A uniqueness result for the two vortex travelling wave in the Nonlinear Schrödinger equation

For the Nonlinear Schrödinger equation in dimension 2, the existence of a global
minimizer of the energy at fixed momentum has been established by
Bethuel-Gravejat-Saut <cit.> (see also <cit.>).
This minimizer is a travelling wave for the Nonlinear Schrödinger equation.
For large momentums, the propagation speed is small and the minimizer behaves like
two well separated vortices. In that limit, we show the uniqueness of this
minimizer, up to the invariances of the problem, hence proving the orbital stability of
this travelling wave. This work is a follow up to
two previous papers <cit.>, <cit.>, where we constructed and
studied a particular travelling wave of the equation. We show a uniqueness
result on this travelling wave in a class of functions that contains in
particular all possible minimizers of the energy.

§ INTRODUCTION AND STATEMENT OF THE RESULTS

We consider the Nonlinear Schrödinger equation

\begin{equation}
\tag{NLS}
%0 = (\tmop{NLS}) (\Psi) \assign
i \partial_t \Psi + \Delta \Psi - (|\Psi |^2 - 1) \Psi = 0
\end{equation}

in dimension 2 for $\Psi: \mathbbm{R}_t \times \mathbbm{R}_x^2
\rightarrow \mathbbm{C}$, also called the Gross-Pitaevskii equation without potential.
The Nonlinear Schrödinger

equation is a physical model for Bose-Einstein condensate (see <cit.>, <cit.>, <cit.>,
<cit.>), superfluidity (<cit.>) and nonlinear Optics (see <cit.>).
The condition at infinity for $(\tmop{NLS})$ will be

\[
\lvert \Psi \rvert \rightarrow 1 \quad \tmop{as} \quad | x | \rightarrow + \infty .
\]

The $(\tmop{NLS})$ equation is associated with the Ginzburg-Landau energy

\[ E (v) \assign \frac{1}{2} \int_{\mathbbm{R}^2} | \nabla v |^2 + \frac{1}{4}
\int_{\mathbbm{R}^2} (1 - | v |^2)^2 , \]

which is formally conserved by the $(\tmop{NLS})$ flow.
We denote by $\mathcal{E}$ the set of functions with finite energy, that is

\[ \mathcal{E} \assign \{ u \in H^1_{\tmop{loc}} (\R^2, \C), E (u) < + \infty \} . \]

The Cauchy problem for (NLS) is globally well-posed in the energy space: see <cit.>,
<cit.>, <cit.>.

Besides the energy, the momentum is another quantity formally conserved
by the $(\tmop{NLS})$ flow, and is associated with the invariance by translation of $(\tmop{NLS})$.
Formally, the momentum of $u$ is $ \frac{1}{2} \int_{\mathbbm{R}^2} \mathfrak{Re} ( i \nabla u \bar{u} ) \in \mathbbm{R}^2 $,
but its precise definition requires some care in the energy space due to the condition at infinity
(see <cit.> in dimension larger than two and <cit.> in dimension two).
If $ u \in 1 + \BC^\ii_c (\R^2 ) $ for instance, then the expression of the momentum reduces to

\[
\vec{P} (u)
= ( P_1( u) , P_2( u) )
= { \frac{1}{2} } \int_{\R^2} \mathfrak{Re} \big( i \nabla u ( \bar{u} - 1 ) \big) .
\]

In addition to the translation invariance, the $(\tmop{NLS})$ equation is also phase shift
invariant, that is invariant by multiplication by a complex of modulus one, and
rotation invariant.

§.§ Travelling waves for $(\tmop{NLS})$

Following the works in the physical literature of Jones and Roberts
(see <cit.>, <cit.>), there has been a large amount of
mathematical works on the question of existence and properties of travelling
waves solutions in the $(\tmop{NLS})$ equation, that is solution of

\[ 0 = (\tmop{TW}_c) (u) \assign - i c \partial_{x_2} u - \Delta u - (1 - | u |^2) u \]

for some $ c > 0 $, corresponding to particular solutions of $(\tmop{NLS})$ of the form
$ \Psi (t, x) = u ( x_1 , x_2 + ct ) $ (due to the rotational invariance, we may always
assume that the traveling wave moves along the direction $ - \overrightarrow{e_2} $).
We refer to <cit.> for an overview on
these problems in several dimensions. A natural approach is to look at the
minimizing problem for $ \mathfrak{p} > 0$

\[ E_{\min} ( \mathfrak{p} ) \assign \inf_{u \in \mathcal{E}} \{ E (u), P_2 (u) = \mathfrak{p} \} .
\]

It was shown by Bethuel-Gravejat-Saut in <cit.> that there exists a
minimizer to this problem.

For any $ \mathfrak{p} > 0$, there exists a non constant function $ u_\mathfrak{p} \in \mathcal{E}$
and $ c (u_\mathfrak{p}) > 0 $ such that $ P_2(u_\mathfrak{p}) = \mathfrak{p}$,
$u_\mathfrak{p}$ is a solution to $(\tmop{TW}_{c(u_\mathfrak{p})}) (u_\mathfrak{p}) = 0$ and

\[ E (u_\mathfrak{p}) = E_{\min} (\mathfrak{p}) . \]

Furthermore, any minimizer for $ E_{\min} (\mathfrak{p}) $ is, up to a translation in $ x_1 $, even in $ x_1 $.

The strategy is to look at the corresponding minimization problem on tori (this avoids the problems
with the definition of the momentum) larger and larger, and then pass to the limit.
For the minimizing problem $ E_{\min} (\mathfrak{p}) $, the compactness of minimizing sequences
has been shown later on in <cit.> for the natural semi-distance on $ \mathcal{E} $

\[
D_0 ( u, v) \assign \| \nabla u - \nabla v \|_{L^2(\R^2)} + \| \lvert u \rvert - \lvert v \rvert \|_{L^2(\R^2)} .
\]

For any $ \mathfrak{p} > 0 $, and any minimizing sequence $ (u_n)_{n \in \N} $ for $ E_{\min} (\mathfrak{p}) $,
there exists a subsequence $ (u_{n_j})_{j \in \N} $, a sequence of translations $ ( y_j )_{j \in \N} $ and a non constant
function $ u_\mathfrak{p} \in \mathcal{E} $ such that $ D_0 ( u_{n_j} , u_\mathfrak{p} ) \to 0 $,
$ P_2(u_{n_j}) \to P_2(u_\mathfrak{p}) = \mathfrak{p} $ and $ E(u_{n_j}) \to E(u_\mathfrak{p}) = E_{\min} (\mathfrak{p}) $
as $ j \to + \ii $. In particular, there exists $ c (u_\mathfrak{p}) > 0 $ such that $ P_2( u_\mathfrak{p}) = \mathfrak{p}$,
$u_\mathfrak{p}$ is a solution to $(\tmop{TW}_{c(u_\mathfrak{p})}) (u_\mathfrak{p}) = 0$ and

\[ E (u_\mathfrak{p}) = E_{\min} (\mathfrak{p}) . \]

Furthermore, the set $ \mathcal{S}_\mathfrak{p} \assign \{ v \in \mathcal{E} , \, P_2 ( v ) = \mathfrak{p} \ and \ E( v) = E_{\min} (\mathfrak{p}) \} $
of minimizers for $ E_{\min} (\mathfrak{p}) $ is orbitally stable for the semi-distance $ D_0 $.

An open and difficult question is to show, up to the invariances of the
problem, the uniqueness of the energy minimizer at fixed momentum. In other words, the problem is to
determine if $ \mathcal{S}_\mathfrak{p} $ consists of a single orbit under phase shift and space translation, that
is: do we have, for some minimizer $ U_\mathfrak{p} $,

\[
\mathcal{S}_\mathfrak{p} = \{ U_\mathfrak{p} (. - X) \ex^{i \gamma} , \, \gamma \in \R , \, X \in \R^2 \}?
\]

The main consequence of our work is to solve this open problem of uniqueness for large momentum.

There exists $ \mathfrak{p}_0 > 0$ such that, for any $ \mathfrak{p} > \mathfrak{p}_0 $,
if $u, v \in \mathcal{E}$ with $ P_2(u) = P_2(v) = \mathfrak{p} $ satisfy

\[ E (u) = E (v) = E_{\min} (\mathfrak{p}), \]

then, there exist $X \in \mathbb{R}^2 $ and $ \gamma \in \mathbb{R}$ such that

\[ u = v (. - X) \ex^{i \gamma} . \]

In fact, we will be able to show slightly stronger results than Theorem <ref>,
see Theorem <ref> below.

Even though we focus on the Ginzburg-Landau nonlinearity, it is plausible that our results
hold true (still for large momentum) for more general nonlinearities, provided vortices exist.
For the Ginzburg-Landau (cubic) nonlinearity, it is also possible that uniqueness of minimizers
holds true for $ E_{\min} (\mathfrak{p}) $ for any $ \mathfrak{p} > 0 $. However, the numerical
results given in <cit.> suggest that this may no longer be the case for more
general nonlinearities.

In the analysis of the minimization problem in <cit.> (and also <cit.>),
the following properties of $ E_{\min} $ play a key role.

The function $ E_{\min } : \R_+ \to \R $ is concave, nondecreasing and $ \sqrt{2} $-Lipschitz continuous.
In addition, there exists $K\gs 0 $ such that, for any $ \mathfrak{p} \gs 1 $, we have

\begin{equation}
\label{upper_bound_mini}
E_{\min}(\mathfrak{p} )
\ls 2 \pi \ln \mathfrak{p} + K .
\end{equation}

§.§ A smooth branch of travelling waves for large momentum

There have been several ways of constructing travelling waves of the
$(\tmop{NLS})$ equation, with different approaches. For instance, we may use variational methods,
such as a mountain pass argument in <cit.> and in <cit.>, or by minimizing the
energy at fixed kinetic energy (<cit.>, <cit.>).
Also, we have constructed in <cit.> a travelling
wave by perturbative methods, taking for ansatz a pair of vortices, by following the
Lyapounov-Schmidt reduction method as initiated in <cit.>. Vortices
are stationary solutions of $(\tmop{NLS})$ of degrees $n \in
\mathbb{Z}^{\ast}$ (see <cit.>, <cit.>, <cit.>,
<cit.>, <cit.>):

\[ V_n (x) = \rho_n (r) \ex^{i n \theta}, \]

where $x = r \ex^{i \theta}$, solving

\[ \left\{\begin{array}{l}
\Delta V_n - (| V_n |^2 - 1) V_n = 0\\
| V_n | \rightarrow 1 \tmop{as} | x | \rightarrow \infty .
\end{array}\right. \]

In the previous paper <cit.>, we constructed solutions of
$(\tmop{TW}_c)$ for small values of $c > 0$ as a perturbation of two
well-separated vortices (the distance between their centers is large when $c$
is small). We have shown the following result.

There exists $c_0 > 0$ a small constant such that for any $0 < c
\leqslant c_0$, there exists a solution of $(\tmop{TW}_c)$ of the form

\[ Q_c = V_1 (. - d_c \overrightarrow{e_1}) V_{- 1} (. + d_c
\overrightarrow{e_1}) + \Gamma_c, \]

where $d_c = \frac{1 + o_{c \rightarrow 0} (1)}{c}$ is a continuous function
of c. This solution has finite energy, that is $ Q_c \in \mathcal{E} $,

and $Q_c \rightarrow 1$ at infinity.

Furthermore, for all $ 2 < p \leqslant + \infty $, there exists $c_0 (p) > 0$
such that, if $0 < c \leqslant c_0 (p)$, for the norm

\[ \| h \|_p \assign \| h \|_{L^p (\mathbb{R}^2)} + \| \nabla h \|_{L^{p -
1} (\mathbb{R}^2)} \]

and the space $X_p \assign \{ f \in L^p (\mathbb{R}^2), \nabla f \in L^{p -
1} (\mathbb{R}^2) \}$, one has

\[ \| \Gamma_c \|_p = o_{c \rightarrow 0} (1) . \]

In addition,

\[ c \mapsto Q_c - 1 \in C^1 (] 0, c_0 (p) [, X_p), \]

with the estimate

\[ \left\| \partial_c Q_c + \left( \frac{1 + o_{c \rightarrow 0} (1)}{c^2}
\right) \partial_d (V_1 (. - d \overrightarrow{e_1}) V_{- 1} (. + d
\overrightarrow{e_1}))_{| d = d_c \nobracket} \right\|_p = o_{c
\rightarrow 0} \left( \frac{1}{c^2} \right) . \]

Finally, we have

\[
\frac{d}{dc} \big( P_2 (Q_c) \big) = \frac{- 2 \pi + o_{c \rightarrow 0} (1)}{c^2} < 0 ,
\]

hence the $ C^1 $ mapping

\[
\mathcal{P} : ] 0, c_0 ] \ni c \mapsto P_2 (Q_c) \in \mathbbm{R}
\]

is a strictly decreasing diffeomorphism from $ ]0 ,c_0] $ onto $ [ P_2( Q_{c_0} ) , +\ii [ $.

With the same kind of approach, <cit.> also provides
an existence result of travelling waves for $(\tmop{NLS}) $, including some cases with
more than two vortices. Our result has the advantage of showing the smoothness of the branch with respect to the speed.
In particular, with the last part of Theorem <ref>, we see that we may also parametrize the
branch $ c \mapsto Q_c $ by its momentum $ \mathcal{P} $.

It is conjectured that all these constructions yield the same branch of
travelling waves (for large momentum) when they are all defined, and that they are the solutions
numerically observed in <cit.> and <cit.> for more general
nonlinearities (see also <cit.>).
We will show here that the construction of Theorem <ref> are the unique, up to the natural
translation and phase invariances, constrained energy minimizers.

§.§ A uniqueness result for symmetric functions

We have shown in <cit.> several coercivity results for the travelling waves constructed in
Theorem <ref>. This will allow us to show the following uniqueness result for symmetric functions
close to the branch constructed in Theorem <ref>.

There exists a small constant $\varepsilon > 0$ and a large constant $ \lambda >  0$ such that,
if a function $u \in \mathcal{E}$ satisfies

* $\forall (x_1, x_2) \in \mathbb{R}^2, u (x_1, x_2) = u (- x_1, x_2)$,

* $u = V_1 (x - d \overrightarrow{e_1}) V_{- 1} (x + d
\overrightarrow{e_1}) + \Gamma$, with $d > \frac{1}{\varepsilon}$, $\|
\Gamma \|_{L^{\infty} (\{ \tilde{r}_d \leqslant 2 \lambda \})} \leqslant \varepsilon$,

* $\| | u | - 1 \|_{L^{\infty} (\{ \tilde{r}_d \geqslant \lambda \})} \leqslant \varepsilon$,

* $(\tmop{TW}_c) (u) = 0$ for some $c > 0$ such that $| d c - 1 | \leqslant \varepsilon$,

then, there exist $X \in \mathbb{R} $ and $ \gamma \in \mathbb{R}$ such that
$u = Q_c (. - X \vec{e}_2) \ex^{i \gamma}$, where $Q_c$ is defined in Theorem

In view of the symmetry assumption, we may replace the second hypothesis by

\[
\| u - V_1 (\cdot - d \overrightarrow{e_1}) \| _{L^\infty ( D ( d \overrightarrow{e_1} , 2 \lambda )) }
\ls \e .
\]

We will discuss the main arguments of the proof of Proposition <ref> in
the next section. This result can be seen as a local uniqueness result, but
the uniqueness turns out to be in a rather large class of function. Indeed, two functions that
satisfies the hypothesis of Proposition <ref> can be very far from each
other, for two main reasons. First, in condition 2., the vortices that compose one of them
have no reason to be close to the ones composing the other function since $d$ depends on $u$:
their centers $\pm d \overrightarrow{e_1}$ only need to satisfy $| d c - 1 |
\leqslant \varepsilon$, but for instance both $d = \frac{1}{c}$ and $d =
\frac{1}{c} + \frac{1}{\sqrt{c}}$ satisfy these conditions for $c > 0$ small enough.
Secondly, we only have that far from the vortices, the modulus is close to $1$ from condition 3., but we have no information
on the phase. The proof of Proposition <ref> will rely on methods used in <cit.>
in order to prove some coercivity, and we shall need to be very precise
to take into account all these cases.

