C^{\alpha}(D)}$
$\displaystyle=\|f\|_{C^{0,\alpha}(D)},\qquad[f]_{\alpha,D}=\sup_{x,y\in
D,\,x\not=y}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}},$
$\displaystyle\|f\|_{C^{k,\alpha}(D)}$
$\displaystyle=\sum_{j=0}^{k}\|D^{j}f\|_{L^{\infty}(D)}+[D^{k}f]_{\alpha,D},\quad\text{for}~{}f:\mathbb{R}^{2}\rightarrow\mathbb{R}.$

######  Proposition 3.1.

Recall that

${\mathcal{R}}=({\mathcal{R}}^{+},{\mathcal{R}}^{-})\quad\text{with}\ \
{\mathcal{R}}^{\pm}=-i\frac{S^{\pm}(v_{d})}{v_{d}^{\pm}}.$

The following estimate for ${\mathcal{R}}$ holds

$\|{\mathcal{R}}\|_{**}\leq\frac{C}{|\ln\epsilon|}.$ (3.4)

###### Proof.

Without loss of generality, we assume $x_{1}>0$. For $S^{\pm}(v_{d})$, we
recall that

$\displaystyle S^{\pm}(v_{d})=S^{\pm}_{0}(v_{d})+S^{\pm}_{1}(v_{d}),$

with

$\displaystyle S_{0}^{\pm}(v_{d})=\Delta
v_{d}^{\pm}+\Big{[}A_{\pm}\big{(}{t^{+}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{-}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}v_{d}^{\pm},$

and

$\displaystyle S_{1}^{\pm}(v_{d})$
$\displaystyle=\epsilon^{2}(\partial^{2}_{ss}v_{d}^{\pm}-4i\partial_{s}v_{d}^{\pm}-4v_{d}^{\pm}).$

Denote

$w^{\pm}_{a}(z):=w^{\pm}(z-\tilde{d}),\quad
w^{\pm}_{b}(z):=w^{\pm}(z+\tilde{d}).$

Since $x_{1}>0$, it follows from Lemma 1.2 that

$({t^{\pm}})^{-1}|w^{\pm}_{b}(z)|=1+O_{\epsilon}(\epsilon^{2}|\ln\epsilon|).$

Substituting (3.1) into the expressions of $S_{0}^{\pm}(v_{d})$, we get

$\displaystyle S_{0}^{\pm}(v_{d})$
$\displaystyle=({t^{\pm}})^{-1}\Bigg{\\{}\,\Delta(w^{\pm}_{a}w^{\pm}_{b})+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-({t^{\pm}})^{-2}|w^{\pm}_{a}|^{2}|w^{\pm}_{b}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-({t^{\mp}})^{-2}|w^{\mp}_{a}|^{2}|w^{\mp}_{b}|^{2}\big{)}\Big{]}w^{\pm}_{a}w^{\pm}_{b}\Bigg{\\}}$
$\displaystyle=({t^{\pm}})^{-1}\Bigg{\\{}\Big{(}w^{\pm}_{b}\Delta
w^{\pm}_{a}+w^{\pm}_{a}\Delta w^{\pm}_{b}+2\nabla w^{\pm}_{a}\nabla
w^{\pm}_{b}\Big{)}$
$\displaystyle\qquad\qquad+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-({t^{\pm}})^{-2}|w^{\pm}_{a}|^{2}|w^{\pm}_{b}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-({t^{\mp}})^{-2}|w^{\mp}_{a}|^{2}|w^{\mp}_{b}|^{2}\big{)}\Big{]}w^{\pm}_{a}w^{\pm}_{b}\Bigg{\\}}$
$\displaystyle=-({t^{\pm}})^{-1}\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|w^{\pm}_{a}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|w^{\mp}_{a}|^{2}\big{)}\Big{]}w^{\pm}_{a}w^{\pm}_{b}$
$\displaystyle\quad-({t^{\pm}})^{-1}\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|w^{\pm}_{b}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|w^{\mp}_{b}|^{2}\big{)}\Big{]}w^{\pm}_{a}w^{\pm}_{b}$
$\displaystyle\quad+({t^{\pm}})^{-1}\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-({t^{\pm}})^{-2}|w^{\pm}_{a}|^{2}|w^{\pm}_{b}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-({t^{\mp}})^{-2}|w^{\mp}_{a}|^{2}|w^{\mp}_{b}|^{2}\big{)}\Big{]}w^{\pm}_{a}w^{\pm}_{b}$
$\displaystyle\quad+({t^{\pm}})^{-1}\Big{(}{W^{\pm}}^{\prime}(\ell_{1})e^{i\theta_{1}}\nabla{\ell_{1}}+i{W^{\pm}}(\ell_{1})e^{i\theta_{1}}\nabla{\theta_{1}}\Big{)}\cdot\Big{(}{W^{\pm}}^{\prime}(\ell_{2})e^{i\theta_{2}}\nabla{\ell_{2}}+i{W^{\pm}}(\ell_{2})e^{i\theta_{2}}\nabla{\theta_{2}}\Big{)}$
$\displaystyle=({t^{\pm}})^{-1}\Big{[}A_{\pm}\big{(}|w^{\pm}_{a}|^{2}-{t^{\pm}}^{2}\big{)}\big{(}1-({t^{\pm}})^{-2}|w^{\pm}_{b}|^{2}\big{)}+B\big{(}|w^{\mp}_{a}|^{2}-{t^{\mp}}^{2}\big{)}\big{(}1-({t^{\mp}})^{-2}|w^{\mp}_{b}|^{2}\big{)}\Big{]}w^{\pm}_{a}w^{\pm}_{b}$
$\displaystyle\quad+({t^{\pm}})^{-1}\Bigg{[}{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}^{\prime}(\ell_{2})\big{(}\cos{\theta_{1}}\cos{\theta_{2}}+\sin{\theta_{1}}\sin{\theta_{2}}\big{)}$
$\displaystyle\qquad\qquad\quad-{W^{\pm}}(\ell_{1})W^{\pm}(\ell_{2})\frac{\cos{\theta_{1}}\cos{\theta_{1}}-\sin{\theta_{1}}\sin{\theta_{2}}}{\ell_{1}\ell_{2}}$
$\displaystyle\qquad\qquad\quad+i{W^{\pm}}^{\prime}(\ell_{1})W^{\pm}(\ell_{2})\frac{\cos{\theta_{2}}\sin{\theta_{1}}-\cos{\theta_{1}}\sin{\theta_{2}}}{\ell_{2}}$
$\displaystyle\qquad\qquad\quad+i{W^{\pm}}^{\prime}(\ell_{2})W^{\pm}(\ell_{1})\frac{\cos{\theta_{1}}\sin{\theta_{2}}-\cos{\theta_{2}}\sin{\theta_{1}}}{\ell_{1}}\Bigg{]}e^{i(\theta_{1}+\theta_{2})}.$
(3.5)

On the other hand, we consider the expression of $S_{1}^{\pm}(v_{d})$. We
recall that

$\ell_{1}=\sqrt{(r\cos s-\tilde{d})^{2}+r^{2}\sin^{2}s},\ \ \ \ \
\ell_{2}=\sqrt{(r\cos s+\tilde{d})^{2}+r^{2}\sin^{2}s},$
$e^{i\theta_{1}}=\frac{(r\cos s-\tilde{d})+ir\sin s}{\ell_{1}},\ \ \ \
e^{i\theta_{2}}=\frac{(r\cos s+\tilde{d})+ir\sin s}{\ell_{2}}.$

Similar to the analysis in the proof of Proposition $4.1$ in [12], we give the
expressions of $S_{1}^{\pm}(v_{d})$ in $(\ell_{j},\theta_{j})$ as the
following

$\displaystyle S_{1}^{\pm}(v_{d})$
$\displaystyle=\,\Bigg{\\{}\frac{\hat{d}^{2}}{|\ln\epsilon|}\Bigg{[}{W^{\pm}}^{\prime\prime}(\ell_{1}){W^{\pm}}(\ell_{2})\sin^{2}{\theta_{1}}++{W^{\pm}}^{\prime\prime}(\ell_{2}){W^{\pm}}(\ell_{1})\sin^{2}{\theta_{2}}-{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}^{\prime}(\ell_{2})\sin{\theta_{1}}\sin{\theta_{2}}$
$\displaystyle\qquad\qquad+\frac{\cos^{2}{\theta_{1}}}{\ell_{1}}{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}(\ell_{2})+\frac{\cos^{2}{\theta_{2}}}{\ell_{2}}{W^{\pm}}^{\prime}(\ell_{2}){W^{\pm}}(\ell_{1})-\Big{(}\frac{\cos\theta_{1}}{\ell_{1}}-\frac{\cos\theta_{2}}{\ell_{2}}\Big{)}^{2}{W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})\Bigg{]}$
$\displaystyle\qquad+\frac{\epsilon\hat{d}}{\sqrt{|\ln\epsilon|}}\Bigl{(}\cos{\theta_{1}}{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}(\ell_{2})-\cos{\theta_{2}}{W^{\pm}}^{\prime}(\ell_{2}){W^{\pm}}(\ell_{1})\Bigr{)}\Bigg{\\}}(t^{\pm})^{-1}e^{i({\theta_{1}}+{\theta_{2}})}$
$\displaystyle\quad+i\Bigg{[}\frac{\epsilon\hat{d}}{\sqrt{|\ln\epsilon|}}\Big{(}\frac{\sin{\theta_{2}}}{\ell_{2}}-\frac{\sin{\theta_{1}}}{\ell_{1}}\Big{)}{W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})-\frac{2\hat{d}^{2}}{|\ln\epsilon|}\Big{(}\frac{\sin{\theta_{1}}\cos{\theta_{1}}}{\ell_{1}^{2}}+\frac{\sin{\theta_{2}}\cos{\theta_{2}}}{\ell_{2}^{2}}\Big{)}{W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})$
$\displaystyle\qquad\quad+\frac{2\hat{d}^{2}}{|\ln\epsilon|}\Big{(}\frac{\cos{\theta_{1}}}{\ell_{1}}-\frac{\cos{\theta_{2}}}{\ell_{2}}\Big{)}\Big{(}\sin{\theta_{1}}{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}(\ell_{2})-\sin{\theta_{2}}{W^{\pm}}(\ell_{2})^{\prime}{W^{\pm}}(\ell_{1})\Big{)}\Bigg{]}(t^{\pm})^{-1}e^{i({\theta_{1}}+{\theta_{2}})}.$
(3.6)

Proceeding as in the proof of Proposition $4.1$ in [12], we can show that
(3.4) holds. ∎

From the expression of $S_{1}^{\pm}(v_{d})$ in (3.1), we deduce the following:

######  Lemma 3.2.

In the region $B_{\tilde{d}}(e_{1})$, we have the decompositions

$\displaystyle
S_{1}^{\pm}(v_{d})=\frac{\hat{d}^{2}}{|\ln\epsilon|}w^{\pm}_{a,x_{2}x_{2}}w_{b}^{\pm}+\frac{\hat{d}\epsilon}{\sqrt{|\ln\epsilon|}}w^{\pm}_{a,x_{1}}w_{b}^{\pm}+\Gamma^{\pm},$
(3.7)

where $\Gamma^{\pm}$ are the terms given in (3.9). Moreover, there hold

${\rm{Re}}\int_{B_{\tilde{d}}(e_{1})}w^{\pm}_{a,x_{2}x_{2}}\overline{w}^{\pm}_{a,x_{1}}=0\quad\mbox{
and
}\quad{\rm{Re}}\int_{B_{\tilde{d}}(e_{1})}\frac{\Gamma^{\pm}}{w^{\pm}_{b}}\overline{w}^{\pm}_{a,x_{1}}=O_{\epsilon}\left(\frac{\epsilon}{\sqrt{|\ln\epsilon|}}\right).$
(3.8)

###### Proof.

We can show easily that the first equality of (3.8) holds by orthogonality.
From (3.1), we have