A way to see that Proposition <ref> is a strong unicity result is that it implies the following local
uniqueness result in $L^{\infty}$ for even functions in $x_1$.

There exists $c_0, \varepsilon > 0$ such that, for $0 < c < c_0$,
if a function $u \in \mathcal{E}$ satisfies

* $\forall (x_1, x_2) \in \mathbb{R}^2, u (x_1, x_2) = u (- x_1, x_2)$,

* $(\tmop{TW}_c) (u) = 0$ in the distributional sense,

* $ \| u - Q_c \|_{L^{\infty} (\mathbb{R}^2)} \leqslant \varepsilon $,

then, there exists $X \in \mathbb{R}$ and $ \gamma \in \mathbb{R}$ such that
$u = Q_c (. - X \vec{e}_2) \ex^{i \gamma}$.

We may now state our main result. It establishes that any travelling wave solution
which is even in $ x_1 $ and within $\mathcal{O} (1) $ of the minimizing energy must be,
for large momentum, the $ Q_c $ travelling wave constructed in Theorem <ref>,
up to the natural translation and phase invariances.

For any $ \Lambda_0 > 0$ exists $ \mathfrak{p}_0 (\Lambda_0) > 0$ such that,
if $u \in \mathcal{E}$ satisfies

* $ \forall (x_1, x_2) \in \mathbb{R}^2, u (x_1, x_2) = u (- x_1, x_2)$,

* $ (\tmop{TW}_c) (u) = 0$ for some $c > 0$,

* $ P_2 (u) \gs \mathfrak{p}_0 (\Lambda_0) $,

* $ E (u) \leqslant 2 \pi \ln P_2 (u) + \Lambda_0 $,

then, there exist $X \in \mathbb{R} $ and $ \gamma \in \mathbb{R} $ such that

\[ u = Q_c (. - X \vec{e}_2) \ex^{i \gamma}, \]

where $Q_c$ is defined in Theorem <ref>.
In particular, $ P_2( u) = \mathcal{P} ( c ) $ (where $\mathcal{P}$ is defined in Theorem <ref>).

Section <ref> is devoted to the proof of this result. We show there that a function satisfying the hypothesis of Theorem <ref> also satisfies the hypothesis of Proposition <ref>.
Our result applies in particular to the constraint minimizers for the problem $ E_{\min} (\mathfrak{p}) $,
for large $ \mathfrak{p} $.

There exists $ \mathfrak{p}_0 > 0 $ such that, for any $ \mathfrak{p} \gs \mathfrak{p}_0 $, any minimizer $ U $
for $ E_{\min} (\mathfrak{p}) $, there exists $ \gamma \in \R $ and $ X \in \R^2 $ such that,
with $ c = \mathcal{P}^{-1} (\mathfrak{p} ) $,

\[
U = Q_c ( \cdot - X ) \ex^{ i \gamma } .
\]

Moreover, $ (\tmop{TW}_c) ( U ) = 0 $.

By a first translation in $ x_1 $, we may assume, by Theorem <ref>, that this
minimizer $U$ is even in $ x_1 $. By Proposition <ref>, the last hypothesis 4 of
Theorem <ref> is satisfied hence we may translate in $x_2 $ and use phase shift and
get that this minimizer $U$ is $ Q_c $. Necessarily, $P_2 ( U) = \mathfrak{p} = P_2 (Q_c ) $,
thus $ c = \mathcal{P}^{-1} (\mathfrak{p} ) $.

Theorem <ref> is a direct consequence of this corollary.
This allows to derive several interesting consequences on the function $ E_{\min} $.
This also shows that the branch of Theorem <ref> coincides with the global
energy minimizer at fixed momentum (up to translation and phase shift).

There exists $ c_* > 0 $ such that, for $ 0 < c \ls c_* $, $ Q_c $ is a minimizer for
$ E_{\min} ( P_2 ( Q_c ) ) $. Moreover, there exists $ \mathfrak{p}_0 > 0 $ such that the following statements hold.

* The function $ E_{\min} $ is of class $ C^2 $ in $ [ \mathfrak{p}_0 , +\ii [ $ and

\[
0 > E_{\min}'' ( \mathfrak{p} ) \sim - \frac{2\pi}{\mathfrak{p}^2 } ,
\quad \quad
0 < E_{\min}' ( \mathfrak{p} ) \sim \frac{2\pi}{\mathfrak{p} } ,
\quad \quad
E_{\min} ( \mathfrak{p} ) = 2\pi \ln \mathfrak{p} + \mathcal{O}(1)
. \]

* For $ \mathfrak{p} \gs \mathfrak{p}_0 $
$ \mathcal{S}_\mathfrak{p} = \{ Q_{\mathcal{P}^{-1} (\mathfrak{p})} (. - X) \ex^{i \gamma} , \, \gamma \in \R , \, X \in \R^2 \} $,
hence, for any $ \mathfrak{p} \gs \mathfrak{p}_0 $, $ E_{\min}' ( \mathfrak{p} ) $ is the speed of any
minimizer for $ E_{\min} ( \mathfrak{p} ) $.

* For any $ \mathfrak{p} \gs \mathfrak{p}_0 $, $ Q_{ \mathcal{P}^{-1} ( \mathfrak{p} ) } $ is orbitally stable
for the semi-distance $ D_0 $ (or, equivalently, for $ 0 < c \ls c_* $, $ Q_c $ is orbitally stable
for the semi-distance $ D_0 $).

* For $ \mathfrak{p} \gs \mathfrak{p}_0 $ and any minimizer $u$ for $ E_{\min} (\mathfrak{p} ) $, then,
up to a space translation and a phase shift, $u$ enjoys the symmetry

\[
\forall (x_1, x_2 ) \in \R^2 , \quad \quad
u ( x_1, -x_2 ) = \bar{u} (x_1, x_2 )
\]

in addition to the symmetry in $ x_1 $.

* For any $ \Lambda > 0$, there exists $ \mathfrak{p}_0 (\Lambda) > 0$ such that,
if $u \in \mathcal{E}$ satisfies $(\tmop{TW}_c) (u) = 0$ for some $c > 0$,
$ P_2 (u) \geqslant \mathfrak{p}_0 (\Lambda )$ and $u$ is even in $x_1$, then either
$ E (u) = E_{\min} ( P_2 (u) ) $, or $E (u) \geqslant E_{\min} (P_2 (u) ) + \Lambda $.

By Theorems <ref> and <ref>, we have existence of at least one minimizer
$ U_\mathfrak{p} $ for $ E_{\min} (\mathfrak{p} ) $, whatever is $ \mathfrak{p} > 0 $.
For large $\mathfrak{p} $, by applying Corollary <ref>, we have $ U_\mathfrak{p} = Q_c ( \cdot - X ) \ex^{i \gamma } $
for some $ X \in \R^2 $ and $ \gamma \in \R $, thus proving that $ Q_c $ is a minimizer for $ E_{\min} (\mathfrak{p} ) $
and that $ P_2 ( Q_c ) = \mathcal{P} ( c) = \mathfrak{p} $.

For 1., it suffices to notice that, in view of Corollary <ref> applied
to any minimizer (we have existence by Theorems <ref> and <ref>)
$ E_{\min} (\mathfrak{p} ) = E ( Q_{ \mathcal{P}^{-1} (\mathfrak{p} ) } ) $.
We then conclude by using that $ \mathcal{P} $ is a $ C^1 $ diffeomorphism and that
$ c \mapsto E (Q_c) $ is also of class $ C^1 $ (see <cit.>, Proposition 1.2)
that $ E_{\min} $ is of class $ C^1 $ in $ [\mathfrak{p}_0 , +\ii [ $ and that

\[
E_{\min}' (\mathfrak{p} )
= \frac{d}{ dc} E ( Q_{ c } )_{|c = \mathcal{P}^{-1}(\mathfrak{p})} \times \frac{1}{ \mathcal{P}' ( \mathcal{P}^{-1} (\mathfrak{p} )) }
= \mathcal{P}^{-1}(\mathfrak{p}) ,
\]

in view of the Hamilton like relation (formally shown in <cit.> and rigorously proved for the
branch constructed in Theorem <ref> in <cit.>)

\[
\frac{d}{ dc} E ( Q_{ c } ) = c \frac{d}{ dc} P_2 ( Q_{ c } ) .
\]

Since $ \mathcal{P} $ is a $ C^1 $ diffeomorphism, we deduce that $ E_{\min}' $ is of class $ C^1 $.
The asymptotics for $ E_{\min}' $ and $ E_{\min}'' $ then follow from Proposition 1.2 in <cit.>.
Integration would yield $ E_{\min} ( \mathfrak{p} ) \sim 2\pi \ln \mathfrak{p} $, but we may slightly improve
this estimate. Indeed, Proposition <ref> gives $ E_{\min} ( \mathfrak{p} ) \ls 2\pi \ln \mathfrak{p} + \mathcal{O} (1) $,
and the lower bound is a straightforward consequence of Theorem <ref> $ (i) $ and
the study in subsection <ref>.

Statement 2. is a rephrasing of Corollary <ref> combined with the existence of at least
one constrained minimizer. Statement 3. is then a direct consequence of Theorem <ref>.
Statement 4. simply follows from the fact that $ Q_c $ enjoys by construction this symmetry
(see <cit.>). Finally, statement 5. is also a rephrasing of Theorem <ref>.

Concerning the stability stated in statement 3. in the above theorem, we quote the work <cit.>,
where a linear "spectral" stability result is proved (through ad hoc hypotheses that have been checked in <cit.>),
namely that the linearized equation $ i \p_t v = L_{Q_c} (v) $ does not have exponentially growing solutions
(in $ \dot{H}_ 1(\R^2; \C) $, say). Statement 3. in the above theorem does not rely on the result in <cit.>,
and is for the nonlinear (orbital) stability (following the Cazanave-Lions approach).

Let us conclude this section with several comments on our result.
First, let us explain the relevance of the symmetry hypothesis, namely
that we restrict to mappings even in $x_1 $. This symmetry is used in the
coercivity of the branch of Theorem <ref>, along the following arguments. The
quadratic form around the travelling wave $Q_c$ is decomposed in three areas,
close to the two vortices, and far from them. In the latter region, the
coercivity can be shown without any orthogonality condition. Close to the
vortices, the quadratic form is close to the one of a single vortex, that has been
studied in <cit.>. Its coercivity requires three orthogonality
conditions, two for the translation, and one for the phase. Therefore, we can
show the coercivity of the full quadratic form with six orthogonality
conditions, three for each vortex. However, the family of travelling waves
of Theorem <ref> has only five parameters (two for the speed, two for the
translation, and one for the phase). The symmetry is then used to reduce the
problem to three orthogonality conditions into a family with three parameters.
With this symmetry, both orthogonality conditions on the phase for the two
vortices become the same condition. It is possible to prove a coercivity result with
only five orthogonality conditions without symmetry (see <cit.>),
but then the coercivity constant goes to $0$ when $ c \rightarrow 0 $.
This would pose a problem for the uniqueness result. The last statement in Theorem
<ref> shows that, when restricting ourselves to symmetric travelling waves,
there is an energy threshold under which there is no travelling wave
except the $Q_c $ branch.

Secondly, the proof of the fact that $ Q_c $ is a minimizer of the energy for fixed
momentum relies on the existence of such minimizers. In particular, we have not
been able to use our coercivity results in <cit.> in order to prove
directly that $ Q_c $ is orbitally stable (for small $ c$).

Thirdly, the symmetry in $ x_2 $ for the minimizers (statement 4) is
established as a consequence of the uniqueness result and not in itself.
Notice that the numerical studies in <cit.>, <cit.> and
<cit.> assume the two symmetries.

§.§ The travelling wave $ Q_c $ and two other variational characterizations

The travelling wave $ Q_c $ as a mountain pass solution.
Thanks to the results in Theorem <ref>, it is easy to show that we have locally,
near $ Q_c $, a mountain-pass geometry. Indeed, let $ c_* > 0 $ be small, and define

\[
\Upsilon_{c_*}
\assign \{ \upsilon : [-1 , +1 ] \to \BE \ {\rm continuous}, \, S(-1) = Q_{3c_*/2} , \, S(+1) = Q_{c_*/2} \}
\]

the set of paths from $ Q_{3c_*/2} $ to $ Q_{ c_*/2} $ in $ \BE $. Then, we claim that

\begin{equation}
\label{critic_val}
\inf_{ \upsilon \in \Upsilon_{c_*} } \max_{ t \in [-1,+1] } ( E - c_* P_2 ) ( \upsilon(t) )
= ( E - c_* P_2 ) ( Q_{c_*} ) .
\end{equation}

Indeed, let $ \upsilon \in \Upsilon_{c_*} $. By the intermediate value theorem, there
exists $ t_* \in [ -1, + 1 ] $ such that $ P_2 ( \upsilon(t) ) = P_2 (Q_{c_*} ) $. Since
$ Q_{c_*} $ is a minimizer for $ E_{\rm min} ( Q_{c_*} ) $, we infer

\[
\max_{ t \in [-1,+1] } ( E - c_* P_2 ) ( \upsilon (t) )
\gs E ( S(t_*) ) - c_* P_2 ( Q_{c_*} )
\gs E ( Q_{c_*} ) - c_* P_2 ( Q_{c_*} ) .
\]

Moreover, considering the particular $ \BC^1 $ path $ \upsilon_* : [-1, +1 ] \to \BE $
defined by $ \upsilon ( t) \assign Q_{c_* - t c_*/2 } $, we see that

\[
\frac{d}{dt} ( E - c_* P_2 ) ( \upsilon_* (t) )
= - \frac{c_*}{2} \Big( \frac{d}{dc} E(Q_c) - c_* \frac{d}{dc} P_2(Q_c) \Big)_{|c = c_*- t c_*/2 }
= \frac{c_*^2 t }{4} \Big( \frac{d}{dc} P_2(Q_c) \Big)_{|c = c_*- t c_*/2 }
\]

in view of the Hamilton group relation $ \frac{d}{dc} E(Q_c) = c \frac{d}{dc} P_2(Q_c) $ (see
Proposition 1.2 in <cit.>). Since $ \frac{d}{dc} P_2(Q_c) < 0 $, we deduce that
$ ( E - c_* P_2 ) ( \upsilon_*(t) ) $ increases in $ [-1 , 0 ] $ and decreases in $ [0,+1 ] $, hence
has maximal value $ E ( Q_{c_*} ) - c_* P_2 ( Q_{c_*} ) $, as wished.

Furthermore, by the asymptotics given in <cit.> and the above mentioned
Hamilton group relation $ \frac{d}{dc} E(Q_c) = c \frac{d}{dc} P_2(Q_c) $, we have

\[
( E - c_* P_2 ) ( Q_{c_*} ) - ( E - c_* P_2 ) ( Q_{c_*/2} )
= \int_{c_*/2}^{c_*} (c - c_*) \frac{d}{dc} P_2(Q_c) \, dc
> 0
\]

since $ c - c_* < 0 $ and $ \frac{d}{dc} P_2(Q_c) < 0 $. Similarly, we prove that
$ ( E - c_* P_2 ) ( Q_{c_*} ) - ( E - c_* P_2 ) ( Q_{3c_*/2} ) < 0 $.