$\displaystyle S_{1}^{\pm}(v_{d})$
$\displaystyle=\,\frac{\hat{d}^{2}{W^{\pm}}(\ell_{2})e^{i({\theta_{1}}+{\theta_{2}})}}{|\ln\epsilon|\,t^{\pm}}\Bigg{[}{W^{\pm}}^{\prime\prime}(\ell_{1})\sin^{2}{\theta_{1}}+\Big{(}\cos^{2}\theta_{1}+2i\cos{\theta_{1}}\sin{\theta_{1}}\Big{)}\Big{(}\frac{{W^{\pm}}^{\prime}(\ell_{1})}{\ell_{1}}-\frac{{W^{\pm}}(\ell_{1})}{\ell_{1}^{2}}\Big{)}\Bigg{]}$
$\displaystyle\quad+\frac{\hat{d}\epsilon}{\sqrt{|\ln\epsilon|\,t^{\pm}}}\Bigl{[}\cos{\theta_{1}}{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}(\ell_{2})-i{W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})\frac{\sin{\theta_{1}}}{\ell_{2}}\Bigr{]}+\Gamma^{\pm},$

where

$\displaystyle\Gamma^{\pm}$
$\displaystyle=\,\Bigg{\\{}\frac{\hat{d}^{2}}{|\ln\epsilon|}\Bigg{[}{W^{\pm}}^{\prime\prime}(\ell_{2}){W^{\pm}}(\ell_{1})\sin^{2}{\theta_{2}}-{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}^{\prime}(\ell_{2})\sin{\theta_{1}}\sin{\theta_{2}}$
$\displaystyle\qquad\qquad\qquad+\frac{\cos^{2}{\theta_{2}}}{\ell_{2}}{W^{\pm}}^{\prime}(\ell_{2}){W^{\pm}}(\ell_{1})-\left(\frac{-2\cos\theta_{1}\cos\theta_{2}}{\ell_{1}\ell_{2}}+\frac{\cos^{2}\theta_{2}}{\ell_{2}^{2}}\right){W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})\Bigg{]}$
$\displaystyle\qquad+\frac{\epsilon\hat{d}}{\sqrt{|\ln\epsilon|}}\Bigl{(}-\cos{\theta_{2}}{W^{\pm}}^{\prime}(\ell_{2}){W^{\pm}}(\ell_{1})\Bigr{)}\Bigg{\\}}(t^{\pm})^{-1}e^{i({\theta_{1}}+{\theta_{2}})}$
$\displaystyle\quad+i\Bigg{[}\frac{\epsilon\hat{d}}{\sqrt{|\ln\epsilon|}}\frac{\sin{\theta_{2}}}{\ell_{2}}{W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})-\frac{2\hat{d}^{2}}{|\ln\epsilon|}\left(\frac{\sin{\theta_{2}}\cos{\theta_{2}}}{\ell_{2}^{2}}\right){W^{\pm}}(\ell_{1}){W^{\pm}}(\ell_{2})$
$\displaystyle\qquad-\frac{2\hat{d}^{2}}{|\ln\epsilon|}\frac{\cos{\theta_{2}}}{\ell_{2}}\Big{(}\sin{\theta_{1}}{W^{\pm}}^{\prime}(\ell_{1}){W^{\pm}}(\ell_{2})-\sin{\theta_{2}}{W^{\pm}}^{\prime}(\ell_{2}){W^{\pm}}(\ell_{1})\Big{)}$
$\displaystyle\qquad-\frac{2\hat{d}^{2}}{|\ln\epsilon|}\frac{\cos{\theta_{1}}}{\ell_{1}}\sin{\theta_{2}}{W^{\pm}}^{\prime}(\ell_{2}){W^{\pm}}(\ell_{1})\Bigg{]}(t^{\pm})^{-1}e^{i({\theta_{1}}+{\theta_{2}})}.$
(3.9)

From the decompositions above, we know that (3.7) and (3.8) hold. ∎

In Lemma 3.2, we decompose $S^{+}_{1}(v_{d})$ into two parts: one is
$\frac{\hat{d}^{2}}{|\ln\epsilon|}w^{+}_{a,x_{2}x_{2}}w^{+}_{b}$ which is in
big size but has a symmetry that makes it orthogonal to the kernel
${w}^{+}_{a,x_{1}}$ locally, the other one is
$\frac{\hat{d}\epsilon}{\sqrt{|\ln\epsilon|}}w^{+}_{a,x_{1}}w^{+}_{b}+\Gamma^{+}$
which has smaller size. Similar decomposition also holds for
$S^{-}_{1}(v_{d})$. Lemma 3.2 will be used in the reduction process in Section
7.

###  3.2. Fourier series for the error

Next, we shall make a more precise decomposition for the error
${\mathcal{R}}$. Recall the relations

$z-e_{j}=\ell_{j}e^{i\theta_{j}}\quad\text{for}~{}j=1,2,$

where $e_{j}$’s are given in (2.11). For given
${\mathcal{H}}=({\mathcal{H}}^{+},{\mathcal{H}}^{-})\in\mathbb{C}^{2}$
satisfying

${\mathcal{H}}(\overline{z})=-\overline{{\mathcal{H}}}(z),\quad\text{i.e,}\quad{\mathcal{H}}^{\pm}(\overline{z})=-\overline{{\mathcal{H}}^{\pm}}(z),$

we have the following local decomposition for ${\mathcal{H}}^{\pm}$ when near
$e_{j}$, $j=1,2$,

$\displaystyle{\mathcal{H}}^{\pm}(z)=\sum_{k=0}^{\infty}{\mathcal{H}}^{\pm}_{k,j}(\ell_{j},\theta_{j}),$
(3.10)

and

$\displaystyle{\mathcal{H}}^{\pm}_{k,j}(\ell_{j},\theta_{j})={\mathcal{H}}_{k,j}^{1,\pm}(\ell_{j})\sin(k\theta_{j})+i{\mathcal{H}}_{k,j}^{2,\pm}(\ell_{j})\cos(k\theta_{j}),\quad{\mathcal{H}}_{k,j}^{1,\pm}(\ell_{j}),{\mathcal{H}}_{k,j}^{2,\pm}(\ell_{j})\in\mathbb{R}.$
(3.11)

Then define

$\displaystyle{\mathcal{H}}^{\pm}_{e,j}=\sum_{k~{}\text{is
even}}{\mathcal{H}}^{\pm}_{k,j},\qquad{\mathcal{H}}^{\pm}_{o,j}=\sum_{k~{}\text{is
odd}}{\mathcal{H}}^{\pm}_{k,j},$ (3.12)
$\displaystyle{\mathcal{H}}_{e,j}=({\mathcal{H}}^{+}_{e,j},{\mathcal{H}}^{-}_{e,j}),\qquad{\mathcal{H}}_{o,j}=({\mathcal{H}}^{+}_{o,j},{\mathcal{H}}^{-}_{o,j}).$
(3.13)

Let $\mathscr{R}_{j}z$ denote the reflection across the real line
${{\rm{Re}}(z)}=(-1)^{j+1}\tilde{d}$, then

$\displaystyle\mathscr{R}_{j}z=2e_{j}-{{\rm{Re}}(z)}+i{{\rm{Im}}(z)},\quad\text{or}\quad\mathscr{R}_{j}z=\ell_{j}e^{i(\pi-\theta_{j})}+e_{j}.$
(3.14)

Combining (3.14) and (3.11)-(3.13), we get

$\displaystyle{\mathcal{H}}^{\pm}_{e,j}\big{(}\mathscr{R}_{j}z\big{)}=-\overline{{\mathcal{H}}^{\pm}_{e,j}}(z),\qquad{\mathcal{H}}^{\pm}_{o,j}\big{(}\mathscr{R}_{j}z\big{)}=\overline{{\mathcal{H}}^{\pm}_{o,j}}(z),$
$\displaystyle{\mathcal{H}}_{e,j}\big{(}\mathscr{R}_{j}z\big{)}=-\overline{{\mathcal{H}}_{e,j}}(z),\qquad{\mathcal{H}}_{o,j}\big{(}\mathscr{R}_{j}z\big{)}=\overline{{\mathcal{H}}_{o,j}}(z),$

and thus we can define equivalently

$\displaystyle{\mathcal{H}}_{o,j}^{\pm}(z)=\frac{1}{2}\Big{[}{\mathcal{H}}^{\pm}(z)+\overline{{\mathcal{H}}^{\pm}(\mathscr{R}_{j}z)}\Big{]},\qquad{\mathcal{H}}_{e,j}^{\pm}(z)=\frac{1}{2}\Big{[}{\mathcal{H}}^{\pm}(z)-\overline{{\mathcal{H}}^{\pm}(\mathscr{R}_{j}z)}\Big{]},$
$\displaystyle{\mathcal{H}}_{o,j}(z)=\frac{1}{2}\Big{[}{\mathcal{H}}(z)+\overline{{\mathcal{H}}(\mathscr{R}_{j}z)}\Big{]},\qquad\quad\,\,\,{\mathcal{H}}_{e,j}(z)=\frac{1}{2}\Big{[}{\mathcal{H}}(z)-\overline{{\mathcal{H}}(\mathscr{R}_{j}z)}\Big{]}.$

Note that, in the coordinates $(\ell_{j},\theta_{j})$, the functions
${\mathcal{H}}_{o,j}$ introduced in (3.11)-(3.13) are the odd parts of
${\mathcal{H}}$ in local domains $B_{\tilde{d}}(e_{j}),j=1,2$. Furthermore, we
would define functions ${\mathcal{H}}^{\pm}_{o},{\mathcal{H}}_{o}$ which stand
for the odd part of ${\mathcal{H}}^{\pm},{\mathcal{H}}$ globally. Define the
cut-off functions $\eta_{j,R}$ for $j=1,2$ as the following

$\displaystyle\eta_{j,R}(z)=\eta_{j}\Big{(}\frac{|z-e_{j}|}{2}\Big{)},$ (3.15)

where $\eta_{1}$ is the cut-off functions defined in (2.16) and $R$ is a
constant to be chosen later. For any given
${\mathcal{H}}=({\mathcal{H}}^{+},{\mathcal{H}}^{-})\in\mathbb{C}^{2}$, we
define

$\displaystyle{\mathcal{H}}_{o}=\big{(}{\mathcal{H}}_{o}^{+},{\mathcal{H}}_{o}^{-}\big{)},\quad\text{with}\quad{\mathcal{H}}_{o}^{\pm}=\eta_{1,\mathbb{R}_{\epsilon}}{\mathcal{H}}_{o,1}^{\pm}+\eta_{2,\mathbb{R}_{\epsilon}}{\mathcal{H}}_{o,2}^{\pm},\quad$
(3.16)

where
$\mathbb{R}_{\epsilon}=\frac{c_{0}}{\epsilon|\ln\epsilon|^{\frac{1}{2}}}$ for
some $c_{0}$ not large such that
$\mathbb{R}_{\epsilon}\leq\frac{1}{2}\tilde{d}.$ On the other hand, define the
global functions

${\mathcal{H}}_{e}={\mathcal{H}}-{\mathcal{H}}_{o},\quad\text{with}\quad{\mathcal{H}}_{e}^{\pm}={\mathcal{H}}^{\pm}-{\mathcal{H}}_{o}^{\pm}.$

We note that, using the decompositions introduced above, we could decompose
the error function

${\mathcal{R}}=({\mathcal{R}}^{+},{\mathcal{R}}^{-})=\Bigg{(}\frac{S^{+}(v_{d})}{iv_{d}^{+}},\frac{S^{-}(v_{d})}{iv_{d}^{-}}\Bigg{)}$

in odd Fourier modes and even Fourier modes, see Proposition 3.3. The odd
Fourier modes of the error have much smaller sizes than even ones. However,
there exists one difficulty: some terms of odd Fourier modes have a good size
but decay slowly. Using the idea in [12], we introduce the semi-norm
$|\cdot|_{\sharp\sharp}$ for these slow decay terms. For any
${\mathcal{H}}=({\mathcal{H}}^{+},{\mathcal{H}}^{-})=({\mathcal{H}}^{+}_{1}+i{\mathcal{H}}^{+}_{2},{\mathcal{H}}^{-}_{1}+i{\mathcal{H}}^{-}_{2})\in\mathbb{C}^{2}$,
the semi-norm $|{\mathcal{H}}|_{\sharp\sharp}$ is defined as

$\displaystyle|{\mathcal{H}}|_{\sharp\sharp}=|{\mathcal{H}}^{+}|_{\sharp\sharp_{1}}+|{\mathcal{H}}^{-}|_{\sharp\sharp_{1}},$
(3.17)

where

$\displaystyle|{\mathcal{H}}^{\pm}|_{\sharp\sharp_{1}}=\sum_{j=1}^{2}\|v_{d}^{\pm}{\mathcal{H}}^{\pm}\|_{C^{0,\alpha}(\ell_{j}<4)}\
+\
\sup_{2<\ell_{1}<\mathbb{R}_{\epsilon},\,2<\ell_{2}<\mathbb{R}_{\epsilon}}\Biggl{[}\frac{|{\mathcal{H}}^{\pm}_{1}|}{\ell_{1}^{-1}+\ell_{2}^{-1}}+\frac{|{\mathcal{H}}^{\pm}_{2}|}{\ell_{1}^{-1+\sigma}+\ell_{2}^{-1+\sigma}}\Biggr{]}.$
(3.18)

Now we give a precise decomposition for the error ${\mathcal{R}}$.

######  Proposition 3.3.

Recall that $v_{d}$ is given by (2.9) and

$\mathbb{E}=S(v_{d})=\big{(}iv_{d}^{+}{\mathcal{R}}^{+},iv_{d}^{-}{\mathcal{R}}^{-}\big{)},\qquad{\mathcal{R}}=({\mathcal{R}}^{+},{\mathcal{R}}^{-}).$

Then we can write

${\mathcal{R}}={\mathcal{R}}_{o}+{\mathcal{R}}_{e}=\big{(}{\mathcal{R}}_{o}^{+},{\mathcal{R}}_{o}^{-}\big{)}+\big{(}{\mathcal{R}}_{e}^{+},{\mathcal{R}}_{e}^{-}\big{)},\qquad{\mathcal{R}}_{o}={\mathcal{R}}_{o}^{\alpha}+{\mathcal{R}}_{o}^{\beta}=\big{(}{\mathcal{R}}_{o}^{+,\alpha},{\mathcal{R}}_{o}^{-,\alpha}\big{)}+\big{(}{\mathcal{R}}_{o}^{+,\beta},{\mathcal{R}}_{o}^{-,\beta}\big{)},$

i.e,

${\mathcal{R}}^{\pm}={\mathcal{R}}^{\pm}_{o}+{\mathcal{R}}^{\pm}_{e},\qquad{\mathcal{R}}^{\pm}_{o}={\mathcal{R}}_{o}^{\pm,\alpha}+{\mathcal{R}}_{o}^{\pm,\beta},$

with ${\mathcal{R}}_{o},{\mathcal{R}}_{o}^{\pm}$ defined analogously to
(3.16). Moreover, the symmetry
${\mathcal{R}}_{o}(\mathscr{R}_{j}z)=\overline{{\mathcal{R}}_{o}(z)}$ in
$B_{\mathbb{R}_{\epsilon}}(e_{1})\cup B_{\mathbb{R}_{\epsilon}}(e_{2})$ is
true and the following estimates are valid

$\displaystyle\|{\mathcal{R}}_{e}\|_{**}+\|{\mathcal{R}}_{o}\|_{**}\leq\frac{C}{|\ln\epsilon|},\qquad|{\mathcal{R}}_{o}^{\alpha}|_{\sharp\sharp}$
$\displaystyle\leq
C\frac{\epsilon}{\sqrt{|\ln\epsilon|}},\qquad\|{\mathcal{R}}_{o}^{\beta}\|_{**}\leq
C\epsilon\sqrt{|\ln\epsilon|}.$

###### Proof.

Following Proposition 3.1, we immediately get that

$\displaystyle\|{\mathcal{R}}_{e}\|_{**}+\|{\mathcal{R}}_{o}\|_{**}\leq\frac{C}{|\ln\epsilon|}.$

For every $j\in\\{1,2\\}$, we define

$\displaystyle r_{o,1}^{\pm}$
$\displaystyle:=-i\frac{{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\Bigg{(}\frac{2\cos\theta_{1}\cos\theta_{2}}{\ell_{1}\ell_{2}}+\frac{\cos^{2}\theta_{2}}{\ell_{2}^{2}}\Bigg{)}+\Bigg{(}\frac{{\tilde{d}}\epsilon}{\sqrt{|\ln\epsilon|}(t^{\pm})}\frac{-\sin\theta_{1}}{\ell_{1}}-\frac{2{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\frac{\sin\theta_{2}\cos\theta_{2}}{\ell_{2}^{2}}\Bigg{)},$
$\displaystyle r_{o,2}^{\pm}$
$\displaystyle:=-i\frac{{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\Bigg{(}\frac{2\cos\theta_{1}\cos\theta_{2}}{\ell_{1}\ell_{2}}+\frac{\cos^{2}\theta_{1}}{\ell_{1}^{2}}\Bigg{)}+\Bigg{(}\frac{{\tilde{d}}\epsilon}{\sqrt{|\ln\epsilon|}(t^{\pm})}\frac{+\sin\theta_{2}}{\ell_{2}}-\frac{2{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\frac{\sin\theta_{1}\cos\theta_{1}}{\ell_{1}^{2}}\Bigg{)},$
$\displaystyle
R_{o,j}^{\pm,\alpha}:=\frac{1}{2}\Big{[}r_{o,j}^{\pm}(z)+\overline{r_{o,j}^{\pm}\left(\mathscr{R}_{j}z\right)}\Big{]},$
(3.19)

and

$R_{o}^{\alpha}:=(R_{o}^{+,\alpha},R_{o}^{-,\alpha}),\quad\text{with}~{}R_{o}^{\pm,\alpha}:=\eta_{1,\mathbb{R}_{\epsilon}}R_{o,1}^{\pm,\alpha}+\eta_{2,\mathbb{R}_{\epsilon}}R_{o,2}^{\pm,\alpha}.$

Note that the terms in $r_{o,j}^{\pm}$ are all slow decay terms.

We claim that the components of functions $R_{o,j}^{\pm,\alpha}$ are all odd
mode terms when they lie locally in domain $B_{\mathbb{R}_{\epsilon}}(e_{j})$.
In fact, for example, taking $j=1$, when
${|z-e_{1}|}\leq\mathbb{R}_{\epsilon}$, by rewriting $r_{o,1}^{\pm}$ in
Fourier series in $\theta_{1}$, there hold

$\displaystyle\frac{2\cos\theta_{1}\cos\theta_{2}}{\ell_{1}\ell_{2}}$
$\displaystyle=\frac{2\cos\theta_{1}(x_{1}+\tilde{d})}{\ell_{1}\ell_{2}^{2}}=\frac{2\cos\theta_{1}\cos\theta_{1}}{\ell_{2}^{2}}+\frac{4\cos\theta_{1}\tilde{d}}{\ell_{1}\ell_{2}^{2}}$
$\displaystyle=\frac{1}{\ell_{1}^{2}+{\tilde{d}}^{2}+4\tilde{d}\ell_{1}\cos\theta_{1}}\Big{(}2\cos^{2}\theta_{1}+\frac{4\cos\theta_{1}\tilde{d}}{\ell_{1}}\Big{)}$
$\displaystyle=\sum_{m=0}^{\infty}a_{m}(\ell_{1})\cos^{m}{\theta_{1}},$

and

$\displaystyle\frac{\cos^{2}\theta_{2}}{\ell_{2}^{2}}$
$\displaystyle=\frac{(x_{1}-\tilde{d}+2\tilde{d})^{2}}{\ell_{2}^{4}}$
$\displaystyle=\Big{(}\frac{1}{\ell_{1}^{2}+{\tilde{d}}^{2}+4\tilde{d}\ell_{1}\cos\theta_{1}}\Big{)}^{2}(\ell_{1}^{2}\cos^{2}\theta_{1}+2\tilde{d}\ell_{1}\cos\theta_{1}+4\tilde{d}^{2})$
$\displaystyle=\sum_{m=0}^{\infty}b_{m}(\ell_{1})\cos^{m}{\theta_{1}},$
$\displaystyle\frac{\sin\theta_{2}\cos\theta_{2}}{\ell_{2}^{2}}$
$\displaystyle=\frac{(x_{1}-\tilde{d}+2\tilde{d})x_{2}}{\ell_{4}^{2}}=\frac{(x_{1}-\tilde{d})x_{2}+2\tilde{d}x_{2}}{\ell_{2}^{4}}$
$\displaystyle=\Big{(}\frac{1}{\ell_{1}^{2}+{\tilde{d}}^{2}+4\tilde{d}\ell_{1}\cos\theta_{1}}\Big{)}^{2}(\ell_{1}^{2}\sin\theta_{1}\cos\theta_{1}+2\tilde{d}\ell_{1}\sin\theta_{1})$
$\displaystyle=\sin\theta_{1}\sum_{m=0}^{\infty}c_{m}(\rho_{1})\cos^{m}{\theta_{1}},$

where $a_{m},b_{m},c_{m}$ are bounded smooth functions. Therefore, we can
rewrite $r_{o,1}^{\pm}$ as

$\displaystyle r_{o,1}^{\pm}(z)$
$\displaystyle=-i\frac{{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\Bigg{(}\sum_{m=0}^{\infty}\Big{[}a_{m}(\ell_{1})\cos^{m}{\theta_{1}}+b_{m}(\ell_{1})\cos^{m}{\theta_{1}}\Big{]}\Bigg{)}$
(3.20)
$\displaystyle\quad+\Bigg{(}\frac{{\tilde{d}}\epsilon}{\sqrt{|\ln\epsilon|}(t^{\pm})}\frac{-\sin\theta_{1}}{\ell_{1}}-\frac{2{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\sin\theta_{1}\sum_{m=0}^{\infty}c_{m}(\rho_{1})\cos^{m}{\theta_{1}}\Bigg{)}.$
(3.21)

By the analysis above and recalling

$\mathscr{R}_{1}z=\ell_{1}e^{i(\pi-\theta_{1})}+e_{1},$

we can easily show that

$\displaystyle\overline{r_{o,1}^{\pm}\left(\mathscr{R}_{1}z\right)}$
$\displaystyle=-i\frac{{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\Bigg{(}\sum_{m=0}^{\infty}(-1)^{m+1}\Big{[}a_{m}(\ell_{1})\cos^{m}{\theta_{1}}+b_{m}(\ell_{1})\cos^{m}{\theta_{1}}\Big{]}\Bigg{)}$
$\displaystyle\quad+\Bigg{(}\frac{{\tilde{d}}\epsilon}{\sqrt{|\ln\epsilon|}(t^{\pm})}\frac{-\sin\theta_{1}}{\ell_{1}}-\frac{2{\tilde{d}}^{2}}{|\ln\epsilon|(t^{\pm})}\sin\theta_{1}\sum_{m=0}^{\infty}(-1)^{m}c_{m}(\rho_{1})\cos^{m}{\theta_{1}}\Bigg{)}.$
(3.22)

Therefore, from (3.19), (3.20), and (3.2), we know that the claim above is
true. Furthermore, we can also check that $R_{o}^{\alpha}$ and
$R_{o}^{\beta}:=R_{o}-R_{o}^{\alpha}$ satisfy the desired properties. ∎

##  4\. The resolution of projected linear problem

For convenience, we first introduce some symmetries for the functions
${\mathcal{H}},\psi\in\mathbb{C}^{2}:$

$\displaystyle{\mathcal{H}}(\overline{z})=-\overline{{\mathcal{H}}}(z),$ (S1)
$\displaystyle{\mathcal{H}}(\mathscr{R}_{j}z)=-\overline{{\mathcal{H}}(z)},\quad|z-e_{j}|<2\mathbb{R}_{\epsilon},\quad
j=1,2,$ (S2)
$\displaystyle{\mathcal{H}}(\mathscr{R}_{j}z)=\overline{{\mathcal{H}}(z)},\quad|z-e_{j}|<2\mathbb{R}_{\epsilon},\quad
j=1,2,$ (S3) $\displaystyle\psi(x_{1},-x_{2})=-\overline{\psi}(x_{1},x_{2}),\
\ \ \ \ \psi(-x_{1},x_{2})=-\overline{\psi}(x_{1},x_{2}).$ (S4)

From the analysis in Lemma 2.2, we know that the linear operator
$L^{\pm}_{d,j}(\Phi):=iw^{\pm}(z-e_{j})\mathcal{L}_{\epsilon}^{\pm}(\psi)$ has
one nontrivial kernel for $\psi$ which satisfies the symmetry (S4) in the
vortex region. Therefore, in order to solve the problem (2.22), we should
consider the nonlinear projected problem for $\psi$ satisfies the symmetry
(S4)

$\displaystyle\left\\{\begin{aligned}
&{\mathcal{L}}^{\pm}_{\epsilon}(\psi)={\mathcal{R}}^{\pm}-\mathcal{N}^{\pm}(\psi)+c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{\pm}(z-e_{j})}(-1)^{j}w^{\pm}_{x_{1}}(z-e_{j})\quad\text{in
}\mathbb{R}^{2},\\\\[5.69054pt]
&{\rm{Re}}\int_{B(0,4)}\chi\Big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\Big{]}=0,\text{
with }\phi^{\pm}_{j}(z)=iw^{\pm}(z)\psi^{\pm}(z+e_{j}),\quad
j=1,2,\\\\[5.69054pt] &\psi\text{ satisfies the symmetry
}\eqref{s4},\end{aligned}\right.$ (4.1)

where

$\chi(z):=\eta_{1}\Big{(}\frac{|z|}{2}\Big{)},\qquad\chi_{j}(z):=\eta_{1}\Big{(}\frac{\ell_{j}}{2}\Big{)}=\eta_{1}\Big{(}\frac{|z-e_{j}|}{2}\Big{)},$

and $\eta_{1}$ is a smooth cut-off function defined in (2.16). The resolution
theory of a linear projected problem will be first provided for solving (4.1).
Then an application of the Contraction Mapping Principle will give the
existence of solutions.

###  4.1. The linear resolution theory

The main objective of this part is to set up the resolution theory of a linear
projected problem. For any ${\mathcal{H}}$ satisfies (S4), we first consider
the projected problem

$\displaystyle\left\\{\begin{aligned}
&{\mathcal{L}}^{\pm}_{\epsilon}(\psi)={\mathcal{H}}^{\pm}+c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{\pm}(z-e_{j})}(-1)^{j}w^{\pm}_{x_{1}}(z-e_{j})\quad\text{in
}\mathbb{R}^{2},\\\\[5.69054pt]
&{\rm{Re}}\int_{B(0,4)}\chi\Big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\Big{]}=0,\text{
with }\phi^{\pm}_{j}(z)=iw^{\pm}(z)\psi^{\pm}(z+e_{j}),\quad
j=1,2,\\\\[5.69054pt] &\psi\text{ satisfies the symmetry
}\eqref{s4}.\end{aligned}\right.$ (4.2)

For solving (4.2), at first we shall get a priori estimates expressed in
suitable norms. Define for
$\psi=(\psi^{+},\psi^{-}):\mathbb{R}^{2}\to{\mathbb{C}}^{2}$ the norms for
fixed small $\alpha>0,\sigma>0$,