We now claim that if $ u \in \mathcal{E} $ is such that $ (\tmop{TW}_{c_*}) (u) = 0 $ and

\begin{equation}
\label{critic_val_atteinte}
( E - c_* P_2 ) ( u )
= \inf_{ \upsilon \in \Upsilon_{c_*} } \max_{ t \in [-1,+1] } ( E - c_* P_2 ) ( \upsilon(t) )
= ( E - c_* P_2 ) ( Q_{c_*} ) ,
\end{equation}

by (<ref>), that is if $u $ is a critical point of $ E - c_* P_2 $ at the good critical value,
then we must have $ P_2 ( u) = P_2 ( Q_{c_*} ) $. Indeed, by the Pohozaev identity (<ref>),
we have

\[
c_* P_2( u) = \frac{1}{2} \int_{ \R^2 } ( 1- \lvert u \rvert^2 )^2 \, dx
\gs 0 ,
\]

hence $ P_2 (u) \gs 0 $. Furthermore, we know that $ E_{\min} $ is concave in $ \R_+ $
(Proposition <ref>), and that $ E_{\min} $ is of class $ C^1 $ and strictly
concave on $ [ \mathfrak{p}_0, +\ii [ $ (by statement 1. of Theorem <ref>).
Therefore, if $ P_2 ( u) \neq P_2 ( Q_{c_*} ) $, then

\[
\frac{ E_{\min} ( P_2 ( u) ) - E_{\min} ( P_2 ( Q_{c_*} )) }{P_2 ( u) - P_2 ( Q_{c_*} ) }
\neq c_*
= E_{\min} '( P_2 ( Q_{c_*} ) ),
\]

in contradiction with (<ref>).

As a consequence, we have

\[
E( u) = E ( Q_{c_*} ) = E_{\min} ( P_2 ( u) ) = E_{\min} ( P_2 (Q_{c_*} ) ) ,
\]

implying that $ u $ is a minimizer for $ E_{\min} ( P_2 (Q_{c_*} ) ) $, hence there exist
$ \gamma \in \R $ and $ X \in \R^2 $ such that $ u = Q_{c_*} ( \cdot - X ) \ex^{i \gamma } $,
hence proving a uniqueness result for mountain pass type travelling wave solutions.
However, stating rigorously a useful uniqueness result for this kind of variational solution is not
so easy: in <cit.>, the mountain pass is implemented in the space $ 1 + H^1 (\R^2 ) $
whereas we know (by the result in <cit.>) that the nontrivial traveling wave
do not belong to this affine space; in <cit.>, the solution is constructed by
working first on $ [-N,+N ] \times \R $ and then passing to the limit, and it is then not
immediate to compute the functional $ E - c P $ on the solution; in addition, the method
does not provide easily some explicit bounds on the energy or the momentum.
We shall then not go further in this discussion even though the previous arguments
indicate that mountain pass solutions are (at least for small $c$) only the orbit of $Q_c$.

The travelling wave $ Q_c $ as a minimizer of $ E - c P_2 $ for fixed kinetic energy.
In <cit.>, for $ \kappa \gs 0 $, the following variational problem is investigated:

\[
I_{\min} ( \kappa ) = \inf \Big\{ \frac{1}{4} \int_{\R^2} ( 1 - \lvert v \rvert^2 )^2 \, dx - P_2 (v), \,
v \in \mathcal{E} \textrm{ s.t. } \frac{1}{2} \int_{\R^2} | \nabla v |^2 \, d x = \kappa \Big\} .
\]

Any minimizer $v $ for $ I_{\min} ( \kappa ) $ is such that there exists $c >0 $ satisfying
$ (\tmop{TW}_{c}) ( v ( \cdot / c ) ) = 0 $. In 2d and for the Ginzburg-Landau nonlinearity,
existence of minimizers for $ \kappa > 0 $ is established in Theorem 1.2 there.
Furthermore, it is shown in <cit.> (see Proposition 8.4 there) that if $ \mathfrak{p} > 0 $
and if $ U $ is a minimizer for
$ E_{\min} ( \mathfrak{p}) $ with speed $ c $, then $ U ( c\, \cdot ) $ is a minimizer
for $ I_{\min} ( \kappa ) $ with $ \kappa = \frac{1}{2} \int_{\R^2} | \nabla U |^2 \, d x $ (this last quantity
is scale-invariant in 2d) and $ I_{\min} $ is differentiable at this $ \kappa $, with
$ I_{\min} ' ( \kappa ) = - 1 / c^2 $. Since $ Q_c $ is a minimizer for
$ E_{\min} ( P_2 (Q_c) ) $, if we prove that $ c \mapsto \frac{1}{2} \int_{\R^2} | \nabla Q_c |^2 \, d x $
is a decreasing $ C^1 $-diffeomorphism from $ ]0, c_0 ] $, for some small $ c_0 $,
onto $ [ \kappa_0 , +\ii [ $, with $ \kappa_0 \assign \frac{1}{2} \int_{\R^2} | \nabla Q_{c_0} |^2 \, d x $,
then we shall conclude that $ I_{\min} $ is of class $ C^1 $ on $ [ \kappa_0 , +\ii [ $,
and that (by the arguments in <cit.>), the only minimizer for
$ \kappa = \frac{1}{2} \int_{\R^2} | \nabla Q_c |^2 \, d x $ (for some suitable $ c \in ]0, c_0 ] $) is
$ Q_c ( c \, \cdot ) $ up to the natural translation and phase invariances and, in
addition, $ I_{\min} ' ( \kappa ) = - 1 / c^2 $. In order to prove that statement, it suffices to
use the Pohozaev identity (<ref>) and deduce

\[
\frac{1}{2} \int_{\R^2} | \nabla Q_c |^2 \, d x
= E ( Q_c) - \frac{1}{4} \int_{\R^2} ( 1 - \lvert Q_c \rvert^2 )^2 \, dx
= E ( Q_c) - \frac{ c P_2 (Q_c) }{2 } .
\]

Therefore, by using the Hamilton $ \frac{d}{ dc} E ( Q_{ c } ) = c \frac{d}{ dc} P_2 ( Q_{ c } ) $
and then the asymptotics of obtained in <cit.>, we arrive at

\[
\frac{d}{2 dc}
\int_{\R^2} | \nabla Q_c |^2 \, d x
= \frac{d}{dc} ( E ( Q_c) ) - \frac{ c }{2} \frac{d}{dc} P_2 (Q_c) - \frac{1}{2} P_2 (Q_c)
= \frac{ c }{2} \frac{d}{dc} P_2 (Q_c) - \frac{1}{2} P_2 (Q_c)
\sim - \frac{2\pi}{c} < 0
\]

and this concludes.

Acknowledgement: E.P. is supported by Tamkeen under the NYU Abu Dhabi
Research Institute grant CG002.

§ PROOF OF THE LOCAL UNIQUENESS RESULT (PROPOSITION <REF>)

This section is devoted to the proof of Proposition <ref> and Corollary
<ref>. The proof of Proposition <ref> uses arguments from the
proof of Theorem 1.14 of <cit.>, another local uniqueness result of this
problem, but in different spaces. We explain here the core ideas of the proof.

Let us explain schematically the proof of Proposition <ref>. We first pick $c' $, $X $, $ \gamma' $
in such a way that $ Q = Q_c'(.-X) \ex^{i \gamma}$ has the same vortices as $u$. This is possible because $c \rightarrow d_c$,
the position of the vortices, is smooth. We then decompose $u=Q \ex^{\psi} $, where $\psi $ is the error term.
This can not be done near the zeros of $Q$, but we focus here on the domain far from the vortices.

The equation satisfied by $\psi$ is then $ (\tmop{TW}_c) (u) = 0 =(\tmop{TW}_c)(Q)+\it{L}(\psi)+\it{NL}(\psi)$,
where we regroup the linear terms in $\it{L}$ and the nonlinear terms in $\it{NL}$, and $(\tmop{TW}_c)(Q) \neq 0$
because $c \neq c'$. We then take the scalar product of this equation with $\psi$, and we get
$ 0 = \langle (\tmop{TW}_c)(Q), \psi \rangle + B_Q(\psi) + \langle \it{NL}(\psi) , \psi \rangle $.
Now, the coercivity of $B_Q$ has been studied in <cit.>. It holds (for even function in $x_1$)
up to three orthogonality conditions, that can be satisfied by changing slightly the modulation parameters
$c',X,\gamma$. We deduce that $B_Q(\psi) \geqslant K \| \psi \|^2_1$ for some norm $ \| \cdot \|_1$.

There are two main difficulties at this point. First, since the hypothesis on $u$ in Proposition <ref>
are weak, we simply have $||\psi||_1 < + \infty$, but not the fact that it is small. Therefore, an estimate
of the form $ \lvert \langle \it{NL}(\psi) , \psi \rangle \rvert \leqslant K \| \psi \|^3_1$ would not be enough to conclude.
Secondly, the norm $ \| \cdot \|_1$ is rather weak, and in fact $\langle \it{NL}(\psi) , \psi \rangle$ can not be controlled
by powers of $ \| \psi \|_1$.

Concerning the term $\langle (TW_c)(Q), \psi \rangle $, we may show that we always have $|c-c'| \leqslant o(1) \| \psi \|_1$,
thus $ \lvert \langle (TW_c)(Q), \psi \rangle \rvert \leqslant o(1) \| \psi \|^2_1 $. Therefore, we are led to
$ (K/2) \| \psi \|^2_1 \ls \langle (\tmop{TW}_c)(Q), \psi \rangle + B_Q(\psi) = - \langle \it{NL}(\psi) , \psi \rangle $.

Then, even if $\| \psi \|_1$ is not small, by the hypothesis of Proposition <ref>, $\psi$ will be small in other
(non equivalent) norms. Let us write one of them $ \| \cdot \|_2$. Our goal is then to show an estimate of the
form $ \lvert \langle \it{NL}(\psi) , \psi \rangle \rvert \leqslant K \| \psi \|_2 \| \psi \|^2_1$, which would conclude.
This is possible, except for one nonlinear term, which contains two derivatives. We then perform some
integrations by parts on it. When both derivatives fall on the same term, we get a term containing $\Delta \psi$,
which also appear in the equation $0= (\tmop{TW}_c)(Q)+\it{L}(\psi)+\it{NL}(\psi)$ (in $\it{L}(\psi)$).
We thus replace it using the equation, which leads to another term containing two derivatives (from $\it{NL}(\psi)$),
and other terms that can be successfully estimated. After $n$ such integration by parts, we have an estimate
of the form $ \lvert \langle \it{NL}(\psi) , \psi \rangle \rvert \leqslant K\| \psi \|_2 \| \psi \|^2_1 + \| \psi \|_3 \| \psi \|_2^n \| \psi \|_1^2 $,
where $\|\cdot \|_3$ is another (semi-)norm in which $\psi$ is not necessarily small. Now, taking $n$ large enough
(depending on $\psi$), since $ \| \psi \|_2 \ll 1$, we get $| \langle \it{NL}(\psi) , \psi \rangle | \leqslant o(1) \| \psi \|^2_1$,
concluding the proof.

The problem is a lot simpler near the vortices. There, we write $u = Q +\phi$ and the coercivity norm is equivalent
to the $H^1$ norm, and the hypothesis of Proposition <ref> gives us $ \| \phi \|_{L^\infty} = o(1)$.
The estimate of the nonlinear terms then becomes trivial.

As stated in the introduction, the symmetry condition is necessary to have a
coercivity result where the coercivity constant is uniform, see Corollary
<ref> below. This is the only place where the symmetry is used in a crucial way.

§.§ Some properties of the branch of travelling waves from Theorem <ref>

We recall here properties on the branch $c \mapsto Q_c$ from Theorem
<ref>, coming mainly from <cit.> and <cit.>. In this section, we will use the notation

\[ \langle f,g \rangle \assign \int_{\mathbb{R}^2} \mathfrak{Re} (f \bar{g}) . \]

§.§.§ Properties of vortices

We start with some estimates on vortices, that compose the travelling wave
(see Theorem <ref>).

A vortex centered around $0$, $V_1 (x) = \rho_1 (r) \ex^{i
\theta}$, verifies $V_1 (0) = 0$, $E (V_1) = + \infty$ and there exist
constants $K, \kappa > 0$ such that

\[ \forall r > 0, \ 0 < \rho_1 (r) < 1;
\quad \quad
\rho_1 (r) \sim_{r \rightarrow 0} \kappa r ;
\quad \quad
\rho'_1 (r) \sim_{r \rightarrow 0} \kappa ; \]

\[ \forall r > 0, \ \rho_1' (r) > 0 ;
\quad \quad
\rho_1' (r) = O_{r \rightarrow \infty} \left( \frac{1}{r^3} \right);
\quad \quad | \rho_1'' (r) | + | \rho_1''' (r) | \leqslant K ;
\]

\[ 1 - | V_1 (x) | = \frac{1}{2 r^2} + O_{r \rightarrow \infty} \left(
\frac{1}{r^3} \right) ; \]

\[ | \nabla V_1 | \leqslant \frac{K}{1 + r};
\quad \quad | \nabla^2 V_1 | \leqslant
\frac{K}{(1 + r)^2} \]

\[ \nabla V_1 (x) = i V_1 (x) \frac{x^{\bot}}{r^2} + O_{r \rightarrow
\infty} \left( \frac{1}{r^3} \right), \]

where $x^{\perp} \assign (- x_2, x_1)$, $x = r \ex^{i \theta} \in
\mathbb{R}^2$. Furthermore, similar properties hold for $V_{- 1}$, since

\[ V_{- 1} (x) = \overline{V_1 (x)_{}} . \]

§.§.§ Toolbox

We list in this section some results useful for the analysis of travelling waves for not necessarily small speeds.

Assume that $ U \in L^3_{\rm loc} (\R^d) $ solves

\[
\Delta U + i c \p_2 U + U ( 1 - \lvert U \rvert^2 ) = 0 .
\]

\[
\| U \|_{L^\ii(\R^d)} \ls 1+ \frac{c^2}{4} .
\]

There exists $ K > 0 $ such that for any $ c \in [ - \sqrt{2} , + \sqrt{2} ] $ and
any $ U \in L^3_{\rm loc} (\R^d) $ satisfying $ (\textrm{TW}_c)(U)=0  $, we have

\begin{equation}
\label{basik}
\| \nabla U \|_{L^\ii(\R^d)} + \| \nabla^2 U \|_{L^\ii(\R^d)} \ls K .
\end{equation}

The following Pohozaev identity (see <cit.> for instance) will be useful in our analysis. If
$ c \in \R $ and $ U \in \BE $ satisfies $ (\textrm{TW}_c) $, then

\begin{equation}
\label{Pohoz}
\frac{1}{2} \int_{\R^2} (1 - \lvert U \rvert^2 )^2 \, dx
= c P_2 (U) .
\end{equation}

We shall also make use of the algebraic decay of the travelling waves conjectured in
<cit.> and shown in <cit.>.

Let $ c \in [0, \sqrt{2} [ $. Assume that $ U \in \BE $ is a solution of $ (\textrm{TW}_c)(U)=0  $. Up to a phase shift,
we may assume $ U( x) \to 1 $ for $|x| \to +\ii $. Then, there exists
$ M $, depending on $ U$ and $c$ such that, for $ x \in \R^2 $,

\[
\lvert U (x) - 1 \rvert \ls \frac{M}{ 1+ |x|} ,
\quad \quad
\lvert \nabla U (x) \rvert \ls \frac{M}{( 1+ |x|)^2} ,
\quad \quad
\big\lvert \lvert U (x) \rvert - 1 \big\rvert \ls \frac{M}{( 1+ |x|)^2} .
\]

§.§.§ Symmetries of the travelling waves from Theorem

We recall from <cit.> that the travelling wave $Q_c$ constructed in
Theorem <ref> satisfies for all $x = (x_1, x_2) \in \mathbb{R}^2$,

\[ Q_c (x_1, x_2) = Q_c (- x_1, x_2) = \overline{Q_c (x_1, - x_2)} . \]

This implies that for all $x = (x_1, x_2) \in \mathbb{R}^2$,

\[ \partial_c Q_c (x_1, x_2) = \partial_c Q_c (- x_1, x_2) =
\overline{\partial_c Q_c (x_1, - x_2)}, \]

\[ \partial_{x_1} Q_c (x_1, x_2) = - \partial_{x_1} Q_c (- x_1, x_2) =
\overline{\partial_{x_1} Q_c (x_1, - x_2)}, \]

\[ \partial_{x_2} Q_c (x_1, x_2) = \partial_{x_2} Q_c (- x_1, x_2) = -
\overline{\partial_{x_2} Q_c (x_1, - x_2)} \]

\[ \partial_{c^{\bot}} Q_c (x_1, x_2) = - \partial_{c^{\bot}} Q_c (- x_1, x_2)
= - \overline{\partial_{c^{\bot}} Q_c (x_1, - x_2)}, \]

where $\partial_{c^{\bot}} Q_c \assign x^{\bot} . \nabla Q_c$, see subsection
2.2 of <cit.>. Remark that these quantities all have different symmetries.

§.§.§ A coercivity result

From Proposition 1.2 of <cit.>, we recall that $Q_c$ defined in Theorem
<ref> has two zeros, at $\pm \tilde{d}_c \vec{e}_1$, with

\begin{equation}
d_c - \tilde{d}_c = o_{c \rightarrow 0} (1) . \label{31gne}
\end{equation}

We define (as in <cit.>) the symmetric expended energy space by

\[ H_{Q_c}^{\exp, s} \assign \left\{ \varphi \in H^1_{\tmop{loc}}
(\mathbb{R}^2, \mathbb{C}), \| \varphi \|_{H_{Q_c}^{\exp}} < + \infty,
\forall (x_1, x_2) \in \mathbb{R}^2, \varphi (- x_1, x_2) = \varphi (x_1,
x_2) \right\}, \]

where, with $\varphi = Q_c \psi$, $\tilde{r} = \tilde{r}_{\tilde{d}_c} = \min
(\tilde{r}_1, \tilde{r}_{- 1})$, $\tilde{r}_{\pm 1}$ being the distances to
the zeros of $Q_c$ (we use $\tilde{r}$ instead of $\tilde{r}_{\tilde{d}_c}$ to
simplify the notations here), we define

\[ \| \varphi \|^2_{H_{Q_c}^{\exp}} \assign \| \varphi \|^2_{H^1 (\{ \tilde{r}
\leqslant 10 \})} + \int_{\{ \tilde{r} \geqslant 5 \}} | \nabla \psi |^2
+\mathfrak{Re}^2 (\psi) + \frac{| \psi |^2}{\tilde{r}^2 \ln^2 \tilde{r} } . \]

By using (<ref>), we deduce, for any $R > 0$, $\| \varphi \|_{H^1 (\{ \tilde{r} \leqslant R \})}
\leqslant K (R) \| \varphi \|_{H_{Q_c}^{\exp}}$. The linearized operator
around $Q_c$ is

\[ L_{Q_c} (\varphi) \assign - \Delta \varphi - i c \partial_{x_2} \varphi -
(1 - | Q_c |^2) \varphi + 2\mathfrak{Re} (\overline{Q_c}
\varphi) Q_c . \]

We take a smooth cutoff function $\tilde{\eta}$ such that $\tilde{\eta} (x) =
0$ on $B (\pm \widetilde{d_c} \overrightarrow{e_1}, 2 R)$, $\tilde{\eta} (x) =
1$ on $\mathbb{R}^2 \backslash B (\pm \widetilde{d_c} \overrightarrow{e_1}, 2
R + 1)$, where $\pm \widetilde{d_c} \overrightarrow{e_1}$ are the zeros of
$Q_c$ and $R > 0$ will be defined later on (it will be a universal constant,
independent of any parameters of the problem). We define the quadratic form
(as in <cit.>)

\begin{eqnarray}
B^{\exp}_{Q_c} (\varphi) & \assign & \int_{\mathbb{R}^2} (1 - \tilde{\eta})
(| \nabla \varphi |^2 -\mathfrak{Re} (i c \partial_{x_2} \varphi
\bar{\varphi}) - (1 - | Q_c |^2) | \varphi |^2 + 2\mathfrak{Re}^2
(\overline{Q_c} \varphi)) \nonumber\\
& - & \int_{\mathbb{R}^2} \nabla \tilde{\eta} . (\mathfrak{Re}
(\nabla Q_c \overline{Q_c}) | \psi |^2 - 2\mathfrak{I}\mathfrak{m} (\nabla
Q_c \overline{Q_c}) \mathfrak{Re} (\psi) \mathfrak{I}\mathfrak{m}
(\psi)) \nonumber\\
& + & \int_{\mathbb{R}^2} c \partial_{x_2} \tilde{\eta}
\mathfrak{Re} (\psi) \mathfrak{I}\mathfrak{m} (\psi) | Q_c |^2
\nonumber\\
& + & \int_{\mathbb{R}^2} \tilde{\eta} (| \nabla \psi |^2 | Q_c |^2 +
2\mathfrak{Re}^2 (\psi) | Q_c |^4) \nonumber\\
& + & \int_{\mathbb{R}^2} \tilde{\eta} (4\mathfrak{I}\mathfrak{m} (\nabla
Q_c \overline{Q_c}) \mathfrak{I}\mathfrak{m} (\nabla \psi)
\mathfrak{Re} (\psi) + 2 c | Q_c |^2 \mathfrak{I}\mathfrak{m}
(\partial_{x_2} \psi) \mathfrak{Re} (\psi)) .  \label{CP2Btilda}
\end{eqnarray}

We recall from <cit.> (or by integration by parts) that for $\varphi \in
C^{\infty}_c (\mathbb{R}^2, \mathbb{C})$, we have $B^{\exp}_{Q_c} (\varphi) = \langle
L_{Q_c} (\varphi), \varphi \rangle$, and that $B^{\exp}_{Q_c} (\varphi)$ is
well defined for $\varphi \in H_{Q_c}^{\exp, s}$. This last point is the
reason why we write the quadratic form as (<ref>), which is equal,
up to some integration by parts, to the more natural definition

\[ \int_{\mathbb{R}^2} | \nabla \varphi |^2 - (1 - | Q_c |^2) | \varphi |^2 +
2\mathfrak{Re}^2 (\overline{Q_c} \varphi)
-\mathfrak{Re} (i c \partial_{x_2} \varphi \bar{\varphi}), \]

but this integral is not well defined for $\varphi \in H_{Q_c}^{\exp, s}$. See
<cit.> for more details on this point. We now quote a coercivity
result from <cit.>.

There exists $R, K, c_0 > 0$ such that, for $0 < c \leqslant
c_0$, $Q_c$ defined in Theorem <ref>, if a function $\varphi \in
H_{Q_c}^{\exp, s}$ satisfies the three orthogonality conditions:

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_c Q_c \bar{\varphi}
=\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi} = 0, \]

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} i Q_c \bar{\varphi} = 0, \]

\[ \frac{1}{K} \| \varphi \|^2_{H_{Q_c}^{\exp}} \geqslant B_{Q_c}^{\exp}
(\varphi) \geqslant K \| \varphi \|_{H_{Q_c}^{\exp}}^2 . \]

We will use a slight variation of this result, given in the next corollary.

There exists $R, K, c_0 > 0$ such that, for $0 < c \leqslant
c_0$, $Q_c$ defined in Theorem <ref>, if a function $\varphi \in
H_{Q_c}^{\exp, s}$ satisfies the three orthogonality conditions:

\[ \mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} \partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. + d
\vec{e}_1))_{| d = d_c \nobracket} \bar{\varphi}
=\mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi} = 0, \]

\[ \mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} i Q_c \bar{\varphi} = 0, \]

\[ \frac{1}{K} \| \varphi \|^2_{H_{Q_c}^{\exp}} \geqslant B_{Q_c}^{\exp}
(\varphi) \geqslant K \| \varphi \|_{H_{Q_c}^{\exp}}^2 . \]

Remark, with Theorem <ref> (for $p = + \infty$), that $\frac{1}{c^2}
\partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. + d \vec{e}_1))_{| d = d_c
\nobracket}$ is the first order of $\partial_c Q_c$ when $c \rightarrow 0$ in
$L^{\infty} (\mathbb{R}^2, \mathbb{C})$, and that (with Lemma <ref>)
they both have the same symmetries. We need to change the quantity
$\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_c Q_c \bar{\varphi}$ in the orthogonality
conditions because we will differentiate it with respect to $c$, and

\[ c \mapsto \partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. + d \vec{e}_1))_{| d
= d_c \nobracket} = - \partial_{x_1} V_1 (. - d_c \vec{e}_1) V_{- 1} (. +
d_c \vec{e}_1) + \partial_{x_1} V_{- 1} (. + d_c \vec{e}_1) V_1 (. - d_c
\vec{e}_1) \]

is a $C^1$ function ($c \mapsto d_c \in C^1 (] 0, c_0 [, \mathbb{R})$ for $c_0
> 0$ a small constant, see subsection 4.6 of <cit.>), but it is not
clear that $c \mapsto \partial_c Q_c$ can be differentiated with respect to
$c$. Precise estimates on $\partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. + d
\vec{e}_1))_{| d = d_c \nobracket}$ can be found in Lemma 2.6 of <cit.>.
Furthermore, we changed, in the area of the integrals, $\tilde{d}_c$ by $d_c$
(they are close when $c \rightarrow 0$, see (<ref>)).

Step 1: changing the integrand but not the integration domain.

Take a function $\varphi \in H_{Q_c}^{\exp, s}$ satisfying

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. +
d \vec{e}_1))_{| d = d_c \nobracket} \bar{\varphi}
=\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi} = 0, \]

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} i Q_c \bar{\varphi} = 0. \]

Let us show that it satisfies $\frac{1}{K} \| \varphi \|^2_{H_{Q_c}^{\exp}}
\geqslant B_{Q_c}^{\exp} (\varphi) \geqslant K \| \varphi
\|_{H_{Q_c}^{\exp}}$. For $\mu \in \mathbb{R}$, we define

\[ \varphi^{\ast} = \varphi + c^2 \mu \partial_c Q_c . \]

We have that $\partial_c Q_c \in H_{Q_c}^{\exp, s}$. We want to choose $\mu
\in \mathbb{R}$ such that $\varphi^{\ast}$ satisfied the hypothesis of Theorem <ref>.
By the symmetries of subsection <ref> and the hypotheses on
$\varphi$, we have that

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} i Q_c \overline{\varphi^{\ast}}
=\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_{x_2} Q_c \overline{\varphi^{\ast}} =
0, \]

and we compute, using $\mathfrak{Re} \int_{B (\tilde{d}_c
\vec{e}_1, R) \cup B (- \tilde{d}_c \vec{e}_1, R)} \partial_d (V_1 (. - d
\vec{e}_1) V_{- 1} (. + d \vec{e}_1))_{| d = d_c \nobracket} \bar{\varphi} =
0$, that

\begin{eqnarray*}
&  & \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B
(- \tilde{d}_c \vec{e}_1, R)} c^2 \partial_c Q_c
\overline{\varphi^{\ast}}\\
& = & \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B
(- \tilde{d}_c \vec{e}_1, R)} c^2 \partial_c Q_c \bar{\varphi} + \mu
\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} c^4 | \partial_c Q_c |^2\\
& = & \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B
(- \tilde{d}_c \vec{e}_1, R)} (c^2 \partial_c Q_c - \partial_d (V_1 (. - d
\vec{e}_1) V_{- 1} (. + d \vec{e}_1))_{| d = d_c \nobracket})
\bar{\varphi}\\
& + & \mu \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R)
\cup B (- \tilde{d}_c \vec{e}_1, R)} c^4 | \partial_c Q_c |^2 .
\end{eqnarray*}

By Theorem <ref> (for $p = + \infty$) and Lemma 2.6 of <cit.>, we

\[ \| c^2 \partial_c Q_c - \partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. + d
\vec{e}_1))_{| d = d_c \nobracket} \|_{L^{\infty} (\mathbb{R}^2)} = o_{c
\rightarrow 0} (1), \]

and also that there exists a universal constant $K > 0$ (we recall that $R >
0$ is a universal constant) such that

\[ \frac{1}{K} \leqslant \mathfrak{Re} \int_{B (\tilde{d}_c
\vec{e}_1, R) \cup B (- \tilde{d}_c \vec{e}_1, R)} c^4 | \partial_c Q_c
|^2 \leqslant K. \]

Now, taking

\[ \mu = \frac{-\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R)
\cup B (- \tilde{d}_c \vec{e}_1, R)} (c^2 \partial_c Q_c - \partial_d
(V_1 (. - d \vec{e}_1) V_{- 1} (. + d \vec{e}_1))_{| d = d_c \nobracket})
\bar{\varphi}}{\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1,
R) \cup B (- \tilde{d}_c \vec{e}_1, R)} c^4 | \partial_c Q_c |^2}, \]

we have

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} c^2 \partial_c Q_c \overline{\varphi^{\ast}} =
0, \]

\[ | \mu | \leqslant o_{c \rightarrow 0} (1) \| \varphi \|_{L^2 (B
(\tilde{d}_c \vec{e}_1, R) \cup B (- \tilde{d}_c \vec{e}_1, R))}
\leqslant o_{c \rightarrow 0} (1) \| \varphi \|_{H_{Q_c}^{\exp}} . \]

Since $\partial_c Q_c \in H_{Q_c}^{\exp, s}$ by Lemma 2.8 of <cit.>,
we deduce that $\varphi^{\ast}$ satisfies all the hypotheses of Theorem
<ref>, therefore

\[ \frac{1}{K} \| \varphi^{\ast} \|^2_{H_{Q_c}^{\exp}} \geqslant
B_{Q_c}^{\exp} (\varphi^{\ast}) \geqslant K \| \varphi^{\ast}
\|^2_{H_{Q_c}^{\exp}} . \]

Now, from Lemma 6.3 of <cit.>, we have $\frac{1}{K} \leqslant \| c^2
\partial_c Q_c \|_{H^{\exp}_{Q_c}} \leqslant K$ for a universal constant $K
> 0$. With $| \mu | \leqslant o_{c \rightarrow 0} (1) \| \varphi
\|_{H_{Q_c}^{\exp}}$, we deduce that, taking $c > 0$ small enough,

\[ \frac{1}{K} \| \varphi \|^2_{H_{Q_c}^{\exp}} \geqslant B_{Q_c}^{\exp}
(\varphi^{\ast}) \geqslant K \| \varphi \|^2_{H_{Q_c}^{\exp}} \]

for some universal constant $K > 0$. Now, we decompose (using Lemmas 6.2 and
6.3 of <cit.>)

\begin{eqnarray*}
B_{Q_c}^{\exp} (\varphi^{\ast}) & = & B_{Q_c}^{\exp} (\varphi + c^2 \mu
\partial_c Q_c)\\
& = & B_{Q_c}^{\exp} (\varphi) + 2 c^2 \mu \langle L_{Q_c} (\partial_c
Q_c), \varphi \rangle + c^4 \mu^2 B_{Q_c}^{\exp} (\partial_c Q_c),
\end{eqnarray*}

and by Lemmas 2.8, 5.4 and 6.1 of <cit.>,

\[ | \langle L_{Q_c} (\partial_c Q_c), \varphi \rangle | = | \langle i
\partial_{x_2} Q_c, \varphi \rangle | \leqslant K \ln \left( \frac{1}{c}
\right) \| \varphi \|_{H_{Q_c}^{\exp}}, \]

\[ | 2 c^2 \mu \langle L_{Q_c} (\partial_c Q_c), \varphi \rangle | \leqslant
K c^2 \ln \left( \frac{1}{c} \right) | \mu | \| \varphi
\|_{H_{Q_c}^{\exp}} \leqslant o_{c \rightarrow 0} (1) \| \varphi
\|_{H_{Q_c}^{\exp}}^2 . \]

By Proposition 1.2 of <cit.>, $B_{Q_c}^{\exp} (\partial_c Q_c) =
\frac{2 \pi + o_{c \rightarrow 0} (1)}{c^2}$, thus

\[ | c^4 \mu^2 B_{Q_c}^{\exp} (\partial_c Q_c) | \leqslant o_{c \rightarrow
0} (1) \| \varphi \|_{H_{Q_c}^{\exp}}^2, \]

which concludes the proof of $\frac{1}{K} \| \varphi \|^2_{H_{Q_c}^{\exp}}
\geqslant B_{Q_c}^{\exp} (\varphi) \geqslant K \| \varphi
\|_{H_{Q_c}^{\exp}}^2 $ by taking $c > 0$ small enough.

Step 2: changing the integration domain.