$\displaystyle\|\psi\|_{*}=\|\psi^{+}\|_{*,1}+\|\psi^{-}\|_{*,1},$ (4.3)
$\displaystyle\|\psi^{\pm}\|_{*,1}=\sum_{j=1}^{2}\|v_{d}^{\pm}\psi^{\pm}\|_{C^{2,\alpha}(\ell_{j}<3)}+\|{\rm{Re}}(\psi^{\pm})\|_{*,1,{Re}}+\|{\rm{Im}}(\psi^{\pm})\|_{*,1,{Im}},$
(4.4)

where, by the relations

${\rm{Re}}(\psi^{\pm})=\psi^{\pm}_{1},\quad{\rm{Im}}(\psi^{\pm})=\psi^{\pm}_{2},$

we have denoted

$\displaystyle\|\psi_{1}^{\pm}\|_{*,1,Re}$
$\displaystyle=\sup_{\ell_{1}>2,\ell_{2}>2}|\psi_{1}^{\pm}|\ +\
\sup_{2<\ell_{1}<\frac{2}{\epsilon},\,2<\ell_{2}<\frac{2}{\epsilon}}\frac{|\nabla\psi_{1}^{\pm}|}{\ell_{1}^{-1}+\ell_{2}^{-1}}\
+\
\sup_{r>\frac{1}{\epsilon}}\Biggl{[}\frac{1}{\epsilon}|\partial_{r}\psi_{1}^{\pm}|\,+\,|\partial_{s}\psi_{1}^{\pm}|\Biggr{]}$
$\displaystyle\qquad\ +\
\sup_{2<\ell_{1}<\mathbb{R}_{\epsilon},\,2<\ell_{2}<\mathbb{R}_{\epsilon}}\frac{|D^{2}\psi_{1}^{\pm}|}{\ell_{1}^{-2}+\ell_{2}^{-2}}\
+\
\sup_{2<\ell_{1}<\mathbb{R}_{\epsilon},\,2<\ell_{2}<\mathbb{R}_{\epsilon}}\frac{[D^{2}\psi_{1}^{\pm}]_{\alpha,B_{|z|/2}(z)}}{\ell_{1}^{-2-\alpha}+\ell_{2}^{-2-\alpha}},$
(4.5) $\displaystyle\|\psi_{2}^{\pm}\|_{*,1,Im}$
$\displaystyle=\sup_{\ell_{1}>2,\ell_{2}>2}\frac{|\psi_{2}^{\pm}|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}+\epsilon^{\sigma-2}}\
+\
\sup_{2<\ell_{1}<\frac{2}{\epsilon},\,2<\ell_{2}<\frac{2}{\epsilon}}\frac{|\nabla\psi_{2}^{\pm}|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}}$
$\displaystyle\qquad\ +\
\sup_{r>\frac{1}{\epsilon}}\Big{[}\epsilon^{\sigma-2}|\partial_{r}\psi_{2}^{\pm}|+\epsilon^{\sigma-1}|\partial_{s}\psi_{2}^{\pm}|\Big{]}$
$\displaystyle\qquad\ +\
\sup_{2<\ell_{1}<\mathbb{R}_{\epsilon},\,2<\ell_{2}<\mathbb{R}_{\epsilon}}\frac{|D^{2}\psi_{2}^{\pm}|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}}\
+\
\sup_{2<\ell_{1}<\mathbb{R}_{\epsilon},\,2<\ell_{2}<\mathbb{R}_{\epsilon}}\frac{[D^{2}\psi_{2}^{\pm}]_{\alpha,B_{1}(z)}}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}}.$
(4.6)

We next give the solvability of linear projected problem (4.2).

######  Proposition 4.1.

There exists a constant $C>0$ depending only on $\alpha,\sigma\in(0,1)$, such
that the following hold: if ${\mathcal{H}}$ satisfies (S4) and

$\|{\mathcal{H}}\|_{**}<+\infty,$

then for $\epsilon>0$ sufficiently small there exists a unique solution
$T_{\epsilon}({\mathcal{H}})=\big{(}T_{\epsilon}^{1}({\mathcal{H}}),T_{\epsilon}^{2}({\mathcal{H}})\big{)}=(\psi_{\epsilon},c_{\epsilon})$
to (4.2). Furthermore, there holds

$\|\psi_{\epsilon}\|_{*}\leq C\|{\mathcal{H}}\|_{**}.$

###### Proof.

We delay the proof of this proposition to § 5.1. ∎

###  4.2. More precise estimates and decompositions for $\psi$

The resolution theory of (4.2) provided in Proposition 4.1 implies that we can
find a solution of (4.1) in the region

$\begin{split}\mathcal{A}:=\Bigg{\\{}&\psi:\;\psi\text{ satisfies
}\eqref{s4},\
{\rm{Re}}\int_{B(0,4)}\chi\big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\big{]}=0,\;\;j=1,2,\;\text{and}\;\|\psi\|_{*}\leq\frac{C}{|\ln\epsilon|}\Bigg{\\}}.\end{split}$

In fact, from Proposition 4.1, the problem (4.1) is equivalent to the
following fixed point problem

$\psi=T_{\epsilon}\Big{(}{\mathcal{R}}+\mathcal{N}(\psi)\Big{)}=:\mathbb{G}_{\epsilon}(\psi).$

The existence of $\psi$ to (4.1) follows by the Contraction Mapping Principle,
see Proposition 6.1 in Section 6 for more details.

Next, to find a real solution of (2.17), we need to solve the reduced problem
by finding a suitable $\hat{d}$ such that the multiplier $c$ in (4.1) is
identical zero when $\epsilon$ is sufficiently small. However, since the
estimates for $\psi$ are not delicate enough, we can not carry out reduction
procedure such that $c$ is zero. In order to get around the technical
difficulty, we are led to using the idea doing more precise decompositions and
estimates for the perturbation $\psi$. This was first introduced by J. Dávila,
M. del Pino, M. Medina and R. Rodiac in [12] and [13] to construct vortex
helical filaments of classical (single complex component) Ginzburg-Landau
equation and Gross-Pitaevskii equation.

For $\psi=\big{(}\psi^{+},\psi^{-}\big{)}\in\mathbb{C}^{2}$ satisfying (S4),
we decompose $\psi$ in Fourier series in $\theta_{j}$ as in (3.10) and define

$\displaystyle\psi_{e,j}^{\pm}=\sum_{k\text{
even}}\psi^{\pm}_{k,j},\qquad\psi_{o,j}^{\pm}=\sum_{k\text{
odd}}\psi_{k,j}^{\pm},\ \ j=1,2.$

Similar to the definitions of ${\mathcal{H}}_{o}$ and ${\mathcal{H}}_{e}$ in
Section 3, we define

$\displaystyle\psi_{o}=(\psi_{o}^{+},\psi_{o}^{-}),\quad\text{with}\ \
\psi^{\pm}_{o}:=\eta_{1,\frac{1}{2}\mathbb{R}_{\epsilon}}\psi_{o,1}^{\pm}+\eta_{2,\frac{1}{2}\mathbb{R}_{\epsilon}}\psi_{o,2}^{\pm},$
(4.7)
$\displaystyle\psi_{e}=(\psi_{e}^{+},\psi_{e}^{-}),\quad\text{with}\quad\psi^{\pm}_{e}:=\psi^{\pm}-\psi^{\pm}_{o},$
(4.8)

where $\eta_{1,\frac{1}{2}\mathbb{R}_{\epsilon}}$ and
$\eta_{2,\frac{1}{2}\mathbb{R}_{\epsilon}}$ are two cut-off functions given by
the form in (3.15).

We recall some facts about the error in the form

${\mathcal{R}}=({\mathcal{R}}^{+},{\mathcal{R}}^{-})=\Bigg{(}\frac{S^{+}(v_{d})}{iv_{d}^{+}},\frac{S^{-}(v_{d})}{iv_{d}^{-}}\Bigg{)}.$

There are some odd Fourier modes terms which have good size but decay slowly.
Therefore, we have introduced the semi-norm $|\cdot|_{\sharp\sharp}$ in (3.17)
for these slow decay terms. As we have stated, by taking
${\mathcal{H}}^{\pm}=-{\mathcal{R}}^{\pm}-\mathcal{N}^{\pm}(\psi)$ in
Proposition 4.1 we can solve the nonlinear projected problem (4.1) in
Proposition 6.1. And then, in order to carry out the reduction process, we
will establish Proposition 4.2. To prove Proposition 4.2, we would have to
solve some problems like $\Delta\psi_{1}^{\pm}\approx
O\Big{(}\frac{1}{\ell_{1}}\Big{)}$ and thus the functions $\psi_{1}^{\pm}$
would grow logarithmically up to a certain distance, see Lemma 5.2 and Lemma
8.5 for more explanations. As a result of these, we need the following semi-
norm to capture the behaviors of
$\psi=\big{(}\psi^{+},\psi^{-}\big{)}=\big{(}\psi^{+}_{1}+i\psi^{+}_{2},\psi^{-}_{1}+i\psi^{-}_{2}\big{)}$:

$\displaystyle|\psi|_{\sharp}$
$\displaystyle=|\psi^{+}|_{\sharp,1}+|\psi^{-}|_{\sharp,1}$ (4.9)

with

$\displaystyle|\psi^{\pm}|_{\sharp,1}$
$\displaystyle=\sum_{j=1}^{2}|\log\epsilon|^{-1}\|V_{d}\psi^{\pm}\|_{C^{2,\alpha}(\ell_{j}<3)}+|\psi_{1}^{\pm}|_{\sharp,1,Re}+|\psi_{2}^{\pm}|_{\sharp,1,Im},$
(4.10)

where, by the relations

${\rm{Re}}(\psi^{\pm})=\psi^{\pm}_{1},\quad{\rm{Im}}(\psi^{\pm})=\psi^{\pm}_{2},$

we have denoted

$\displaystyle|\psi_{1}^{\pm}|_{\sharp,1,Re}$
$\displaystyle=\sup_{2<\ell_{j}<\mathbb{R}_{\epsilon},j=1,2}\Biggl{[}\frac{|\psi_{1}^{\pm}|}{\ell_{1}\log(2\mathbb{R}_{\epsilon}/\ell_{1})+\ell_{2}\log(2\mathbb{R}_{\epsilon}/\ell_{2})}\
+\
\frac{|\nabla\psi_{1}^{\pm}|}{\log(2\mathbb{R}_{\epsilon}/\ell_{1})+\log(2\mathbb{R}_{\epsilon}/\ell_{2})}\Biggr{]},$
(4.11)

and

$\displaystyle|\psi_{2}^{\pm}|_{\sharp,1,Im}$
$\displaystyle=\sup_{2<\ell_{j}<\mathbb{R}_{\epsilon},j=1,2}\Biggl{[}\frac{|\psi_{2}^{\pm}|+|\nabla\psi_{2}^{\pm}|}{\ell_{1}^{-1+\sigma}+\ell_{2}^{-1+\sigma}+\ell_{1}^{-1}\log(2\mathbb{R}_{\epsilon}/\ell_{1})+\ell_{2}^{-1}\log(2\mathbb{R}_{\epsilon}/\ell_{2})}\Biggr{]}.$
(4.12)

We recall the semi-norm $|\cdot|_{\sharp\sharp}$ in (3.17) and the reflection
mapping $\mathscr{R}_{j}z$ in (3.14). We finally give a crucial proposition in
this paper. In some sense, it is the “heart” of the present paper.

######  Proposition 4.2.

Suppose that ${\mathcal{H}}$ satisfies the symmetries (S4) and
$\|{\mathcal{H}}\|_{**}<\infty$. ${\mathcal{H}}_{o}$ is the function defined
in (3.16) and it can be decomposed as

${\mathcal{H}}_{o}={\mathcal{H}}_{o}^{\alpha}+{\mathcal{H}}_{o}^{\beta},$

where $|{\mathcal{H}}_{o}^{\alpha}|_{\sharp\sharp}<\infty$, and the terms
${\mathcal{H}}_{o}^{\alpha}$, ${\mathcal{H}}_{o}^{\beta}$ satisfy (S3), i.e,

$\displaystyle{\mathcal{H}}_{o}^{\kappa}(\mathscr{R}_{j}z)=\overline{{\mathcal{H}}_{o}^{\kappa}(z)},\quad|z-e_{j}|<\mathbb{R}_{\epsilon},\quad
j=1,2\quad\text{and}\quad\kappa=\alpha,\beta,$

and have supports in $B_{2\mathbb{R}_{\epsilon}}(e_{1})\cup
B_{2\mathbb{R}_{\epsilon}}(e_{2})$. Then for any solution
$\psi=\psi_{e}+\psi_{o}$ of (4.2) with $\psi_{o}$ defined by (4.7), $\psi_{o}$
can be decomposed as

$\psi_{o}=\psi_{o}^{\alpha}+\psi_{o}^{\beta},$

where $\psi_{o}^{\kappa}$ are supported in
$B_{\mathbb{R}_{\epsilon}}(e_{1})\cup B_{\mathbb{R}_{\epsilon}}(e_{2})$ with
$\kappa=\alpha,\beta$. Furthermore, the following estimates hold

$\displaystyle|\psi^{\alpha}_{o}|_{\sharp}$
$\displaystyle\lesssim|{\mathcal{H}}^{\alpha}_{o}|_{\sharp\sharp}+\epsilon|\log\epsilon|^{\frac{1}{2}}(\|{\mathcal{H}}^{\alpha}_{o}\|_{**}+\|{\mathcal{H}}-{\mathcal{H}}_{o}\|_{**}),$
$\displaystyle\|\psi^{\beta}_{o}\|_{*}$
$\displaystyle\lesssim\|{\mathcal{H}}_{o}^{\beta}\|_{**},$
$\displaystyle\|\psi^{\alpha}_{o}\|_{*}+\|\psi^{\beta}_{o}\|_{*}$
$\displaystyle\lesssim\|{\mathcal{H}}\|_{**}+\|{\mathcal{H}}^{\alpha}_{o}\|_{**}+\|{\mathcal{H}}^{\beta}_{o}\|_{**},$

and

$\displaystyle\psi_{o}^{\kappa}(\mathcal{R}_{j}z)=\overline{\psi_{o}^{\kappa}(z)},\quad|z-e_{j}|<\mathbb{R}_{\epsilon},\quad
j=1,2,\ \kappa=\alpha,\beta.$

###### Proof.

The proof of Proposition 4.2 will be provided in § 5.2. ∎

##  5\. proof of linear theory

###  5.1. Proof of Proposition 4.1

We now consider the following: for given ${\mathcal{H}}$ satisfying (S4), find
$\psi$ such that

$\displaystyle\left\\{\begin{aligned}
&{\mathcal{L}}^{\pm}_{\epsilon}(\psi)={\mathcal{H}}^{\pm}\quad\text{in
}\mathbb{R}^{2},\\\\[5.69054pt]
&{\rm{Re}}\int_{B(0,4)}\chi\big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\big{]}=0,\text{
with }\phi^{\pm}_{j}(z)=iw^{\pm}(z)\psi^{\pm}(z+e_{j}),\\\\[5.69054pt]
&\psi\text{ satisfies the symmetry }\eqref{s4}.\end{aligned}\right.$ (5.1)

Recall the definitions of norms $\|\cdot\|_{*}$ in (4.3)-(4.6) and
$\|\cdot\|_{**}$ in (3.2)-(3.3). We shall first give a priori estimate for the
problem (5.1).

######  Lemma 5.1.

There exists a constant $C>0$, depending only on
$\alpha\in(0,1),\sigma\in(0,1)$, such that for all $\epsilon$ sufficiently
small and any solution $\psi$ of (5.1) with $\|\psi\|_{*}<\infty$ one has

$\displaystyle\|\psi\|_{*}\leq C\|{\mathcal{H}}\|_{**}.$ (5.2)

###### Proof.

First, we introduce the norms

$\displaystyle\|\psi\|_{\star}=\|\psi^{+}\|_{\star,0}+\|\psi^{-}\|_{\star,0}\,,$

and

$\displaystyle\|{\mathcal{H}}\|_{\star\star}=\|{\mathcal{H}}^{+}\|_{\star\star,0}+\|{\mathcal{H}}^{-}\|_{\star\star,0}\,,$

where we have set the notation by the following

$\displaystyle\|\psi^{\pm}\|_{\star,0}=\sum_{j=1}^{2}\|v_{d}^{\pm}\psi^{\pm}\|_{C^{0}(\ell_{j}<3)}+\|\psi^{\pm}_{1}\|_{\star,0,Re}+\|\psi^{\pm}_{2}\|_{\star,0,Im},$
$\displaystyle\|\psi_{1}^{\pm}\|_{\star,0,Re}$
$\displaystyle=\sup_{\ell_{1}>2,\ell_{2}>2}|\psi_{1}^{\pm}|+\sup_{2<\ell_{1}<\frac{2}{\epsilon},\,2<\ell_{2}<\frac{2}{\epsilon}}\frac{|\nabla\psi_{1}^{\pm}|}{\ell_{1}^{-1}+\ell_{2}^{-1}}+\sup_{r>\frac{1}{\epsilon}}\Bigl{[}\frac{1}{\epsilon}|\partial_{r}\psi_{1}^{\pm}|+|\partial_{s}\psi_{1}|\Bigr{]},$
$\displaystyle\|\psi_{2}^{\pm}\|_{\star,0,Im}$
$\displaystyle=\sup_{\ell_{1}>2,\ell_{2}>2}\frac{|\psi_{2}^{\pm}|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}+\epsilon^{\sigma-2}}+\sup_{2<\ell_{1}<\frac{2}{\epsilon},\,2<\ell_{2}<\frac{2}{\epsilon}}\frac{|\nabla\psi_{2}^{\pm}|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}}$
$\displaystyle\qquad+\sup_{r>\frac{1}{\epsilon}}\left[\epsilon^{\sigma-2}|\partial_{r}\psi_{2}|+\epsilon^{\sigma-1}|\partial_{s}\psi_{2}|\right],$

and

$\displaystyle\|{\mathcal{H}}^{\pm}\|_{\star\star,0}$
$\displaystyle:=\sum_{j=1}^{2}\|v_{d}^{\pm}{\mathcal{H}}^{\pm}\|_{C^{0}(\ell_{j}<3)}+\sup_{\ell_{1}>2,\ell_{2}>2}\Biggl{[}\frac{|{\rm{Re}}({\mathcal{H}}^{\pm})|}{\ell_{1}^{-2}+\ell_{2}^{-2}}+\frac{|{\rm{Im}}({\mathcal{H}}^{\pm})|}{\ell_{1}^{-2+\sigma}+\ell_{2}^{-2+\sigma}}\Biggr{]}.$

We claim that

$\displaystyle\|\psi\|_{\star}\leq C\|{\mathcal{H}}\|_{\star\star}.$ (5.3)

By using Schauder theory, we then can get the full estimate (5.2).

In [12], the authors gave a proof of (5.3) for the complex-valued scalar case
($B=0$). We adopt the methods and technology from [12]. In the present case,
we need to deal with some new problems caused by the coupled terms due to that
$B\neq 0$. For convenience, we only highlight the difference.

We argue by contradiction. Assume that there exist sequences of the parameter
epsilon $\\{\epsilon_{n}\\}$ approaching $0^{+}$ and solutions
$\\{\psi^{n}\\}$ in such a way that (5.1) is valid with
${\mathcal{H}}={\mathcal{H}}^{(n)}$ and the following estimates are true

$\|\psi^{n}\|_{\star}=1,\qquad\|{\mathcal{H}}^{(n)}\|_{\star\star}=o_{n}(1).$
(5.4)

Since the linearized operator
${\mathcal{L}}_{\epsilon}=({\mathcal{L}}^{+}_{\epsilon},{\mathcal{L}}^{-}_{\epsilon})$
has different asymptotic behaviors in different domains, therefore, in order
to obtain a contradiction, we will divide the analysis into two parts.

$\spadesuit$ In the region near the vortices, by a similar method to the proof
of Lemma $5.1$ in [12], we can show that

$\displaystyle\phi_{j}^{n}=\big{(}iw^{+}(\psi^{n})^{+},iw^{-}(\psi^{n})^{-}\big{)}(z+e_{j})\longrightarrow
0,\quad\text{in}~{}C_{loc}^{2}(\mathbb{R}^{2},\mathbb{C}^{2}).$ (5.5)

Here we point out that we have used the non-degeneracy results in Lemma 8.2 in
the proof of (5.5). The linearized operator $L_{0}(\phi)$ does have one kernel
element $w_{x_{1}}$ when the perturbations satisfy some decays and symmetry
constraints in that lemma. Then the orthogonality in (5.1) will imply the
validity of (5.5).

$\spadesuit$ Next we consider the case far away from the vortices. Here we
assume $x_{1}>0$ and $\ell_{1}>R_{0}$. For
${\mathcal{H}}=\big{(}{\mathcal{H}}^{+},{\mathcal{H}}^{-}\big{)}$, we express
${\mathcal{L}}_{\epsilon}^{\pm}(\psi)={\mathcal{H}}^{\pm}$ as the following

$\displaystyle\Delta\psi^{\pm}+\frac{\nabla
v_{d}^{\pm}\nabla\psi^{\pm}}{v_{d}^{\pm}}$
$\displaystyle-2iA_{\pm}|v_{d}^{\pm}|^{2}\psi^{\pm}_{2}-2iB|v_{d}^{\mp}|^{2}\psi^{\mp}_{2}$
$\displaystyle\quad+\epsilon^{2}\Bigl{(}\partial_{ss}^{2}\psi^{\pm}+\frac{2\partial_{s}v_{d}^{\pm}}{v_{d}^{\pm}}\partial_{s}\psi^{\pm}-4i\partial_{s}\psi^{\pm}\Bigr{)}={\mathcal{H}}^{\pm}.$
(5.6)

Noting that

$\psi^{\pm}=\psi^{\pm}_{1}+i\psi^{\pm}_{2},\qquad{\mathcal{H}}^{\pm}={\mathcal{H}}_{1}^{\pm}+i{\mathcal{H}}_{2}^{\pm},$

we now try to simplify the above system (5.1) for $(\psi^{+},\psi^{-})$. We
rewrite the system (5.1) to the following one in $\mathbb{R}^{2}$

$\displaystyle\left\\{\begin{aligned}
&\Delta\psi_{1}^{+}+\epsilon^{2}\partial_{ss}^{2}\psi_{1}^{+}=\tilde{\mathcal{H}}_{1}^{+},\\\\[5.69054pt]
&\Delta\psi_{1}^{-}+\epsilon^{2}\partial_{ss}^{2}\psi_{1}^{-}=\tilde{\mathcal{H}}_{1}^{-},\\\\[5.69054pt]
&\Delta\psi_{2}^{+}+\epsilon^{2}\partial_{ss}^{2}\psi_{2}^{+}-2A_{+}(t^{+})^{2}\psi^{+}_{2}-2B(t^{-})^{2}\psi^{-}_{2}=\tilde{\mathcal{H}}_{2}^{+},\\\\[5.69054pt]
&\Delta\psi_{2}^{-}+\epsilon^{2}\partial_{ss}^{2}\psi_{2}^{-}-2A_{-}(t^{-})^{2}\psi^{-}_{2}-2B(t^{+})^{2}\psi^{+}_{2}=\tilde{\mathcal{H}}_{2}^{-},\end{aligned}\right.$
(5.7)

with

$\displaystyle\tilde{\mathcal{H}}_{1}^{\pm}$
$\displaystyle={\mathcal{H}}_{1}^{\pm}-\left(\frac{\nabla
W^{\pm}(\ell_{1})}{{W^{\pm}(\ell_{1})}}+\frac{\nabla{W^{\pm}(\ell_{2})}}{{W^{\pm}(\ell_{2})}}\right)\nabla\psi_{1}^{\pm}-\nabla(\theta_{1}+\theta_{2})\nabla\psi_{2}^{\pm}$
$\displaystyle\quad-2\epsilon^{2}\left[\left(\frac{\partial_{s}{W^{\pm}(\ell_{1})}}{{W^{\pm}(\ell_{1})}}+\frac{\partial_{s}{W^{\pm}(\ell_{2})}}{{W^{\pm}(\ell_{2})}}\right)\partial_{s}\psi_{1}^{\pm}-\partial_{s}(\theta_{1}+\theta_{2})\partial_{s}\psi_{2}^{\pm}\right]-4\epsilon^{2}\partial_{s}\psi_{2}^{\pm},$

and

$\displaystyle\tilde{\mathcal{H}}_{2}^{\pm}$
$\displaystyle={\mathcal{H}}_{2}^{\pm}-\left(\frac{\nabla
W^{\pm}(\ell_{1})}{{W^{\pm}(\ell_{1})}}+\frac{\nabla{W^{\pm}(\ell_{2})}}{{W^{\pm}(\ell_{2})}}\right)\nabla\psi_{2}^{\pm}-\nabla(\theta_{1}+\theta_{2})\nabla\psi_{1}^{\pm}$
$\displaystyle\quad-2\epsilon^{2}\left[\left(\frac{\partial_{s}{W^{\pm}(\ell_{1})}}{{W^{\pm}(\ell_{1})}}-\frac{\partial_{s}{W^{\pm}(\ell_{2})}}{{W^{\pm}(\ell_{2})}}\right)\partial_{s}\psi_{2}^{\pm}-\partial_{s}(\theta_{1}+\theta_{2})\partial_{s}\psi_{1}^{\pm}\right]+4\epsilon^{2}\partial_{s}\psi_{1}^{\pm}$
$\displaystyle\quad+2A_{\pm}\big{(}|v_{d}^{\pm}|^{2}-{t^{\pm}}^{2}\big{)}\psi^{\pm}_{2}\
+\ 2B\big{(}|v_{d}^{\mp}|^{2}-{t^{\mp}}^{2}\big{)}\psi^{\mp}_{2}.$

Since the matrix $\mathbb{M}=\begin{pmatrix}A_{+}|t^{+}|^{2}&B|t^{-}|^{2}\\\
B|t^{+}|^{2}&A_{-}|t^{-}|^{2}\end{pmatrix}$ is positive definite, there then
exists an invertible matrix ${\bf C}$ such that

$\displaystyle{\bf C}^{T}{\bf C}={\bf I},\quad 2{\bf C}\,\mathbb{M}\,{\bf
C}^{T}=\text{diag}(\lambda^{+},\lambda^{-})\quad\text{with}~{}\lambda^{+},\,\lambda^{-}>0.$
(5.8)

Now we define the new vector functions

$\displaystyle\tilde{\psi}_{1}=(\tilde{\psi}_{1}^{+},\tilde{\psi}_{1}^{-})^{T},$
(5.9)
$\displaystyle\tilde{\psi}_{2}=(\tilde{\psi}_{2}^{+},\tilde{\psi}_{2}^{-})^{T}:={\bf
C}^{T}\,(\psi_{2}^{+},\psi_{2}^{-})^{T},$ (5.10)

and

$\displaystyle\mathbb{H}_{2}=(\mathbb{H}_{2}^{+},\mathbb{H}_{2}^{-}):={\bf
C}^{T}\,(\tilde{\mathcal{H}}_{2}^{+},\tilde{\mathcal{H}}_{2}^{-})^{T},$
$\displaystyle\mathbb{H}_{1}=(\mathbb{H}_{1}^{+},\mathbb{H}_{1}^{-}):=(\tilde{\mathcal{H}}_{1}^{+},\tilde{\mathcal{H}}_{1}^{-})^{T}.$