To change the conditions

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. +
d \vec{e}_1))_{| d = d_c \nobracket} \bar{\varphi}
=\mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi} = 0, \]

\[ \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B (-
\tilde{d}_c \vec{e}_1, R)} i Q_c \bar{\varphi} = 0 \]

\[ \mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} \partial_d (V_1 (. - d \vec{e}_1) V_{- 1} (. + d
\vec{e}_1))_{| d = d_c \nobracket} \bar{\varphi}
=\mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi} = 0, \]

\[ \mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} i Q_c \bar{\varphi} = 0, \]

we use similar arguments, using $| d_c - \tilde{d}_c | = o_{c \rightarrow 0} (1)$ by (<ref>). We
check for instance that

\[ \left| \mathfrak{Re} \int_{B (\tilde{d}_c \vec{e}_1, R) \cup B
(- \tilde{d}_c \vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi}
-\mathfrak{Re} \int_{B (d_c \vec{e}_1, R) \cup B (- d_c
\vec{e}_1, R)} \partial_{x_2} Q_c \bar{\varphi} \right| \leqslant K (R) |
d_c - \tilde{d}_c |  \| \varphi \|_{H_{Q_c}^{\exp}}, \]

and $| d_c - \tilde{d}_c | = o_{c \rightarrow 0} (1)$.

Notice that the integration domain remains symmetric with respect to the $x_2$-axis.

§.§ Proof of Proposition <ref>

In this subsection, we take $\nu \in ] 0, 1 [$ a small but universal constant,
that will be fixed at the end of the proof. We fix $\lambda > 3 R
+ 1$ in the statement of Proposition <ref> (where $R > 0$ is defined in
Corollary <ref>) large such that

\[ \frac{1}{\lambda^{1 / 2}} \leqslant \nu, \]

and then $\varepsilon > 0$ in the statement of Proposition <ref> will be
taken small enough (the smallness will depend only on $\lambda, \nu$). In
particular, we first impose $\varepsilon > 0$ small enough such that the
distance between the vortices (that is $2 d > \frac{2}{\varepsilon}$) is large
compared to $\lambda$ (and thus $R$), for instance, $\frac{2}{\varepsilon} >
100 + 10 \lambda^2$. We take $u$ a function satisfying the hypotheses of
Proposition <ref> for these values of $\varepsilon$ and $\lambda$. In
the rest of the subsection, $K, K' > 0$ denote universal constants,
independent of any parameters of the problem (in particular, $\lambda,
\varepsilon$ and $\nu$).

§.§.§ Modulation on the parameters of the branch

From Theorem 1.1 and the end of section 4.6 of <cit.>, we have that $Q_c
= V_1 (. - d_c \vec{e}_1) V_{- 1} (. + d_c \vec{e}_1) + \Gamma_c$, with $d_c =
\frac{1 + o_{c \rightarrow 0} (1)}{c}$ and

\[ c \mapsto d_c \in C^1 (] 0, c_0 [, \mathbb{R}) . \]

In particular, this implies that there exists $d_0 > 0$ a large constant such
that $\{ d_c, c \in] 0, c_0 [\} \subset] d_0, + \infty [$. By the intermediate
value theorem, given $d > d_0$, there exists $c' > 0$ such that $d_{c'} = d$.
For a function $u$ satisfying the hypotheses of Proposition <ref>,
taking $\varepsilon > 0$ small enough such that $d_0 < \frac{1}{\varepsilon}$,
we define $c' > 0$ a value such that $d = d_{c'}$ (the choice of $c'$ might
not be unique). In particular,

\begin{eqnarray*}
u (x) - Q_{c'} (x) & = & V_1 (x - d \vec{e}_1) V_{- 1} (x + d \vec{e}_1) +
\Gamma (x) - V_1 (x - d_{c'} \vec{e}_1) V_{- 1} (x + d_{c'} \vec{e}_1) -
\Gamma_{c'} (x)\\
& = & \Gamma (x) - \Gamma_{c'} (x) .
\end{eqnarray*}

From the hypotheses on $\Gamma$, and the fact that $\| \Gamma_{c'}
\|_{L^{\infty} (\mathbb{R}^2)} \leqslant \nu$ (since $c' \leqslant \frac{2}{d}
\leqslant 2 \varepsilon$ and taking $\varepsilon > 0$ small enough) by Theorem
<ref> (for $p = + \infty$), we deduce that (we denote $\tilde{r} =
\tilde{r}_d = \tilde{r}_{d_{c'}}$ to simplify the notations)

\[ \| u - Q_{c'} \|_{L^{\infty} (\{ \tilde{r} \leqslant 2 \lambda \})}
\leqslant K \nu . \]

Since $\frac{1 + o_{c' \rightarrow 0} (1)}{c'} = d_{c'} = d$ by Theorem
<ref>, and $| d c - 1 | \leqslant \varepsilon$, we have

\begin{equation}
d | c - c' | \leqslant K \nu \label{CP3ccp}
\end{equation}

by taking $\varepsilon > 0$ small enough. We deduce, from the elliptic
equation satisfied by $u - Q_{c'}$, that is

\[ \Delta (u - Q_{c'}) = - i c \partial_{x_2} (u - Q_{c'}) - (u - Q_{c'}) (1 -
| u |^2) + (| u |^2 - | Q_{c'} |^2) Q_{c'}, \]

that for any $x \in \{ \tilde{r} \leqslant \lambda \}$, ($c \leqslant K \nu$
by (<ref>) and $d_{c'} > \frac{1}{\varepsilon}$, taking $\varepsilon >
0$ small enough, and $1 - | u |^2$ is bounded uniformly by Theorem <ref>)

\[ \| u - Q_{c'} \|_{C^1 (B (x, 1))} \leqslant K \| u - Q_{c'} \|_{L^{\infty}
(\{ \tilde{r} \leqslant 2 \lambda \})} \leqslant K \nu, \]

therefore, for a universal constant $K > 0$,

\begin{equation}
\| u - Q_{c'} \|_{C^1 (\{ \tilde{r} \leqslant \lambda \})} \leqslant K \nu
\label{22CP3} .
\end{equation}

That is, $u$ is close to $Q_{c'}$ near the vortices (in the region $\{
\tilde{r} \leqslant \lambda \}$). Outside of this domain, they are close only
in modulus. Indeed, by equation (2.6) of <cit.> (for $\sigma = 1 / 2$)
and the hypotheses on $u$, on $\{ \tilde{r} \geqslant \lambda \}$,

\[ | | u | - | Q_{c'} | | \leqslant | | u | - 1 | + | | Q_{c'} | - 1 |
\leqslant \varepsilon + \frac{K}{\lambda^{3 / 2}} \leqslant K \nu . \]

Now, we modulate on the parameters of the family of travelling waves to get
the orthogonality conditions of Corollary <ref>. For $c'' > 0$ close
enough to $c'$ and $X, \gamma \in \mathbb{R}$, we define

\begin{equation}
Q \assign Q_{c''} (. - X \vec{e}_2) \ex^{i \gamma} . \label{Qdef}
\end{equation}

For $u$ satisfying the hypotheses of Proposition <ref>
and for values of $\varepsilon, \lambda, \nu$ described above, there exists
$c'' > 0$, $X, \gamma \in \mathbb{R}, K > 0$ such that, for $R > 0$ defined
in Corollary <ref>, and $\vec{d}_{\pm} \assign \pm d_{c''}
\vec{e}_1 + X \vec{e}_2$;

\begin{eqnarray*}
&  & \mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-,
R)} \partial_d (V_1 (. - d \vec{e}_1 - X \vec{e}_2) V_{- 1} (. + d
\vec{e}_1 - X \vec{e}_2) \ex^{i \gamma})_{| d = d_{c''} \nobracket}
\overline{(u - Q)}\\
& = & \mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-,
R)} \partial_{x_2} Q \overline{(u - Q)}\\
& = & \mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-,
R)} i Q \overline{(u - Q)}\\
& = & 0.
\end{eqnarray*}

\[ \frac{| c'' - c' |}{c^{\prime 2}} + | X | + | \gamma | \leqslant K \nu .
\]

We recall here that $\lambda > 2 R + 1$, and that $R > 0$ is a universal constant.

To simplify the notations, in this proof, we define

\[ \partial_d V \assign \partial_d (V_1 (. - d \vec{e}_1 - X \vec{e}_2) V_{-
1} (. + d \vec{e}_1 + X \vec{e}_2) \ex^{i \gamma})_{| d = d_{c''} \nobracket}
. \]

We will keep the notation $\tilde{r}$ for the minimum of the distance to the
zeros of $Q$.

First, from equation (7.5) of <cit.>, there exists a universal
constant $K > 0$ such that, for $c'' < c_0$, $c' / 2 \leqslant c'' \leqslant
2 c'$,

\begin{equation}
\| Q - Q_{c'} \|_{L^{\infty} (\mathbb{R}^2)} \leqslant K \left( | X | +
\frac{| c'' - c' |}{c^{\prime 2}} + | \gamma | \right) . \label{CP3eq21}
\end{equation}

Now, we follow closely the proof of Lemma 7.6 of <cit.>, which is done
in Appendix C.3 there. We define

\[ G \left(\begin{array}{c}
\gamma
\end{array}\right) \assign \left(\begin{array}{c}
\mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
\partial_{x_2} Q \overline{(u - Q)}\\
\mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
\partial_d V \overline{(u - Q)}\\
\mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
i Q \overline{(u - Q)}
\end{array}\right) . \]

Remark that $Q, \partial_d V$ and $\vec{d}_{\pm}$ all depend on $X$ and
$c''$, and $Q$ depends also on $\gamma$. From equation (<ref>) and
$\lambda \geqslant 2 R$, we have $\| u - Q_{c'} \|_{L^{\infty} (\{ \tilde{r}
\leqslant R \})} \leqslant K \nu$, and from Theorem <ref> with $p = +
\infty$ as well as Lemma 2.6 of <cit.>,

\begin{equation}
\| \partial_{x_2} Q_{c'} \|_{L^{\infty} (\mathbb{R}^2)} + \| \partial_d V
\|_{L^{\infty} (\mathbb{R}^2)} + \| i Q_{c'} \|_{L^{\infty}
(\mathbb{R}^2)} \leqslant K \label{CP3A}
\end{equation}

for some universal constant $K > 0$. Therefore, since $Q = Q_{c'}$ for $X =
\gamma = 0$, $c'' = c'$, and $\| u - Q_{c'} \|_{L^{\infty} (\{ \tilde{r}
\leqslant \lambda \})} \leqslant K \nu$ with $\lambda > R$,

\[ \left| G \left(\begin{array}{c}
\end{array}\right) \right| \leqslant K \nu . \]

We want to show that $G$ is invertible in a vicinity of
\end{array}\right)$. With equations (<ref>) and (<ref>), we
check that (we recall that $\tilde{r} = \min (| x - \vec{d}_+ |, | x -
\vec{d}_- |)$)

\begin{eqnarray*}
\| u - Q \|_{L^{\infty} (\{ \tilde{r} \leqslant 2 R \})} & \leqslant & \|
u - Q_{c'} \|_{L^{\infty} (\{ \tilde{r} \leqslant 2 R \})} + \| Q - Q_{c'}
\|_{L^{\infty} (\mathbb{R}^2)}\\
& \leqslant & K \nu + K \left( | X | + \frac{| c'' - c' |}{c^{\prime 2}}
+ | \gamma | \right),
\end{eqnarray*}

and as in Lemma 7.1 of <cit.>, this implies

\begin{equation}
\| u - Q \|_{C^1 (\{ \tilde{r} \leqslant R \})} \leqslant K \nu + K \left(
| X | + \frac{| c'' - c' |}{c^{\prime 2}} + | \gamma | \right) .
\label{CP3eq22}
\end{equation}

Now, we compute

\begin{eqnarray*}
&  & \left| \partial_X \left( \mathfrak{Re} \int_{B
(\vec{d}_+, R) \cup B (\vec{d}_-, R)} \partial_{x_2} Q \overline{(u - Q)}
\right) - \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} | \partial_{x_2} Q
|^2 \right|\\
& \leqslant & \int_{\partial B (\vec{d}_+, R) \cup \partial B (\vec{d}_-,
R)} | \partial_{x_2} Q \overline{(u - Q)} | + \int_{B (\vec{d}_+, R) \cup
B (\vec{d}_-, R)} | \partial_{x_2}^2 Q \overline{(u - Q)} |,
\end{eqnarray*}

therefore, with (<ref>) and (<ref>), we check that

\begin{eqnarray*}
&  & \int_{\partial B (\vec{d}_+, R) \cup \partial B (\vec{d}_-, R)} |
\partial_{x_2} Q \overline{(u - Q)} | + \int_{B (\vec{d}_+, R) \cup B
(\vec{d}_-, R)} | \partial_{x_2}^2 Q \overline{(u - Q)} |\\
& \leqslant & K \nu + K \left( | X | + \frac{| c'' - c' |}{c^{\prime 2}}
+ | \gamma | \right),
\end{eqnarray*}

\begin{eqnarray*}
&  & \left| \partial_X \left( \mathfrak{Re} \int_{B
(\vec{d}_+, R) \cup B (\vec{d}_-, R)} \partial_{x_2} Q \overline{(u - Q)}
\right) - \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} | \partial_{x_2} Q
|^2 \right|\\
& \leqslant & K \nu + K \left( | X | + \frac{| c'' - c' |}{c^{\prime 2}}
+ | \gamma | \right) .
\end{eqnarray*}

With similar computations, using Lemma 2.6 of <cit.>, equations
(<ref>) and (<ref>), we infer that

\[ \left| \partial_X G - \left(\begin{array}{c}
\int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} | \partial_{x_2} Q |^2\\
\mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
\partial_d V \overline{\partial_{x_2} Q}\\
\mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
i Q \overline{\partial_{x_2} Q}
\end{array}\right) \right| \leqslant K \nu + K \left( | X | + \frac{| c''
- c' |}{c^{\prime 2}} + | \gamma | \right) . \]

By the symmetries of $Q (. + X \vec{e}_2) \ex^{- i \gamma}$ and $\partial_d V
(. + X \vec{e}_2) \ex^{- i \gamma}$, we have that

\[ \mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
\partial_d V \overline{\partial_{x_2} Q} = 0, \]

and from Theorem <ref> (with $p = + \infty$), with the symmetries of
$Q_c$ and $V_1$ (see subsections <ref> and <ref>), we have

\[ \left| \mathfrak{Re} \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-,
R)} i Q \overline{\partial_{x_2} Q} - 2\mathfrak{Re} \int_{B
(0, R)} i V_1 \overline{\partial_{x_2} V_1} \right| \leqslant K \nu . \]

By decomposition in harmonics and Lemma <ref>, we check easily
that $\mathfrak{Re} \int_{B (0, R)} i V_1
\overline{\partial_{x_2} V_1} = 0$, thus

\[ \left| \partial_X G - \left(\begin{array}{c}
- \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} | \partial_{x_2} Q
\end{array}\right) \right| \leqslant K \nu + K \left( | X | + \frac{| c''
- c' |}{c^{\prime 2}} + | \gamma | \right) . \]

Similarly, we check that (using $\partial_c (d_c) = \frac{- 1 + o_{c
\rightarrow 0} (1)}{c^2}$ from section 4.6 of <cit.>, and Lemma 2.6 of

\[ \left| c^{\prime 2} \partial_{c''} G - \left(\begin{array}{c}
\int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)}
| \partial_d V |^2\\
\end{array}\right) \right| \leqslant K \nu + K \left( | X | + \frac{| c''
- c' |}{c^{\prime 2}} + | \gamma | \right) \]

(we use here the fact that $c \mapsto \partial_d V$ and $c \mapsto
\vec{d}_{\pm}$ are differentiable) and

\[ \left| \partial_{\gamma} G - \left(\begin{array}{c}
- \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} | Q |^2
\end{array}\right) \right| \leqslant K \nu + K \left( | X | + \frac{| c''
- c' |}{c^{\prime 2}} + | \gamma | \right) . \]

From (<ref>) and Theorem <ref> (for $p = + \infty$) as well as
Lemma 2.6 of <cit.>, there exists a universal constant $K > 0$ such

\[ \frac{1}{K} \leqslant \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} |
\partial_{x_2} Q |^2 \leqslant K, \]

\[ \frac{1}{K} \leqslant \int_{B (\vec{d}_+, R)
\cup B (\vec{d}_-, R)} | \partial_d V |^2 \leqslant K \]

\[ \frac{1}{K} \leqslant \int_{B (\vec{d}_+, R) \cup B (\vec{d}_-, R)} | Q
|^2 \leqslant K. \]

We deduce that $d G$ is invertible in a vicinity of $(0, c', 0)$ of size
independent of $\varepsilon, \nu$ or $\lambda$, hence, there exists $X, c'',
\gamma \in \mathbb{R}$ such that

\[ G \left(\begin{array}{c}
\gamma
\end{array}\right) = 0, \]

\[ \frac{| c'' - c' |}{c^{\prime 2}} + | X | + | \gamma | \leqslant K \nu .
\]

§.§.§ Construction and properties of the perturbation term

We define $\eta$ a smooth cutoff function with $\eta (x) = 0$ for $x \in B
(\vec{d}_{\pm}, 2 R)$ and $\eta (x) = 1$ for $x \in \mathbb{R}^2 \backslash B
(\pm \vec{d}_{\pm}, 2 R + 1)$ even in $x_1$. We infer the following result, where
the space $H_Q^{\exp,s}$ is simply defined by

\[ H_Q^{\exp, s} \assign \left\{ \varphi \in H^1_{\tmop{loc}} (\mathbb{R}^2,
\mathbb{C}), \| \varphi \|_{H_{Q_{}}^{\exp}} < + \infty, \forall (x_1, x_2)
\in \mathbb{R}^2, \varphi (- x_1, x_2) = \varphi (x_1, x_2) \right\}, \]

with, for $\tilde{r}$ the minimum of the distances to the zeros of $Q$,
$\varphi = Q \psi$,

\[ \| \varphi \|^2_{H_Q^{\exp}} \assign \| \varphi \|^2_{H^1 (\{ \tilde{r}
\leqslant 10 \})} + \int_{\{ \tilde{r} \geqslant 5 \}} | \nabla \psi |^2
+\mathfrak{Re}^2 (\psi) + \frac{| \psi |^2}{\tilde{r}^2 \ln^2 \tilde{r} }, \]

and $B_Q^{\exp}$ has the same definition than $B_{Q_c}^{\exp}$, replacing
$\tilde{\eta}$ by $\eta$ and $Q_c$ by $Q$.

Under the hypotheses of Proposition <ref>, there exists
$K, K' > 0$ and a function $\varphi = Q \psi \in H_Q^{\exp, s} \cap C^1
(\mathbb{R}^2, \mathbb{C})$ such that, for $Q$ defined in (<ref>) with
the values of $c'', X, \gamma \in \mathbb{R}$ from Lemma <ref>,

\[ u - Q = (1 - \eta) \varphi + \eta Q (\ex^{\psi} - 1) . \]

\[ B_Q^{\exp} (\varphi) \geqslant K \| \varphi \|_{H_Q^{\exp}}^2 \]

\[ \| \varphi \|_{C^1 (\{ \tilde{r} \leqslant \lambda \})} + \|
\mathfrak{Re} (\psi) \|_{L^{\infty} (\{ \tilde{r} \geqslant
\lambda \})} \leqslant K' \nu . \]

The goal of this lemma is to decompose the error $u - Q$ in a particular form.
In the area $\{ \eta = 1 \}$, that is far from the zeros of $Q$, the error is
written in an exponential form: $u = Q \ex^{\psi}$. This form was already used
in <cit.>, <cit.>, and is useful to have a particular form on the
cubic error terms. Furthermore, we fix the parameters of $Q$ such that
$\varphi$ satisfies the orthogonality conditions of Corollary <ref>,
yielding the coercivity.

Remark that we have no smallness on $\mathfrak{I}\mathfrak{m} (\psi)$
in $\{ \tilde{r} \geqslant \lambda \}$, where $\varphi = Q \psi$. We will
simply be able to show that it is bounded (see equation (<ref>)),
with no a priori bound on it. This lack of smallness is one of the main
difficulties in the proof of Proposition <ref>.

This proof follows some ideas of the proofs of Lemmas 7.2 and 7.3 of
<cit.>. First, in the area $\{ \tilde{r} \leqslant \lambda \}$, the
proof is identical as for Lemma 7.2 of <cit.> for the existence of
$\varphi = Q \psi \in C^1 (\{ \tilde{r} \leqslant \lambda \}, \mathbb{C})$
such that $u - Q = (1 - \eta) \varphi + \eta Q (\ex^{\psi} - 1)$ in $\{
\tilde{r} \leqslant \lambda \}$, with $\| \varphi \|_{C^1 (\{ \tilde{r}
\leqslant \lambda \})} \leqslant K \nu$. The main idea is that $u - Q$ is
small there (in $C^1 (\{ \tilde{r} \leqslant \lambda \}, \mathbb{C})$), and
the equation on $\varphi$ is a perturbation of the identity for functions
$\varphi$ that are small in $C^1 (\{ \tilde{r} \leqslant \lambda \},
\mathbb{C})$. We therefore focus our attention in the area $\{ \tilde{r}
\geqslant \lambda \}$.

We have, by the hypotheses on $u$, that in $\{ \tilde{r} \geqslant \lambda
\}$, taking $\varepsilon > 0$ small enough,

\[ | | u | - 1 | \leqslant \nu, \]

and, still in $\{ \tilde{r} \geqslant \lambda \}$, by equation (2.6) of
<cit.>, $| | Q | - 1 | \leqslant \frac{K}{\lambda} \leqslant K \nu$,
therefore, for some universal constant $C_1 > 0$,

\begin{equation}
\left| \left| \frac{u}{Q} \right| - 1 \right| \leqslant C_1 \nu .
\label{CP3dern}
\end{equation}

In $\{ \eta = 1 \}$, the equation becomes $\ex^{\psi} = \frac{u}{Q}$. From
(<ref>), the function $\eta \frac{u}{Q} + (1 - \eta)$ is close to
$1$ in modulus in $\mathbb{R}^2$ (taking $\nu$ such that $C_1 \nu < 1 / 4$
for instance), and thus, looking at the function

\[ x \mapsto \frac{\eta \frac{u}{Q} + (1 - \eta)}{\left| \eta \frac{u}{Q} +
(1 - \eta) \right|} \in C^1 (\mathbb{R}^2, \mathcal{S}^1), \]

where $\mathcal{S}^1$ is the unit circle, by exponential lifting properties,
we infer that there exists a function $\tilde{\psi} \in C^1 (\mathbb{R}^2,
\mathbb{C})$ such that

\[ \ex^{\tilde{\psi}} = \eta \frac{u}{Q} + (1 - \eta) . \]

Two functions satisfying this equality differ by $2 i k \pi$ for some $k \in
\mathbb{Z}$.

In particular, in $\{ \eta = 1 \} \bigcap \{ \tilde{r} \leqslant \lambda \}
\neq \emptyset$ (since $\lambda > 3 R + 1$), we may choose $\tilde{\psi}$
such that it coincides with the function $\psi$ already constructed (using
the symmetry on $x_1$, it coincides with the construction near both
vortices). We extend the function $\psi$ by $\psi = \tilde{\psi}$ on $\{
\tilde{r} \geqslant \lambda \}$ with this choice. Since $E (u) < + \infty$,
$u$ converges to a constant at infinity (uniformly in all directions) by
<cit.>. Therefore, $\frac{u}{Q}$ converges to a constant at
infinity, and the function $\psi$ converges to a constant, and thus it is
bounded near infinity, that is

\begin{equation}
\| \psi \|_{L^{\infty} (\{ \tilde{r} \geqslant \lambda \})} < + \infty .
\label{CP3dumdum}
\end{equation}

In $\{ \tilde{r} \geqslant \lambda \}$, we have $\eta = 1$ (since $\lambda >
3 R + 1$), thus $\ex^{\psi} = \frac{u}{Q}$ and

\[ \ex^{\mathfrak{Re} (\psi)} = \left| \frac{u}{Q} \right| . \]

From (<ref>), we deduce that

\[ \| \mathfrak{Re} (\psi) \|_{L^{\infty} (\{ \tilde{r} \geqslant
\lambda \})} \leqslant K \nu . \]

Now, we want to show that $\varphi \in H_Q^{\exp, s}$. First, by the
symmetries of $\eta, u$ and $Q$, $\varphi$ satisfies

\[ \forall (x_1, x_2) \in \mathbb{R}^2, \varphi (- x_1, x_2) = \varphi (x_1,
x_2) . \]

Furthermore, to check that $\| \varphi \|_{H_Q^{\exp}} < + \infty$, since
$\varphi \in C^1 (\mathbb{R}^2, \mathbb{C})$, we only have to check the
integrability in $\{ \tilde{r} \geqslant \lambda \}$, where $\ex^{\psi} =
\frac{u}{Q}$. We check that there, with (<ref>),

\[ \int_{\{ \tilde{r} \geqslant \lambda \}} \frac{| \psi |^2}{\tilde{r}^2
\ln^2 (\tilde{r})} < + \infty . \]

Now, using Theorem 11 of <cit.> (we recall that $E (u) < +
\infty, E (Q) < + \infty$),

\[ | \ex^{\mathfrak{Re} (\psi)} - 1 | = \frac{| | u | - | Q | |}{|
Q |} \leqslant 2 (| | u | - 1 | + | | Q | - 1 |) \leqslant \frac{K (u,
Q)}{(1 + r)^2}, \]

where $K (u, Q) > 0$ is a constant depending on $u$ and $Q$, hence $|
\mathfrak{Re} (\psi) | \leqslant \frac{K (u, Q)}{(1 + r)^2}$ and

\[ \int_{\{ \tilde{r} \geqslant \lambda \}} \mathfrak{Re}^2
(\psi) \leqslant \int_{\{ \tilde{r} \geqslant \lambda \}} \frac{K (u,
Q)}{(1 + r)^4} < + \infty . \]

We finally compute that

\[ \nabla \psi = \frac{\nabla u - \nabla Q \ex^{\psi}}{Q \ex^{\psi}} =
\frac{\nabla u}{u} - \frac{\nabla Q}{Q}, \]

and with Theorem 11 of <cit.>, in $\{ \tilde{r} \geqslant
\lambda \}$, we deduce that

\[ (1 + r)^2 | \nabla \psi | \leqslant (1 + r)^2 \left| \frac{\nabla u}{u}
\right| + (1 + r)^2 \left| \frac{\nabla Q}{Q} \right| \leqslant K (u, Q),
\]

\[ \int_{\{ \tilde{r} \geqslant \lambda \}} | \nabla \psi |^2 < + \infty .
\]

This concludes the proof that $\varphi = Q \psi \in H_Q^{\exp, s}$. The fact
that $B_Q^{\exp} (\varphi) \geqslant K \| \varphi \|_{H_Q^{\exp}}^2$ is a
consequence of Corollary <ref> and Lemma <ref>, using in
particular that $B_Q^{\exp} (\varphi) = B_{Q_{c''}}^{\exp} (\varphi (. + X
\vec{e}_2) \ex^{- i \gamma})$ and $\| \varphi \|_{H_Q^{\exp}} = \| \varphi (.
+ X \vec{e}_2) \ex^{- i \gamma} \|_{H_{Q_{c''}}^{\exp}}$.

Remark that we have shown $\varphi \in H_Q^{\exp, s}$, but we have no good
control on $\| \varphi \|_{H_Q^{\exp}}$. This quantity might be a priori very
large at this point.

We now compute the equation satisfied by $\varphi$.

The function $\varphi = Q \psi$ defined in Lemma <ref>
satisfies the equation

\[ L_Q (\varphi) - i (c - c'') \vec{e}_2 .H (\psi) + \tmop{NL}_{\tmop{loc}}
(\psi) + F (\psi) = 0, \]

with $L_Q$ the linearized operator around $Q$: $L_Q (\varphi) = - \Delta
\varphi - i c'' \partial_{x_2} \varphi - (1 - | Q |^2) \varphi +
2\mathfrak{Re} (\bar{Q} \varphi) Q$,

\[ S (\psi) \assign \ex^{2\mathfrak{Re} (\psi)} - 1 -
2\mathfrak{Re} (\psi), \]

\[ F (\psi) \assign Q \eta (- \nabla \psi . \nabla \psi + | Q |^2 S (\psi)),
\]

\[ H (\psi) \assign \nabla Q + \frac{\nabla (Q \psi) (1 - \eta) + Q \nabla \psi
\eta \ex^{\psi}}{(1 - \eta) + \eta \ex^{\psi}} \]

and $\tmop{NL}_{\tmop{loc}} (\psi)$ is a sum of terms at least quadratic in
$\psi$, localized in the area where $\eta \neq 1$. Furthermore,

\[ | \langle \tmop{NL}_{\tmop{loc}} (\psi), Q (\psi + i \gamma) \rangle |
\leqslant K \nu \| \varphi \|_{H^1 (\{ \eta \neq 1 \})}^2 . \]

Notice that $F (\psi)$ (the notation $ X.Y $ for complex vector fields stands for $ X_1 Y_1 + X_2 Y_2 $)
contains all the nonlinear terms far from the zeros of
$^{} Q$, and its structure relies on the fact that the error is written in an exponential form far from the vortices. Close to the zeros of $Q$, this particular form does not hold,
but it will not be necessary, since there the error $\varphi$ is small in the
$C^1$ norm whereas, at infinity, it is small only in a weaker norm.

The proof is identical to the proof of Lemma 7.5 of <cit.>, and it is
in the particular case where all the speeds are along $\vec{e}_2$. The proof
consists simply in decomposing the equation

\[ 0 = (\tmop{TW}_c) (u) = \tmop{TW}_c (Q + (1 - \eta) \varphi + \eta Q
(\ex^{\psi} - 1)) \]

in the different terms.

This result shows in particular that $\psi \in C^2 (\{ \eta \neq 0 \},
\mathbb{C})$, and we can check with it, as in Lemma 7.3 of <cit.>, that
$\| \Delta \psi (1 + r)^2 \|_{L^{\infty} (\{ \tilde{r} \geqslant \lambda \})}
\leqslant K (u, Q)$.

We now infer a critical estimate on the differences of the speeds of the
problem, namely $c$ (the speed of $u$) and $c''$ (the speed of $Q$). The
method for the estimate has been used in <cit.> (we take the scalar
product of the equation of Lemma <ref> with $\partial_c Q$), but since
we have worse estimates on the error term, we need to be more careful ($\| \varphi \|_{H_Q^{\exp}} $ is not a priori small at this point).

Under the previous notations and hypotheses, there exists
a universal constant $K > 0$ such that

\[ | c'' - c | \leqslant K \sqrt{c''} \| \varphi \|_{H_Q^{\exp}} . \]

First, from equation (<ref>) and Lemma <ref>, taking
$\varepsilon > 0$ small enough, we have

\begin{equation}
| c'' - c | \leqslant | c'' - c' | + | c' - c | \leqslant K c'' .
\label{CP3cece}
\end{equation}

We will show the following estimate:

\begin{equation}
| c'' - c | \leqslant K \left( c^{\prime\prime 2} \ln \left( \frac{1}{c''}
\right) \| \varphi \|_{H_Q^{\exp}} + \| \varphi \|_{H_Q^{\exp}}^2 \right)
+ K | c'' - c | \| \varphi \|_{H_Q^{\exp}} . \label{CP3olala}
\end{equation}

This is related to equation (7.13) of <cit.> (its proof is in step 1
in subsection 7.3.1 of <cit.>). With both estimates, we can conclude
the proof of this lemma. Indeed, either $\| \varphi \|_{H_Q^{\exp}}
\geqslant \sqrt{c''}$, and in that case

\[ | c'' - c | \leqslant K c'' \leqslant K \sqrt{c''} \| \varphi
\|_{H_Q^{\exp}}, \]

or $\| \varphi \|_{H_Q^{\exp}} \leqslant \sqrt{c''}$, and then with

\begin{eqnarray*}
| c'' - c | & \leqslant & K \left( c^{\prime\prime 2} \ln \left( \frac{1}{c''}
\right) \| \varphi \|_{H_Q^{\exp}} + \| \varphi \|_{H_Q^{\exp}}^2 \right)
+ K | c'' - c | \| \varphi \|_{H_Q^{\exp}}\\
& \leqslant & K \sqrt{c''} \| \varphi \|_{H_Q^{\exp}} + C_2 \sqrt{c''} |
c'' - c |,
\end{eqnarray*}

therefore, for $c'' > 0$ small enough such that $C_2 \sqrt{c''} < 1 / 2$
(which is implied by taking $\varepsilon > 0$ small enough), we have $| c''
- c | \leqslant K \sqrt{c''} \| \varphi \|_{H_Q^{\exp}}$.