Then if $|z-e_{j}|>R_{0},$ the system (5.7) can be rewritten as

$\displaystyle(\Delta+\epsilon^{2}\partial_{ss}^{2})\tilde{\psi}_{1}=\mathbb{H}_{1},$
(5.11)
$\displaystyle(\Delta+\epsilon^{2}\partial_{ss}^{2})\tilde{\psi}_{2}-\text{diag}(\lambda^{+},\lambda^{-})\tilde{\psi}_{2}=\mathbb{H}_{2}.$
(5.12)

For the system (5.11)-(5.12), we will use the barrier arguments based on the
maximum principle provided in Lemmas 8.3-8.4. We obtain that, by some
calculations,

$\displaystyle|\mathbb{H}_{2}|\leq
C\big{(}\|{\mathcal{H}}\|_{\star\star,0}+R_{0}^{-\sigma}+\epsilon^{\sigma}\big{)}\big{(}\ell_{1}^{\sigma-2}+\epsilon^{2-\sigma}\big{)}.$

By the comparison principle, elliptic estimates, and choosing the barrier as

$\displaystyle\mathcal{B}_{2}=\hat{C}\Big{(}\|{\mathcal{H}}\|_{\star\star}+R_{0}^{-\sigma}+\epsilon^{\sigma}+\|\tilde{\psi}_{2}^{\pm}\|_{L^{\infty}(B_{R_{0}}(\tilde{d}))}\Big{)}\Big{(}\ell_{1}^{\sigma-2}+\epsilon^{2-\sigma}\Big{)},$

for some fixed large constant $\hat{C}>0$, we obtain

$\displaystyle\|\tilde{\psi}_{2}^{\pm}\|_{\star,0,Im}\leq
C\Big{(}\|{\mathcal{H}}\|_{\star\star}+R_{0}^{-\sigma}+\epsilon^{\sigma}+\|\tilde{\psi}_{2}^{\pm}\|_{L^{\infty}(B_{R_{0}}(\tilde{d}))}\Big{)}.$

On the other hand, by a similar argument as in [12], we have

$\displaystyle\|\tilde{\psi}_{1}^{\pm}\|_{\star,0,Re}\leq
C\Big{(}\|{\mathcal{H}}\|_{\star\star}+R_{0}^{\sigma-1}+\epsilon^{1-\sigma}+\|\psi_{1}^{\pm}\|_{L^{\infty}(B_{R_{0}}(\tilde{d}))}\Big{)}.$

Meanwhile, using (5.5) and $\|{\mathcal{H}}^{(n)}\|_{\star\star,0}=o_{n}(1)$,
we can get

$\displaystyle\|\tilde{\psi}\|_{\star}\leq
C\|{\mathcal{H}}\|_{\star\star},\quad\text{with}~{}\tilde{\psi}=(\tilde{\psi}_{1}^{+}+i\tilde{\psi}_{2}^{+},\,\tilde{\psi}_{1}^{-}+i\tilde{\psi}_{2}^{-}).$

Since ${\bf C}^{T}$ is invertible, we directly get

$\displaystyle\|\psi\|_{\star}\leq C\|{\mathcal{H}}\|_{\star\star}.$

Therefore, by combining the above results we derive an estimate contradicting
to (5.4). ∎

We next give the proof of Proposition 4.1.

###### Proof of Proposition 4.1.

For given $M>100\tilde{d}$, we first consider the problem (4.2) locally as the
following

$\displaystyle\left\\{\begin{aligned}
&{\mathcal{L}}^{\pm}_{\epsilon}(\psi)={\mathcal{H}}^{\pm}+c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{\pm}(z-e_{j})}(-1)^{j}w^{\pm}_{x_{1}}(z-e_{j})\quad\text{in
}B_{M}(0),\\\\[5.69054pt]
&{\rm{Re}}\int_{B(0,4)}\chi\big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\big{]}=0,\text{
with }\phi^{\pm}_{j}(z)=iw^{\pm}(z)\psi^{\pm}(z+e_{j}),\\\\[5.69054pt]
&\psi\text{ satisfies the symmetry }\eqref{s4}~{}\text{and
}\psi=0~{}\text{on}~{}\partial{B_{M}(0)}.\end{aligned}\right.$ (5.13)

Let

$\displaystyle{\mathfrak{B}}:=$
$\displaystyle\Bigg{\\{}\phi=(iv_{d}^{+}\psi^{+},iv_{d}^{-}\psi^{-})\in
H^{1}_{0}\big{(}B_{M}(0),\mathbb{C}^{2}\big{)}\,:\,\;\;\psi\text{ satisfies
\eqref{s4}},$
$\displaystyle\qquad\qquad{\rm{Re}}\int_{B(0,4)}\chi\Big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\Big{]}=0,\
j=1,2,\Bigg{\\}}.$

We endow the space ${\mathfrak{B}}$ with the inner product

$[\phi,\varphi]_{\mathfrak{B}}:={\rm{Re}}\int_{B_{M}(0)}\Bigg{[}\big{(}\,\nabla\phi^{+}\overline{\nabla\varphi^{+}}+\epsilon^{2}\partial_{s}\phi^{+}\overline{\partial_{s}\varphi^{+}}\,\big{)}\
+\
\big{(}\,\nabla\phi^{-}\overline{\nabla\varphi^{-}}+\epsilon^{2}\partial_{s}\phi^{-}\overline{\partial_{s}\varphi^{-}}\,\big{)}\Bigg{]},$
(5.14)

for any $\phi,\varphi\in{\mathfrak{B}}.$ Due to the Poincaré inequality, we
know that the space equipped with the topology (5.14) is a subspace of
$H^{1}_{0}\big{(}B_{M}(0),\mathbb{C}^{2}\big{)}$. By some calculations, the
following identities hold

$\displaystyle\Delta\Phi^{\pm}$
$\displaystyle+\Big{[}A_{\pm}\big{(}{t^{\pm}}^{2}-|v_{d}^{\pm}|^{2}\big{)}+B\big{(}{t^{\mp}}^{2}-|v_{d}^{\mp}|^{2}\big{)}\Big{]}\Phi^{\pm}+(\eta_{1}-1)\frac{\mathbb{E}^{\pm}}{v_{d}^{\pm}}\Phi_{j}^{\pm}(z)$
$\displaystyle-\,2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{\pm}\overline{\Phi^{\pm}}\Big{)}v_{d}^{\pm}\,-\,2B{\rm{Re}}\Big{(}v_{d}^{\mp}\overline{\Phi^{\mp}}\Big{)}v_{d}^{\pm}+\epsilon^{2}(\partial^{2}_{ss}\Phi^{\pm}-4i\partial_{s}\Phi^{\pm}-4\Phi^{\pm})$
$\displaystyle=iv_{d}^{\pm}{\mathcal{H}}^{\pm}+iv_{d}^{\pm}c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{\pm}(z-e_{j})}(-1)^{j}w^{\pm}_{x_{1}}(z-e_{j}),\quad\text{in}~{}B_{M}(0),$

with

$\Phi^{\pm}=iv_{d}^{\pm}\psi^{\pm}.$

For any test function $\varphi=(\varphi^{+},\varphi^{-})$, we rewrite these
equations in the sense of distribution

$\displaystyle-{\rm{Re}}\int_{B_{M}(0)}\Big{[}\big{(}\nabla\Phi^{+}\overline{\nabla\varphi^{+}}+\epsilon^{2}\partial_{s}\Phi^{+}\overline{\partial_{s}\varphi^{+}}\big{)}+\big{(}\nabla\Phi^{-}\overline{\nabla\varphi^{-}}+\epsilon^{2}\partial_{s}\Phi^{-}\overline{\partial_{s}\varphi^{-}}\big{)}\Big{]}$
$\displaystyle\qquad\qquad-\epsilon^{2}{\rm{Re}}\int_{B_{M}(0)}\Big{[}\big{(}4i\partial_{s}\Phi^{+}-4\Phi^{+}\big{)}\,\overline{\varphi^{+}}+\big{(}4i\partial_{s}\Phi^{-}-4\Phi^{-}\big{)}\,\overline{\varphi^{-}}\Big{]}$
$\displaystyle\qquad\qquad-{\rm{Re}}\int_{B_{M}(0)}2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{+}\overline{\Phi^{+}}\Big{)}v_{d}^{+}\overline{\varphi^{+}}\,+\,2B{\rm{Re}}\Big{(}v_{d}^{-}\overline{\Phi^{-}}\Big{)}v_{d}^{+}\overline{\varphi^{+}}$
$\displaystyle\qquad\qquad-{\rm{Re}}\int_{B_{M}(0)}2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{-}\overline{\Phi^{-}}\Big{)}v_{d}^{-}\overline{\varphi^{-}}\,+\,2B{\rm{Re}}\Big{(}v_{d}^{+}\overline{\Phi^{+}}\Big{)}v_{d}^{-}\overline{\varphi^{-}}$
$\displaystyle\qquad\qquad+{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}\Big{[}A_{+}\big{(}{t^{+}}^{2}-|v_{d}^{+}|^{2}\big{)}+B\big{(}{t^{-}}^{2}-|v_{d}^{-}|^{2}\big{)}\Big{]}+(\eta_{1}-1)\frac{\mathbb{E}^{+}}{v_{d}^{+}}\Bigg{)}\Phi^{+}\overline{\varphi^{+}}$
$\displaystyle\qquad\qquad+{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}\Big{[}A_{-}\big{(}{t^{-}}^{2}-|v_{d}^{-}|^{2}\big{)}+B\big{(}{t^{+}}^{2}-|v_{d}^{+}|^{2}\big{)}\Big{]}+(\eta_{1}-1)\frac{\mathbb{E}^{-}}{v_{d}^{-}}\Bigg{)}\Phi^{-}\overline{\varphi^{-}}$
$\displaystyle={\rm{Re}}\int_{B_{M}(0)}\Bigg{(}iv_{d}^{+}{\mathcal{H}}^{-}+iv_{d}^{+}c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{+}(z-e_{j})}(-1)^{j}w^{+}_{x_{1}}(z-e_{j})\Bigg{)}\overline{\varphi^{+}}$
$\displaystyle\qquad\qquad+{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}iv_{d}^{-}{\mathcal{H}}^{-}+iv_{d}^{-}c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{-}(z-e_{j})}(-1)^{j}w^{-}_{x_{1}}(z-e_{j})\Bigg{)}\overline{\varphi^{-}}.$
(5.15)

Denote by linear operator
$\big{\langle}\mathcal{K}(\Phi),\cdot\big{\rangle}_{{\mathfrak{B}}}$ on
${\mathfrak{B}}$ as

$\displaystyle-\epsilon^{2}{\rm{Re}}\int_{B_{M}(0)}\Big{[}\big{(}4i\partial_{s}\Phi^{+}-4\Phi^{+}\big{)}\,\overline{\varphi^{+}}+\big{(}4i\partial_{s}\Phi^{-}-4\Phi^{-}\big{)}\,\overline{\varphi^{-}}\Big{]}$
$\displaystyle\qquad\qquad-{\rm{Re}}\int_{B_{M}(0)}2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{+}\overline{\Phi^{+}}\Big{)}v_{d}^{+}\overline{\varphi^{+}}\,+\,2B{\rm{Re}}\Big{(}v_{d}^{-}\overline{\Phi^{-}}\Big{)}v_{d}^{+}\overline{\varphi^{+}}$
$\displaystyle\qquad\qquad-{\rm{Re}}\int_{B_{M}(0)}2A_{\pm}{\rm{Re}}\Big{(}v_{d}^{-}\overline{\Phi^{-}}\Big{)}v_{d}^{-}\overline{\varphi^{-}}\,+\,2B{\rm{Re}}\Big{(}v_{d}^{+}\overline{\Phi^{+}}\Big{)}v_{d}^{-}\overline{\varphi^{-}}$
$\displaystyle\qquad\qquad+{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}\Big{[}A_{+}\big{(}{t^{+}}^{2}-|v_{d}^{+}|^{2}\big{)}+B\big{(}{t^{-}}^{2}-|v_{d}^{-}|^{2}\big{)}\Big{]}+(\eta_{1}-1)\frac{\mathbb{E}^{+}}{v_{d}^{+}}\Bigg{)}\Phi^{+}\overline{\varphi^{+}}$
$\displaystyle\qquad\qquad+{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}\Big{[}A_{-}\big{(}{t^{-}}^{2}-|v_{d}^{-}|^{2}\big{)}+B\big{(}{t^{+}}^{2}-|v_{d}^{+}|^{2}\big{)}\Big{]}+(\eta_{1}-1)\frac{\mathbb{E}^{+}}{v_{d}^{+}}\Bigg{)}\Phi^{-}\overline{\varphi^{-}}$
$\displaystyle=:\big{\langle}\mathcal{K}\Phi,\varphi\big{\rangle}_{{\mathfrak{B}}}.$

Also, we denote by the linear operator
$\big{\langle}\mathcal{S}(\Phi),\cdot\big{\rangle}_{{\mathfrak{B}}}$ on
${\mathfrak{B}}$ as

$\displaystyle{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}iv_{d}^{+}{\mathcal{H}}^{-}+iv_{d}^{+}c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{+}(z-e_{j})}(-1)^{j}w^{+}_{x_{1}}(z-e_{j})\Bigg{)}\overline{\varphi^{+}}$
$\displaystyle\quad+{\rm{Re}}\int_{B_{M}(0)}\Bigg{(}iv_{d}^{-}{\mathcal{H}}^{-}+iv_{d}^{-}c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{-}(z-e_{j})}(-1)^{j}w^{-}_{x_{1}}(z-e_{j})\Bigg{)}\overline{\varphi^{-}}=:\big{\langle}\mathcal{S},\varphi\big{\rangle}_{{\mathfrak{B}}}.$