We now focus on the proof of (<ref>). We take the scalar product
of the equation

\[ L_Q (\varphi) - i (c - c'') \vec{e}_2 .H (\psi) + \tmop{NL}_{\tmop{loc}}
(\psi) + F (\psi) = 0 \]

with $c^{\prime\prime 2} \partial_{c''} Q$. We estimate, as in subsection
7.3.1 of <cit.>, that

\[ | \langle L_Q (\varphi), c^{\prime\prime 2} \partial_{c''} Q \rangle | =
c^{\prime\prime 2} | \langle \varphi, L_Q (\partial_{c''} Q) \rangle | =
c^{\prime\prime 2} | \langle \varphi, i \partial_{x_2} Q \rangle |
\leqslant K c^{\prime\prime 2} \ln \left( \frac{1}{c''} \right) \|
\varphi \|_{H_Q^{\exp}} . \]

We recall that

\[ i \vec{e}_2 .H (\psi) = i \partial_{x_2} Q + i \frac{\partial_{x_2} (Q
\psi) (1 - \eta) + Q \partial_{x_2} \psi \eta \ex^{\psi}}{(1 - \eta) + \eta
\ex^{\psi}}, \]

and we check (estimating the local terms in the area where $\eta \neq 1$ by
Cauchy-Schwarz and $\| c^{\prime\prime 2} \partial_{c''} Q \|_{L^{\infty}
(\mathbb{R}^2)} \leqslant K$ from Theorem <ref> for $p = + \infty$ and
Lemma 2.6 of <cit.>)

\begin{eqnarray*}
&  & | (c - c'') \langle i \vec{e}_2 .H (\psi), c^{\prime\prime 2}
\partial_{c''} Q \rangle - (c - c'') \langle i \partial_{x_2} Q,
c^{\prime\prime 2} \partial_{c''} Q \rangle |\\
& \leqslant & K (| c - c'' | \| \varphi \|_{H^1 (\{ \eta \neq 1 \})} + |
(c - c'') \langle \eta Q i \partial_{x_2} \psi, c^{\prime\prime 2}
\partial_{c''} Q \rangle |)\\
& \leqslant & K (| c - c'' | \| \varphi \|_{H_Q^{\exp}} + | (c - c'')
\langle \eta Q i \partial_{x_2} \psi, c^{\prime\prime 2} \partial_{c''} Q
\rangle |) .
\end{eqnarray*}

We recall from subsection 7.3.1 of <cit.> (using decay estimates on
$c^{\prime\prime 2} \partial_{c''} Q \bar{Q}$ and integrations by parts),

\[ | (c - c'') \langle \eta Q i \partial_{x_2} \psi, c^{\prime\prime 2}
\partial_{c''} Q \rangle | \leqslant K | c - c'' | \| \varphi
\|_{H_Q^{\exp}} \]

and, from Proposition 1.2 of <cit.> (we check easily that the
translation and phase on $Q$ instead of $Q_{c''}$ do not change the

\[ (c - c'') \langle i \partial_{x_2} Q, c^{\prime\prime 2} \partial_{c''} Q
\rangle = (2 \pi + o_{c'' \rightarrow 0} (1)) (c - c'') = (2 \pi +
o_{\varepsilon \rightarrow 0} (1)) (c - c'') . \]

We deduce that, taking $\varepsilon > 0$ small enough, that

\[ | c - c'' | \leqslant K c^{\prime\prime 2} \ln \left( \frac{1}{c''}
\right) \| \varphi \|_{H_Q^{\exp}} + K | c - c'' | \| \varphi
\|_{H_Q^{\exp}} + K | \langle \tmop{NL}_{\tmop{loc}} (\psi) + F (\psi),
c^{\prime\prime 2} \partial_{c''} Q \rangle | . \]

Since $\tmop{NL}_{\tmop{loc}} (\psi)$ contains terms at least quadratic in
$\varphi$, $\| \varphi \|_{C^1 (\{ \eta \neq 1 \})} \leqslant C_3 \nu$ from
Lemma <ref> and $\| c^{\prime\prime 2} \partial_{c''} Q
\|_{L^{\infty} (\mathbb{R}^2)} \leqslant K$, we compute, taking $\nu > 0$
small enough such that $C_3 \nu \leqslant 1$, that

\[ | \langle \tmop{NL}_{\tmop{loc}} (\psi), c^{\prime\prime 2}
\partial_{c''} Q \rangle | \leqslant K \| \varphi \|_{H^1 (\{ \eta \neq 1
\})}^2 \leqslant K \| \varphi \|_{H_Q^{\exp}}^2 . \]

Finally, we estimate, using $\| c^{\prime\prime 2} \partial_{c''} Q
\|_{L^{\infty} (\mathbb{R}^2)} \leqslant K$, that

\[ | \langle Q \eta \nabla \psi . \nabla \psi, c^{\prime\prime 2}
\partial_{c''} Q \rangle | \leqslant K \int_{\mathbb{R}^2} \eta | \nabla
\psi |^2 \| c^{\prime\prime 2} \partial_{c''} Q \|_{L^{\infty}
(\mathbb{R}^2)} \leqslant K \| \varphi \|_{H_Q^{\exp}}^2 . \]

Similarly, since $| \eta \mathfrak{Re} (\psi) | \leqslant \eta
C_4 \nu$ by Lemma <ref>, taking $C_4 \nu < 1 / 2$, we deduce

\[ | Q \eta | Q |^2 S (\psi) | = | Q \eta | Q |^2
(\ex^{2\mathfrak{Re} (\psi)} - 1 - 2\mathfrak{Re}
(\psi)) | \leqslant K \eta \mathfrak{Re}^2 (\psi), \]

\[ | \langle Q \eta | Q |^2 S (\psi), c^{\prime\prime 2} \partial_{c''} Q
\rangle | \leqslant K \int_{\mathbb{R}^2} \eta \mathfrak{Re}^2
(\psi) \| c^{\prime\prime 2} \partial_{c''} Q \|_{L^{\infty}
(\mathbb{R}^2)} \leqslant K \| \varphi \|_{H_Q^{\exp}}^2 . \]

This concludes the proof of (<ref>), and therefore of the lemma.

§.§.§ Proof of Proposition <ref> completed

We want to take the scalar product of the equation of Lemma <ref>
with $\varphi$. It is however not clear at this point that every term is
integrable. In subsection 7.3 of <cit.>, we took the scalar product of
the equation with $\varphi + i \gamma' Q$ for some $\gamma' \in \mathbb{R}$,
using a decay estimate $\| \mathfrak{I}\mathfrak{m} (\psi + i \gamma') (1 +
r) \|_{L^{\infty} (\{ \tilde{r} \leqslant \lambda \})} \leqslant K (u, Q)$
to justify that some terms are well defined, and to do some integration by
parts. Here, we change a little our approach. We require better decay
estimates on $\psi$. We recall that we have shown that

\begin{eqnarray*}
&  & \| \Delta \psi (1 + r)^2 \|_{L^{\infty} (\{ \tilde{r} \geqslant
\lambda \})} + \| (1 + r)^2 \nabla \psi \|_{L^{\infty} (\{ \tilde{r}
\geqslant \lambda \})}\\
& + & \| \psi \|_{L^{\infty} (\{ \tilde{r} \geqslant \lambda \})} + \| (1
+ r)^2 \mathfrak{Re} (\psi) \|_{L^{\infty} (\{ \tilde{r}
\geqslant \lambda \})}\\
& \leqslant & K (u, Q) .
\end{eqnarray*}

Now, let us show the following improvements:

\begin{equation}
\| \mathfrak{I}\mathfrak{m} (\Delta \psi) (1 + r)^3 \|_{L^{\infty} (\{
\tilde{r} \geqslant \lambda \})} + \| (1 + r)^3 \mathfrak{Re}
(\nabla \psi) \|_{L^{\infty} (\{ \tilde{r} \geqslant \lambda \})}
\leqslant K (u, Q) . \label{piz}
\end{equation}

The proof of $\| (1 + r)^3 | \mathfrak{Re} (\nabla \psi) |
\|_{L^{\infty} (\{ \tilde{r} \geqslant \lambda \})} \leqslant K (u, Q)$ is
identical to the one for the same result in Lemma 7.3 of <cit.>. We
focus on the esimate on $\mathfrak{I}\mathfrak{m} (\Delta \psi)$. In $\{
\tilde{r} \geqslant \lambda \}$, we have $u = Q \ex^{\psi}$, therefore,

\[ \Delta \psi = - \frac{\Delta Q}{Q} + \frac{\Delta u}{u} - 2 \frac{\nabla
Q}{Q} . \nabla \psi - \nabla \psi . \nabla \psi . \]

With the previous estimates and <cit.>, we have

\[ \left\| \left( - 2 \frac{\nabla Q}{Q} . \nabla \psi - \nabla \psi .
\nabla \psi \right) (1 + r)^4 \right\|_{L^{\infty} (\{ \tilde{r}
\geqslant \lambda \})} \leqslant K (u, Q), \]

and since $(\tmop{TW}_{c''}) (Q) = 0$,

\[ \frac{\Delta Q}{Q} = i c''  \frac{\partial_{x_2} Q}{Q} - (1 - | Q |^2),
\]

therefore, with <cit.> ($E (Q) < + \infty$),

\[ \left| \mathfrak{I}\mathfrak{m} \left( \frac{\Delta Q}{Q} \right) \right|
\leqslant c'' \left| \mathfrak{Re} \left( \frac{\partial_{x_2}
Q}{Q} \right) \right| \leqslant \frac{K (Q)}{(1 + r)^3} . \]

Similarly, since $(\tmop{TW}_c) (u) = 0$ and $E (u) < + \infty$,

\[ \left| \mathfrak{I}\mathfrak{m} \left( \frac{\Delta u}{u} \right) \right|
\leqslant c \left| \mathfrak{Re} \left( \frac{\partial_{x_2}
u}{u} \right) \right| \leqslant \frac{K (u)}{(1 + r)^3}, \]

\[ \| \mathfrak{I}\mathfrak{m} (\Delta \psi) (1 + r)^3 \|_{L^{\infty}
(\{ \tilde{r} \geqslant \lambda \})} \leqslant K (u, Q) . \]

We infer, with these two additional estimates on $\psi$, that we can do the
same computations as in the proof of Lemma 7.4 of <cit.>, with
$\gamma' = 0$. The only difference is that, when we used $\|
\mathfrak{I}\mathfrak{m} (\psi + i \gamma') (1 + r) \|_{L^{\infty} (\{
\tilde{r} \geqslant \lambda \})} \leqslant K (u, Q)$, we can use (<ref>)
instead to get the same decay for these terms, with $\|
\mathfrak{I}\mathfrak{m} (\psi) \|_{L^{\infty} (\{ \tilde{r} \leqslant
\lambda \})} \leqslant K (Z, Q)$. We deduce, taking the scalar product of
the equation of Lemma <ref> with $\varphi$, that

\begin{equation}
B_Q^{\exp} (\varphi) - \langle i (c - c'') \vec{e}_2 .H (\psi), \varphi
\rangle + \langle \tmop{NL}_{\tmop{loc}} (\psi), \varphi \rangle + \langle
F (\psi), \varphi \rangle = 0. \label{CP3eqeq}
\end{equation}

From Lemma <ref>,

\begin{equation}
B_Q^{\exp} (\varphi) \geqslant K \| \varphi \|_{H_{Q_c}^{\exp}}^2,
\label{CP3eq210}
\end{equation}

and from Lemma <ref>,

\begin{equation}
| \langle \tmop{NL}_{\tmop{loc}} (\psi), \varphi \rangle | \leqslant K \nu
\| \varphi \|^2_{H^1 (\{ \eta \neq 1 \})} \leqslant K \nu \| \varphi
\|^2_{H_{Q_c}^{\exp}} . \label{CP3eq211}
\end{equation}

Let us now show that

\begin{equation}
| \langle i (c - c'') \vec{e}_2 .H (\psi), \varphi \rangle | \leqslant K
\nu \| \varphi \|^2_{H_{Q_c}^{\exp}} . \label{CP3eq212}
\end{equation}

We recall that

\[ i \vec{e}_2 .H (\psi) = i \partial_{x_2} Q + i \frac{\partial_{x_2} (Q
\psi) (1 - \eta) + Q \partial_{x_2} \psi \eta \ex^{\psi}}{(1 - \eta) + \eta
\ex^{\psi}} . \]

We compute with Lemma <ref> and Lemma 5.4 of <cit.> that
taking $c'' > 0$ small enough (which is done by taking $\varepsilon > 0$
small enough) such that $\sqrt{c''} \ln \left( \frac{1}{c''} \right)
\leqslant \nu$,

\[ | (c - c'') \langle i \partial_{x_2} Q, \varphi \rangle | \leqslant K
\sqrt{c''} \| \varphi \|_{H_Q^{\exp}} | \langle i \partial_{x_2} Q,
\varphi \rangle | \leqslant K \sqrt{c''} \ln \left( \frac{1}{c''} \right)
\| \varphi \|_{H_Q^{\exp}}^2 \leqslant K \nu \| \varphi \|^2_{H_Q^{\exp}}
. \]

Now, with $\| \varphi \|_{H^1 (\{ \eta \neq 1 \})} \leqslant K \nu$ from
Lemmas <ref> and <ref>, we compute easily that

\[ \left| \left\langle i \frac{\partial_{x_2} (Q \psi) (1 - \eta) + Q
\partial_{x_2} \psi \eta \ex^{\psi}}{(1 - \eta) + \eta \ex^{\psi}}, \varphi
\right\rangle - \langle i Q \partial_{x_2} \psi \eta, \varphi \rangle
\right| \leqslant K \nu \| \varphi \|_{H_Q^{\exp}} \]

since the left hand side is supported in $\{ \eta \neq 1 \}$, therefore

\[ | \langle i (c - c'') \vec{e}_2 .H (\psi), \varphi \rangle | \leqslant K
\nu \| \varphi \|^2_{H_{Q_c}^{\exp}} + | (c - c'') \langle i Q
\partial_{x_2} \psi \eta, \varphi \rangle | . \]

With the same computations as in subsection 7.3.2 of <cit.> (taking
$\gamma' = 0$), we check that

\[ | \langle i Q \partial_{x_2} \psi \eta, \varphi \rangle | \leqslant K \|
\varphi \|_{H_Q^{\exp}}^2, \]

therefore, using Lemma <ref> and equation (<ref>), for
$\varepsilon > 0$ small enough,

\begin{eqnarray*}
&  & | (c - c'') \langle i Q \partial_{x_2} \psi \eta, \varphi \rangle
& \leqslant & K | c - c'' | \| \varphi \|_{H_Q^{\exp}}^2\\
& \leqslant & K \nu \| \varphi \|_{H_Q^{\exp}}^2 .
\end{eqnarray*}

This completes the proof of equation (<ref>). We focus now on the
proof of

\begin{equation}
| \langle F (\psi), \varphi \rangle | \leqslant K \nu \| \varphi
\|^2_{H_Q^{\exp}} . \label{CP3eq213}
\end{equation}