Then (5.1) is rewritten into the following form

$\displaystyle-[\phi,\varphi]_{\mathfrak{B}}+\big{\langle}\mathcal{K}\Phi,\varphi\big{\rangle}_{{\mathfrak{B}}}=\big{\langle}\mathcal{S},\varphi\big{\rangle}_{{\mathfrak{B}}},\quad\text{for
any}~{}\varphi\in C_{0}^{\infty}\big{(}B_{M}(0),\mathbb{C}^{2}\big{)}.$

Riesz representation theory and compact Sobolev embedding imply that there
exists a compact bounded linear operator $\mathbb{K}$ on ${\mathfrak{B}}$, and
$\mathbb{S}\in{\mathfrak{B}}$ such that

$\displaystyle\Phi-\mathbb{K}(\Phi)=\mathbb{S}.$ (5.16)

In order to get the existence of a solution to (5.16), we need to show the
homogeneous equation has only trivial solutions via the Fredholm’s theory.
Therefore, we shall first establish the a priori estimate of multiplier $c$.
We consider the equivalent form of (5.13) with translated variables

$\displaystyle L^{\pm}_{d,j}(\Phi_{j})$
$\displaystyle=iw^{\pm}(z){\mathcal{H}}^{\pm}(z+e_{j})+c\chi_{j}w^{\pm}_{x_{1}}(z),$
$\displaystyle:={\mathcal{H}}^{\pm}_{j}+c\chi_{j}w^{\pm}_{x_{1}}(z)\quad\text{in}~{}B_{\tilde{d}}(0),$

with

$\Phi_{j}=\big{(}\Phi_{j}^{+},\Phi_{j}^{-}\big{)},\qquad\Phi_{j}^{\pm}(z)=iw^{\pm}(z)\psi^{\pm}(z+e_{j}),$

and where the operator $L^{\pm}_{d,j}(\Phi_{j})$ is defined in (2.3). We can
test the equations against $w^{\pm}_{x_{1}}$ respectively to find

$\displaystyle
c=-\frac{1}{c_{*}}\Bigg{[}{\rm{Re}}\int_{B_{\tilde{d}}(0)}{\mathcal{H}}^{\pm}_{j}w^{\pm}_{x_{1}}-{\rm{Re}}\int_{B_{\tilde{d}}(0)}L^{\pm}_{d,j}(\Phi_{j})w^{\pm}_{x_{1}}\Bigg{]},$

for some $c^{*}\sim C$. Using the expressions of $L^{\pm}_{d,j}(\Phi_{j})$ in
(2.3), by some tedious but straightforward analysis, we can get the expression
for $c$

$c=-\frac{1}{c_{*}}{\rm{Re}}\int_{B_{\tilde{d}}(0)}{\mathcal{H}}^{\pm}_{j}w^{\pm}_{x_{1}}+O_{\epsilon}(\epsilon\sqrt{|\ln\epsilon|})\frac{\|\psi\|_{*}}{c_{*}}.$
(5.17)

When ${\mathcal{H}}^{\pm}_{j}=0$, the expression for $c$ in (5.17) and the
estimate in Lemma 5.1 imply that

$\displaystyle\|\psi\|_{*}\leq
C\|c\sum_{j=1}^{2}\chi_{j}(z)(-1)^{j}\frac{w^{\pm}_{x_{1}}(z-e_{j})}{iw^{\pm}(z-e_{j})}\|_{**}\leq
C\epsilon\sqrt{|\ln\epsilon|}\|\psi\|_{*},$

and thus there exists only trivial solution when ${\mathcal{H}}^{\pm}_{j}=0$.
It is worth noting that we now apply the norms $\|\cdot\|_{*}$ and
$\|\cdot\|_{**}$ restricted on the bounded domains. Then the existence of
(5.13) holds and from Lemma 5.1 there exists a constant $C$ independent of $M$
such that the following estimate holds

$\|\psi_{M}\|_{*}\leq C\|{\mathcal{H}}\|_{**}.$

The real $\psi$ solving (4.2) can be obtain by extracting a subsequence such
that $\psi_{M}\rightharpoonup\psi$ in $H^{1}_{\text{loc}}(\mathbb{R}^{2})$.
And the estimate $\|\psi\|_{*}\leq C\|{\mathcal{H}}\|_{**}$ is a direct result
of Lemma 5.1. ∎

###  5.2. Proof of Proposition 4.2

Consider the linear projected problem (5.1) and recall the semi-norms
$|\cdot|_{\sharp\sharp}$, $|\cdot|_{\sharp}$ in (3.17) and (4.9). We first
give the following a priori estimate.

######  Lemma 5.2.

Suppose $\alpha\in(0,1),\sigma\in(0,1)$. There exists a constant $C$ depending
only on $\alpha,\sigma$, such that for all $\epsilon$ small enough, and any
solution of (5.1), we have the estimate

$|\psi|_{\sharp}\leq
C(|{\mathcal{H}}|_{\sharp\sharp}+\epsilon|\log\epsilon|^{\frac{1}{2}}\|{\mathcal{H}}\|_{**}).$
(5.18)

###### Proof.

We give some technical essentials and the outline of the proof. The reader can
refer to the proof of Lemma 5.2 in [12] for more details.

Step 1: Similar to the proof of Lemma 5.1, we make a transformation

$(\tilde{\psi}_{2}^{+},\tilde{\psi}_{2}^{-})^{T}:={\bf
C}^{T}\,(\psi_{2}^{+},\psi_{2}^{-})^{T}$

such that the main terms in the linear operator of
$\tilde{\psi}_{2}=(\tilde{\psi}_{2}^{+},\tilde{\psi}_{2}^{-})$ can be
decoupled in the region far away from the vortices, see (5.12).

Step 2: When near the vortices, we just proceed as in the proof of Lemma $5.2$
in [12]. We point out that in this step we use Lemma 2.2 and the non-
degeneracy results in Lemma 8.2 . On the other hand, when far away from the
vortices, we handle the system (5.11)-(5.12) by the maximum principle, see
Lemmas 8.3-8.4 in the Appendix. In this step, we establish the estimate

$|\tilde{\psi}|_{\sharp}\leq
C(|{\mathcal{H}}|_{\sharp\sharp}+\epsilon|\log\epsilon|^{\frac{1}{2}}\|{\mathcal{H}}\|_{**}).$

Step 3: Finally, since ${\bf C}^{T}$ is invertible, we get that (5.18) holds.
∎

In order to prove Proposition 4.2, we also need the following lemma. Here we
recall the reflection $\mathscr{R}_{j}(z)$ defined in (3.14) and suppose that
the function ${\mathcal{H}}$ enjoys the local symmetry (S4) and the following

$\displaystyle{\mathcal{H}}(\mathscr{R}_{j}z)=-\overline{{\mathcal{H}}(z)},\quad|z-e_{j}|<2\mathbb{R}_{\epsilon},\quad\text{for}~{}j=1,2.$
(S2)

We note that the function ${\mathcal{H}}$ satisfies the local symmetry (S2)
consisting in even Fourier modes.

######  Lemma 5.3.

Suppose that ${\mathcal{H}}$ enjoys the symmetry (S2) and (S4) and

$\|{\mathcal{H}}\|_{**}\leq+\infty.$

For any solution $\psi$ of (5.1) with $\|\psi\|_{*}\leq+\infty$, there exist
$\psi_{s},\psi_{*}$ such that

$\displaystyle\psi=\psi_{s}+\psi_{*},$

where $\psi_{s}$ satisfies the following symmetry

$\displaystyle\psi_{s}(\mathcal{R}_{j}z)$
$\displaystyle=-\overline{\psi_{s}(z)},\quad|z-e_{j}|<\mathbb{R}_{\epsilon}.$

Moreover, there exists a constant $C$ such that for all $\epsilon$ small, the
following estimates hold

$\displaystyle\|\psi_{s}\|_{*}+\|\psi_{*}\|_{*}\leq C\|{\mathcal{H}}\|_{**},$
$\displaystyle|\psi_{*}|_{\sharp}\leq
C\epsilon|\log\epsilon|^{\frac{1}{2}}\|{\mathcal{H}}\|_{**}.$

###### Proof.

We decompose the operator
${\mathcal{L}}_{\epsilon}=\Big{(}{\mathcal{L}}^{+}_{\epsilon},{\mathcal{L}}^{-}_{\epsilon}\Big{)}$
as

$\displaystyle{\mathcal{L}}_{\epsilon}^{\pm}={\mathcal{L}}_{\epsilon,s,j}^{\pm}+{\mathcal{L}}_{\epsilon,r,j}^{\pm},$

with ${\mathcal{L}}_{\epsilon,s,j}$ preserving the symmetry (S2). More
specifically, we have the expressions of
${\mathcal{L}}_{\epsilon,s,j},{\mathcal{L}}_{\epsilon,r,j}$ in the coordinates
$(\ell_{j},\theta_{j})$

$\displaystyle{\mathcal{L}}_{\epsilon,s,1}^{\pm}(\psi)$
$\displaystyle=\Delta\psi^{\pm}+2\frac{\nabla
w_{a}\nabla\psi^{\pm}}{w_{a}}-2iA_{\pm}|w_{a}^{\pm}|^{2}{\rm{Im}}(\psi^{\pm})-2iB|w_{a}^{\mp}|^{2}{\rm{Im}}(\psi^{\mp})$
$\displaystyle\quad+\epsilon^{2}\Biggl{[}\tilde{d}^{2}\partial_{\ell_{1}\ell_{1}}^{2}\psi\sin^{2}\theta_{1}+\frac{\tilde{d}^{2}}{\ell_{1}}\partial_{\ell_{1}\theta_{1}}^{2}\psi^{\pm}\sin\theta_{1}\cos\theta_{1}+\partial_{\theta_{1}\theta_{1}}^{2}\psi^{\pm}\Bigl{(}1+\frac{\tilde{d}^{2}}{\ell_{1}^{2}}\cos^{2}\theta_{1}\Bigr{)}$
$\displaystyle\qquad\qquad+\partial_{\ell_{1}}\psi^{\pm}\frac{\tilde{d}^{2}}{\ell_{1}}\cos^{2}\theta_{1}-2\partial_{\theta_{1}}\psi^{\pm}\frac{\tilde{d}^{2}}{\ell_{1}^{2}}\sin\theta_{1}\cos\theta_{1}\Biggr{]}$
$\displaystyle\quad+\frac{\hat{d}^{2}\eta_{1}}{|\ln\epsilon|}{W^{\pm}}^{\prime}(\ell_{1})\sin^{2}\theta_{1}\psi^{\pm},$

and

$\displaystyle{\mathcal{L}}_{\epsilon,r,1}^{\pm}(\psi)$
$\displaystyle=2\frac{\nabla
w_{b}\nabla\psi^{\pm}}{w_{b}}-2iA_{\pm}(|v_{d}^{\pm}|^{2}-|w_{a}^{\pm}|^{2}){\rm{Im}}(\psi^{\pm})-2iB(|v_{d}^{\mp}|^{2}-|w_{a}^{\mp}|^{2}){\rm{Im}}(\psi^{\mp})$
$\displaystyle\quad+\epsilon^{2}\Biggl{[}2\tilde{d}\partial_{\ell_{1}\theta_{1}}^{2}\psi\sin\theta_{1}+2\partial_{\theta_{1}\theta_{1}}^{2}\psi^{\pm}\frac{\tilde{d}}{\ell_{1}}\cos\theta_{1}+\partial_{\ell_{1}}\psi^{\pm}\tilde{d}\cos\theta_{1}-\partial_{\theta_{1}}\psi^{\pm}\frac{\tilde{d}}{\ell_{1}}\sin\theta_{1}\Biggr{]}$
$\displaystyle\quad+\epsilon^{2}\Bigl{(}\frac{2\partial_{s}v_{d}^{\pm}}{v_{d}^{\pm}}-4i\Bigr{)}\Biggl{[}\partial_{\ell_{1}}\psi^{\pm}\tilde{d}\sin\theta_{1}+\Big{(}1+\frac{\tilde{d}}{\ell_{1}}\cos\theta_{1}\partial_{\theta_{1}}\Big{)}\psi\Biggr{]}$
$\displaystyle\quad+\eta_{1}\Bigg{(}\frac{S^{\pm}(v_{d})}{v_{d}^{\pm}}-\frac{\hat{d}^{2}}{|\ln\epsilon|}{W^{\pm}}^{\prime}(\ell_{1})\sin^{2}\theta_{1}\Bigg{)}\psi^{\pm},$

with

$w^{\pm}_{a}(z)=W^{\pm}(\ell_{1})e^{i\theta_{1}},\qquad
w^{\pm}_{b}(z)=W^{\pm}(\ell_{2})e^{i\theta_{2}}.$

The expressions of ${\mathcal{L}}_{\epsilon,s,2}^{\pm}(\psi)$ and
${\mathcal{L}}_{\epsilon,r,2}^{\pm}(\psi)$ are similar.

Next, we consider the following problem

$\displaystyle\left\\{\begin{aligned}
&{\mathcal{L}}_{\epsilon,s,j}^{\pm}(\psi_{s,j})={\mathcal{H}}^{\pm}\eta_{j,2\mathbb{R}_{\epsilon}}\quad\text{in
}\mathbb{R}^{2},\\\\[5.69054pt]
&{\rm{Re}}\int_{B(0,4)}\chi\big{[}\overline{\phi_{s,j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{s,j}^{-}}w_{x_{1}}^{-}\big{]}=0,\text{
with }\phi_{s,j}^{\pm}(z)=iw^{\pm}(z)\psi_{s,j}^{\pm}(z+e_{j}),\\\\[5.69054pt]
&\psi_{s,j}^{\pm}\text{ satisfies
}\psi_{s,j}^{\pm}(\bar{z})=-\psi_{s,j}^{\pm}(z).\end{aligned}\right.$

Note that, due to the local symmetry (S2), ${\mathcal{H}}$ is orthogonal to
the kernel locally, thus the multiplier $c$ vanishes. Recall that
${\mathcal{L}}_{\epsilon,s,j}=\big{(}{\mathcal{L}}_{\epsilon,s,j}^{+},{\mathcal{L}}_{\epsilon,s,j}^{-}\big{)}$
preserves symmetry (S2) and ${\mathcal{H}}$ satisfies (S2). Then for
$\psi_{s,j}$, the symmetry (S2) and the following estimate both hold

$\displaystyle\|\psi_{s,j}\|_{*}\leq C\|{\mathcal{H}}\|_{**}.$

Let

$\displaystyle\psi_{s}=\eta_{1,\frac{1}{2}\mathbb{R}_{\epsilon}}\psi_{s,1}+\eta_{2,\frac{1}{2}\mathbb{R}_{\epsilon}}\psi_{s,2},$

then

$\displaystyle\|\psi_{s}\|_{*}\leq C\|{\mathcal{H}}\|_{**}.$

Define $\hat{\mathcal{H}}=(\hat{\mathcal{H}}^{\pm},\hat{\mathcal{H}}^{\pm})$
with

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle:={\mathcal{H}}-{\mathcal{L}}_{\epsilon,r,1}(\eta_{1,\frac{1}{2}R_{\epsilon}}\psi_{s,1})-{\mathcal{L}}_{\epsilon,s,1}(\eta_{1,\frac{1}{2}R_{\epsilon}}\psi_{s,1})-{\mathcal{L}}_{\epsilon,r,2}(\eta_{2,\frac{1}{2}R_{\epsilon}}\psi_{s,2})-{\mathcal{L}}_{\epsilon,s,2}(\eta_{2,\frac{1}{2}R_{\epsilon}}\psi_{s,2}),$

and consider the following

$\displaystyle\left\\{\begin{aligned}
&{\mathcal{L}}_{\epsilon}^{\pm}(\hat{\psi}^{\pm})=\hat{\mathcal{H}}^{\pm}+c\sum_{j=1}^{2}\frac{\chi_{j}}{iw^{\pm}(z-e_{j})}(-1)^{j}w^{\pm}_{x_{1}}(z-e_{j})\quad\text{in
}\mathbb{R}^{2},\\\\[5.69054pt]
&{\rm{Re}}\int_{B(0,4)}\chi\big{[}\overline{\hat{\phi}_{j}^{+}}w_{x_{1}}^{+}+\overline{\hat{\phi}_{j}^{-}}w_{x_{1}}^{-}\big{]}=0,\text{
with
}\hat{\phi}^{\pm}_{j}(z)=iw^{\pm}(z)\hat{\psi}^{\pm}(z+e_{j}),\\\\[5.69054pt]
&\hat{\psi}^{\pm}\text{ satisfies the symmetry
}\eqref{s2}.\end{aligned}\right.$ (5.19)

By the solvability in Proposition 4.1 and the a priori estimate in Lemma 5.2,
we can get a solution $\hat{\psi}$ of (5.19) with the estimates

$\displaystyle\|\hat{\psi}\|_{*}$ $\displaystyle\leq
C\|\hat{\mathcal{H}}\|_{**},$ $\displaystyle|\hat{\psi}|_{\sharp}$
$\displaystyle\leq
C(|\hat{\mathcal{H}}|_{\sharp\sharp}+\epsilon|\log\epsilon|^{\frac{1}{2}}\|{\mathcal{H}}\|_{**}),$
$\displaystyle|\hat{\mathcal{H}}|_{\sharp\sharp}$ $\displaystyle\leq
C\epsilon|\log\epsilon|^{\frac{1}{2}}\|{\mathcal{H}}\|_{**}.$

In fact, we have

$\psi=\psi_{s}+\hat{\psi},$

so we take

$\psi_{*}:=\hat{\psi}.$

From the analysis above, $\psi_{s}$ and $\psi_{*}$ satisfy the estimates in
this Lemma. ∎

Finally, we give the proof of Proposition 4.2.

###### Proof of Proposition 4.2.

By combining the a priori estimates in Lemma 5.2 and Lemma 5.3, and the
decompositions in Lemma 5.3, we can prove Proposition 4.2. More technical
details can be found in [12]. ∎

##  6\. Solving the projected nonlinear problem

In this section, we would deal with the projected problem (4.1). The
resolution theory is provided in the following:

######  Proposition 6.1.

There exists a number $D>0$, depending only on $\alpha,\sigma\in(0,1)$, such
that for any $\epsilon$ sufficiently small, problem (4.1) has a solution
$(\psi_{\epsilon},c)$ which satisfies

$\displaystyle\|\psi_{\epsilon}\|_{*}\leq D\frac{1}{|\ln\epsilon|},$

and $\psi_{\epsilon}$ is a continuous function of the parameter $\hat{d}$.
Furthermore, we have the decomposition

$\psi_{\epsilon}=\psi_{\epsilon o}+\psi_{\epsilon e}=\psi_{\epsilon
o}^{\alpha}+\psi_{\epsilon o}^{\beta}+\psi_{\epsilon e}$

and the estimates

$\displaystyle|\psi^{\beta}_{\epsilon o}|_{\sharp}+\|\psi^{\beta}_{\epsilon
o}\|_{*}\leq D\epsilon\sqrt{|\ln\epsilon|},\qquad|\psi_{\epsilon
o}^{\alpha}|_{\sharp}\leq D\frac{\epsilon}{\sqrt{|\ln\epsilon|}}.$

###### Proof.

We consider the following closed, bounded domain for functions with the form
$\psi_{\epsilon}=\psi^{\alpha}_{\epsilon o}+\psi^{\beta}_{\epsilon
o}+\psi_{\epsilon e}$:

$\displaystyle\mathcal{\tilde{A}}:=$ $\displaystyle\Bigg{\\{}\psi:\;\psi\text{
satisfies
}\eqref{s4},\quad{\rm{Re}}\int_{B(0,4)}\chi\big{[}\overline{\phi_{j}^{+}}w_{x_{1}}^{+}+\overline{\phi_{j}^{-}}w_{x_{1}}^{-}\big{]}=0,\;\;j=1,2,\;$
$\displaystyle\qquad\;\|\psi\|_{*}\leq\frac{C}{|\ln\epsilon|},\quad|\psi^{\alpha}_{\epsilon
o}|_{\sharp}\leq\frac{D\epsilon}{\sqrt{|\ln\epsilon|}},\quad|\psi^{\alpha}_{\epsilon
o}|_{\sharp}+\|\psi^{\beta}_{\epsilon o}\|_{*}\leq
D\epsilon\sqrt{|\ln\epsilon|}\Bigg{\\}}.$

From Proposition 4.1, the problem (4.1) is equivalent to finding a fixed point
of

$\psi=T_{\epsilon}\Big{(}{\mathcal{R}}+\mathcal{N}(\psi)\Big{)}=:\mathbb{G}_{\epsilon}(\psi)$

in the domain $\mathcal{\tilde{A}}$. Now what we shall show is that, for
$\epsilon$ small enough, the map $\mathbb{G}_{\epsilon}$ is a contraction from
$\mathcal{\tilde{A}}$ to itself. We divide this proof into several steps.

Step 1: We claim that

$\|\mathbb{G}_{\epsilon}(\psi)\|_{*}\leq\frac{D}{|\ln\epsilon|},\quad\text{for
any}~{}\psi\in\mathcal{\tilde{A}}.$

The readers can refer to [12] for the proof of this step.

Step 2: We claim that, for any $\psi\in\mathcal{\tilde{A}}$, we have the
decompositions and estimates

$\displaystyle\mathbb{G}_{\epsilon}(\psi)=\mathbb{G}_{\epsilon
o}(\psi)+\mathbb{G}_{\epsilon e}(\psi)=\mathbb{G}_{\epsilon
o}^{\alpha}(\psi)+\mathbb{G}_{\epsilon o}^{\beta}(\psi)+\mathbb{G}_{\epsilon
e}(\psi),$ (6.1) $\displaystyle|\mathbb{G}_{\epsilon
o}^{\alpha}(\psi)|_{\sharp\sharp}\leq\frac{D\epsilon}{\sqrt{|\ln\epsilon|}},\quad\|\mathbb{G}_{\epsilon
o}^{\beta}(\psi)\|_{*}\leq D\epsilon\sqrt{|\ln\epsilon|}.$ (6.2)

The proofs of (6.1)-(6.2) rely on the following two facts.

$\clubsuit$ From Proposition 3.3, we have

$\displaystyle{\mathcal{R}}={\mathcal{R}}_{o}+{\mathcal{R}}_{e}={\mathcal{R}}_{o}^{\alpha}+{\mathcal{R}}_{o}^{\beta}+{\mathcal{R}}_{e},$

with

$\displaystyle|{\mathcal{R}}_{o}^{\alpha}|_{\sharp\sharp}\leq\frac{D\epsilon}{\sqrt{|\ln\epsilon|}},\quad\|{\mathcal{R}}_{o}^{\beta}\|_{*}\leq
D\epsilon\sqrt{|\ln\epsilon|}.$

$\clubsuit$ Next, we claim that we can decompose $\mathcal{N}(\psi)$ into the
following three parts

$\displaystyle\mathcal{N}(\psi)=\big{(}\mathcal{N}(\psi)\big{)}_{o}^{\alpha}+\big{(}\mathcal{N}(\psi)\big{)}_{o}^{\beta}+\big{(}\mathcal{N}(\psi)\big{)}_{e},$
(6.3)

with the estimates

$\displaystyle|\big{(}\mathcal{N}(\psi)\big{)}_{o}^{\alpha}|_{\sharp\sharp}\leq\frac{D\epsilon}{\sqrt{|\ln\epsilon|}},\qquad\|\big{(}\mathcal{N}(\psi)\big{)}_{o}^{\beta}\|_{**}\leq
D\epsilon\sqrt{|\ln\epsilon|},\quad\forall\,\psi\in\mathcal{\tilde{A}}.$ (6.4)

These can be verified in the following way. Recall that

$\mathcal{N}(\psi)=\mathcal{N}_{1}(\psi)+i\mathcal{N}_{2}(\psi),\quad\text{with}\
\mathcal{N}_{j}(\psi)=\big{(}\mathcal{N}_{j}^{+}(\psi),\mathcal{N}_{j}^{-}(\psi)\big{)}\quad
j=1,2.$

Without loss of generality, express
$\mathcal{N}^{\pm}_{1},\mathcal{N}^{\pm}_{2}$ in the coordinates
$(\ell_{1},\theta_{1})=(\ell,\theta)$ when $|z-\tilde{d}|>3$ and
${\rm{Re}(z)}>0$

$\displaystyle\mathcal{N}^{\pm}_{1}$
$\displaystyle=\,2(\partial_{\ell}\psi^{\pm}_{1})(\partial_{\ell}\psi^{\pm}_{2})+2(\partial_{\theta}\psi^{\pm}_{1})(\partial_{\theta}\psi^{\pm}_{2})\Big{(}\epsilon^{2}+\frac{1}{\ell^{2}}\Big{)}$
$\displaystyle\quad+\epsilon^{2}\tilde{d}\Big{(}\sin\theta\,\big{[}\partial_{\ell}\psi^{\pm}_{1}\partial_{\theta}\psi^{\pm}_{2}+\partial_{\theta}\psi^{\pm}_{1}\partial_{\ell}\psi^{\pm}_{2}\big{]}+\frac{2\cos\theta}{\ell}\partial_{\theta}\psi^{\pm}_{1}\partial_{\theta}\psi^{\pm}_{2}\Big{)}$
$\displaystyle\quad+\epsilon^{2}\tilde{d}^{2}\left(2\sin^{2}\theta\partial_{\ell}\psi^{\pm}_{1}\partial_{\ell}\psi^{\pm}_{2}+\frac{4\sin\theta\cos\theta}{\ell}\big{[}\partial_{\ell}\psi^{\pm}_{1}\partial_{\theta}\psi_{2}+\partial_{\theta}\psi^{\pm}_{1}\partial_{\ell}\psi^{\pm}_{2}\big{]}\right.+\frac{2\cos^{2}\theta}{\ell^{2}}\partial_{\theta}\psi^{\pm}_{1}\partial_{\theta}\psi^{\pm}_{2}\Big{)},$
(6.5)

and

$\displaystyle\mathcal{N}^{\pm}_{2}$
$\displaystyle=-(\partial_{\ell}\psi^{\pm}_{1})^{2}+(\partial_{\ell}\psi^{\pm}_{2})^{2}-\big{(}\epsilon^{2}+\frac{1}{\ell^{2}}\big{)}\Big{(}(\partial_{\theta}\psi^{\pm}_{2})^{2}-(\partial_{\theta}\psi^{\pm}_{1})^{2}\Big{)}$