We compute

\[ \int_{\mathbb{R}^2} \mathfrak{Re} (Q \eta (| Q |^2 S (\psi))
\bar{\varphi}) = \int_{\mathbb{R}^2} | Q |^4 \eta
(\ex^{2\mathfrak{Re} (\psi)} - 1 - 2\mathfrak{Re}
(\psi)) \mathfrak{Re} (\psi), \]

and since, in $\{ \eta \neq 0 \}$, $| \mathfrak{Re} (\psi) |
\leqslant C_5 \nu$ from Lemma <ref>, we have, for $\nu > 0$ small
enough (such that $C_5 \nu < 1 / 2$),

\[ | \ex^{2\mathfrak{Re} (\psi)} - 1 - 2\mathfrak{Re}
(\psi) | \leqslant K\mathfrak{Re}^2 (\psi) \]

\[ \left| \int_{\mathbb{R}^2} \mathfrak{Re} (Q \eta (| Q |^2 S
(\psi)) \bar{\varphi}) \right| \leqslant K \int_{\mathbb{R}^2} \eta
\mathfrak{Re}^3 (\psi) \leqslant K \nu \int_{\mathbb{R}^2}
\eta \mathfrak{Re}^2 (\psi) \leqslant K \nu \| \varphi
\|^2_{H_Q^{\exp}} . \]

We are left with the estimation of $\int_{\mathbb{R}^2}
\mathfrak{Re} (Q \eta (- \nabla \psi . \nabla \psi)
\bar{\varphi})$, which will be slightly more delicate. First, we compute,
using $\varphi = Q \psi$

\begin{eqnarray*}
&  & \int_{\mathbb{R}^2} \mathfrak{Re} (Q \eta (- \nabla \psi
. \nabla \psi) \bar{\varphi})\\
& = & - \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\nabla
\psi . \nabla \psi \bar{\psi})\\
& = & - \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\nabla
\psi . \nabla \psi) \mathfrak{Re} (\psi)\\
& - & \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{I}\mathfrak{m} (\nabla
\psi . \nabla \psi) \mathfrak{I}\mathfrak{m} (\psi)\\
& = & - \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\nabla
\psi . \nabla \psi) \mathfrak{Re} (\psi)\\
& - & 2 \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\nabla
\psi) .\mathfrak{I}\mathfrak{m} (\nabla \psi) \mathfrak{I}\mathfrak{m}
(\psi) .
\end{eqnarray*}

We estimate, using $| \mathfrak{Re} (\psi) | \leqslant K \nu$ in
$\{ \eta \neq 0 \}$ that

\[ \left| - \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re}
(\nabla \psi . \nabla \psi) \mathfrak{Re} (\psi) \right|
\leqslant K \nu \int_{\mathbb{R}^2} \eta | \nabla \psi |^2 \leqslant K
\nu \| \varphi \|^2_{H_Q^{\exp}} . \]

Now, by integration by parts (that can be justified as in <cit.>), we

\begin{eqnarray*}
&  & \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\nabla
\psi) .\mathfrak{I}\mathfrak{m} (\nabla \psi) \mathfrak{I}\mathfrak{m}
& = & - \int_{\mathbb{R}^2} \nabla (| Q |^2) \eta
\mathfrak{Re} (\psi) .\mathfrak{I}\mathfrak{m} (\nabla \psi)
\mathfrak{I}\mathfrak{m} (\psi)\\
& - & \int_{\mathbb{R}^2} | Q |^2 \nabla \eta \mathfrak{Re}
(\psi) .\mathfrak{I}\mathfrak{m} (\nabla \psi) \mathfrak{I}\mathfrak{m}
& - & \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\psi)
\mathfrak{I}\mathfrak{m} (\Delta \psi) \mathfrak{I}\mathfrak{m} (\psi)\\
& - & \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\psi)
\mathfrak{I}\mathfrak{m} (\nabla \psi) .\mathfrak{I}\mathfrak{m} (\nabla
\psi),
\end{eqnarray*}

and with $| \mathfrak{Re} (\psi) | \leqslant K \nu$ in $\{ \eta
\neq 0 \}$ and $| \nabla (| Q |^2) | \leqslant \frac{K}{(1 + \tilde{r})^{5 /
2}}$ from equation (2.9) of <cit.> (for $\sigma = 1 / 2$), we have by
Cauchy-Schwarz that

\begin{eqnarray*}
&  & \left| \int_{\mathbb{R}^2} \nabla (| Q |^2) \eta
\mathfrak{Re} (\psi) .\mathfrak{I}\mathfrak{m} (\nabla \psi)
\mathfrak{I}\mathfrak{m} (\psi) \right|\\
& \leqslant & K \nu \sqrt{\int_{\mathbb{R}^2} \eta | \nabla \psi |^2
\int_{\mathbb{R}^2} \eta \frac{| \psi |^2}{(1 + \tilde{r})^5}}\\
& \leqslant & K \nu \| \varphi \|^2_{H_Q^{\exp}}
\end{eqnarray*}

\[ \left| \int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\psi)
\mathfrak{I}\mathfrak{m} (\nabla \psi) .\mathfrak{I}\mathfrak{m} (\nabla
\psi) \right| \leqslant K \nu \int_{\mathbb{R}^2} \eta | \nabla \psi |^2
\leqslant K \nu \| \varphi \|^2_{H_{Q_c}^{\exp}} . \]

Since $\nabla \eta$ is supported in $\{ \eta \neq 0 \}$, we check easily
with $\| \varphi \|_{C^1 (\{ \eta \neq 0 \})} \leqslant K \nu$ from Lemma
<ref> that

\[ \left| \int_{\mathbb{R}^2} | Q |^2 \nabla \eta \mathfrak{Re}
(\psi) .\mathfrak{I}\mathfrak{m} (\nabla \psi) \mathfrak{I}\mathfrak{m}
(\psi) \right| \leqslant K \nu \| \varphi \|^2_{H_Q^{\exp}} . \]

We focus now on the estimation of the last remaining term,
$\int_{\mathbb{R}^2} | Q |^2 \eta \mathfrak{Re} (\psi)
\mathfrak{I}\mathfrak{m} (\Delta \psi) \mathfrak{I}\mathfrak{m} (\psi)$. For
that purpose, we define more generally for $n \geqslant 1$

\[ A_n \assign \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) \mathfrak{I}\mathfrak{m} (\Delta \psi) \mathfrak{I}\mathfrak{m}
(\psi) . \]

Remark that we want to estimate $A_1$.

We compute, using that $(\tmop{TW}_{c''}) (Q) = 0$, that

\[ L_Q (\varphi) = Q \left( - \Delta \psi - i c'' \partial_{x_2} \psi - 2
\frac{\nabla Q}{Q} . \nabla \psi + 2\mathfrak{Re} (\psi) | Q
|^2 \right), \]

therefore, by Lemma <ref>, in $\{ \eta \neq 0 \}$,

\begin{eqnarray*}
\mathfrak{I}\mathfrak{m} (\Delta \psi) & = & \mathfrak{I}\mathfrak{m}
\left( - i c'' \partial_{x_2} \psi - 2 \frac{\nabla Q}{Q} . \nabla \psi +
2\mathfrak{Re} (\psi) | Q |^2 + \frac{- i (c - c'') \vec{e}_2
.H (\psi) + \tmop{NL}_{\tmop{loc}} (\psi) + F (\psi)}{Q} \right)\\
& = & - c'' \mathfrak{Re} (\partial_{x_2} \psi) -
2\mathfrak{I}\mathfrak{m} \left( \frac{\nabla Q}{Q} . \nabla \psi \right)
+\mathfrak{I}\mathfrak{m} \left( \frac{- i (c - c'') \vec{e}_2 .H (\psi) +
\tmop{NL}_{\tmop{loc}} (\psi) + F (\psi)}{Q} \right) .
\end{eqnarray*}

We compute, by integration by parts, with $\mathfrak{Re}^n (\psi)
\mathfrak{Re} (\partial_{x_2} \psi) = \frac{1}{n + 1}
\partial_{x_2} (\mathfrak{Re}^{n + 1} (\psi))$, that

\begin{eqnarray*}
&  & \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n (\psi)
c'' \mathfrak{Re} (\partial_{x_2} \psi)
\mathfrak{I}\mathfrak{m} (\psi)\\
& = & \frac{- 1}{n + 1} \int_{\mathbb{R}^2} (\partial_{x_2} | Q |^2)
\eta^n \mathfrak{Re}^{n + 1} (\psi) c''
\mathfrak{I}\mathfrak{m} (\psi)\\
& - & \frac{n}{n + 1} \int_{\mathbb{R}^2} | Q |^2 \partial_{x_2} \eta
\eta^{n - 1} \mathfrak{Re}^{n + 1} (\psi) c''
\mathfrak{I}\mathfrak{m} (\psi)\\
& - & \frac{1}{n + 1} \int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^{n + 1} (\psi) c'' \mathfrak{I}\mathfrak{m}
(\partial_{x_2} \psi) .
\end{eqnarray*}

Since $| c'' | \leqslant \nu$ by equation (<ref>) (taking
$\varepsilon > 0$ small enough), Lemma <ref> and the hypotheses of
Proposition <ref>, $\| \varphi \|_{C^1 (\{ \tilde{r} \leqslant \lambda
\})} + \| \mathfrak{Re} (\psi) \|_{L^{\infty} (\{ \tilde{r}
\geqslant \lambda \})} \leqslant K \nu$ by Lemma <ref> and $| \nabla
(| Q |^2) | \leqslant \frac{K}{(1 + \tilde{r})^{5 / 2}}$ from equation (2.9)
of <cit.>, we infer by Cauchy-Schwarz that

\begin{eqnarray}
&  & \left| \int_{\mathbb{R}^2} (\partial_{x_2} | Q |^2) \eta^n
\mathfrak{Re}^{n + 1} (\psi) c'' \mathfrak{I}\mathfrak{m}
(\psi) \right| \nonumber\\
& \leqslant & c'' \sqrt{\int_{\mathbb{R}^2} \eta
\mathfrak{I}\mathfrak{m}^2 (\psi) (\partial_{x_2} | Q |^2)^2
\int_{\mathbb{R}^2} \eta \mathfrak{Re}^2 (\psi)} ( K\| \varphi
\|_{C^1 (\{ \tilde{r} \leqslant \lambda \})} + K \| \mathfrak{Re}
(\psi) \|_{L^{\infty} (\{ \tilde{r} \geqslant \lambda \})})^n \nonumber\\
& \leqslant & (K \nu)^n \| \varphi \|^2_{H_Q^{\exp}},  \label{320ziz}
\end{eqnarray}

\begin{equation}
\left| \int_{\mathbb{R}^2} | Q |^2 \partial_{x_2} \eta \eta^{n - 1}
\mathfrak{Re}^{n + 1} (\psi) c'' \mathfrak{I}\mathfrak{m}
(\psi) \right| \leqslant (K \nu)^n \| \varphi \|^2_{H_Q^{\exp}}
\end{equation}

\begin{equation}
\left| \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^{n + 1}
(\psi) c'' \mathfrak{I}\mathfrak{m} (\partial_{x_2} \psi) \right|
\leqslant (K \nu)^n \sqrt{\int_{\mathbb{R}^2} \eta | \nabla \psi |^2
\int_{\mathbb{R}^2} \eta \mathfrak{Re}^2 (\psi)} \leqslant (K
\nu)^n \| \varphi \|^2_{H_Q^{\exp}} .
\end{equation}

We deduce that

\begin{equation}
\left| \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) c'' \mathfrak{Re} (\partial_{x_2} \psi)
\mathfrak{I}\mathfrak{m} (\psi) \right| \leqslant (K \nu)^n \| \varphi
\|^2_{H_Q^{\exp}} . \label{CP32177}
\end{equation}

For $\int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n (\psi)
\mathfrak{I}\mathfrak{m} \left( \frac{\nabla Q}{Q} . \nabla \psi \right)
\mathfrak{I}\mathfrak{m} (\psi)$, we compute

\[ \mathfrak{I}\mathfrak{m} \left( \frac{\nabla Q}{Q} . \nabla \psi \right)
=\mathfrak{Re} \left( \frac{\nabla Q}{Q} \right)
.\mathfrak{I}\mathfrak{m} (\nabla \psi) +\mathfrak{Re} (\nabla
\psi) .\mathfrak{I}\mathfrak{m} \left( \frac{\nabla Q}{Q} \right), \]

and with previous estimates, we check easily that

\begin{eqnarray}
&  & \left| \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) \mathfrak{Re} \left( \frac{\nabla Q}{Q} \right)
.\mathfrak{I}\mathfrak{m} (\nabla \psi) \mathfrak{I}\mathfrak{m} (\psi)
\right| \nonumber\\
& \leqslant & (K \nu)^n \sqrt{\int_{\mathbb{R}^2} \eta | \nabla \psi |^2
\int_{\mathbb{R}^2} \eta \mathfrak{I}\mathfrak{m}^2 (\psi)
\mathfrak{Re}^2 \left( \frac{\nabla Q}{Q} \right)} \leqslant (K
\nu)^n \| \varphi \|^2_{H_Q^{\exp}},
\end{eqnarray}

and by integration by parts, with computations similar to those for the proof of
(<ref>), using

\[ \left| \nabla .\mathfrak{I}\mathfrak{m} \left( \frac{\nabla Q}{Q} \right)
\right| \leqslant \frac{K}{(1 + \tilde{r})^{3 / 2}} \]

from (2.9) to (2.11) of <cit.> (for $\sigma = 1 / 2$) and Lemma
<ref>, we infer that

\begin{equation}
\left| \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) \mathfrak{Re} (\nabla \psi) .\mathfrak{I}\mathfrak{m}
\left( \frac{\nabla Q}{Q} \right) \mathfrak{I}\mathfrak{m} (\psi) \right|
\leqslant (K \nu)^n \| \varphi \|^2_{H_Q^{\exp}},
\end{equation}

and we check easily that

\begin{equation}
\left| \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) \mathfrak{I}\mathfrak{m} \left( \frac{\tmop{NL}_{\tmop{loc}}
(\psi)}{Q} \right) \mathfrak{I}\mathfrak{m} (\psi) \right| \leqslant (K
\nu)^n \| \varphi \|^2_{H_Q^{\exp}} .
\end{equation}

Now, we look at $\int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^n (\psi) \mathfrak{I}\mathfrak{m} \left( \frac{- i
(c - c'') \vec{e}_2 .H (\psi)}{Q} \right) \mathfrak{I}\mathfrak{m} (\psi)$,
for the part of $\vec{e}_2 .H (\psi)$ related to the cutoff, the estimation
can be done as previously, and we are left with the estimation of

\begin{eqnarray*}
&  & (c - c'') \int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^n (\psi) \mathfrak{I}\mathfrak{m} \left( - i
\frac{\partial_{x_2} Q}{Q} - i \partial_{x_2} \psi \right)
\mathfrak{I}\mathfrak{m} (\psi)\\
& = & (c - c'') \int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^n (\psi) \mathfrak{Re} \left(
\frac{\partial_{x_2} Q}{Q} + \partial_{x_2} \psi \right)
\mathfrak{I}\mathfrak{m} (\psi) .
\end{eqnarray*}

From equation (<ref>) and Lemma <ref>, we have $| c - c'' |
\leqslant \nu$ (for $\varepsilon > 0$ small enough), and from equation (2.9)
of <cit.>, $\left| \mathfrak{Re} \left(
\frac{\partial_{x_2} Q}{Q} \right) \right| \leqslant \frac{K}{(1 +
\tilde{r})^{5 / 2}}$, therefore

\begin{eqnarray}
&  & \left| (c - c'') \int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^n (\psi) \mathfrak{Re} \left(
\frac{\partial_{x_2} Q}{Q} \right) \mathfrak{I}\mathfrak{m} (\psi) \right|
\nonumber\\
& \leqslant & (K \nu)^n \sqrt{\int_{\mathbb{R}^2} \eta
\mathfrak{Re}^2 (\psi) \int_{\mathbb{R}^2} \eta
\mathfrak{Re}^2 \left(  \frac{\partial_{x_2} Q}{Q} \right)
\mathfrak{I}\mathfrak{m}^2 (\psi)} \nonumber\\
& \leqslant & (K \nu)^n \| \varphi \|^2_{H_Q^{\exp}},
\end{eqnarray}

and we estimate

\begin{equation}
\left| (c - c'') \int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^n (\psi) \mathfrak{Re} (\partial_{x_2}
\psi) \mathfrak{I}\mathfrak{m} (\psi) \right| \leqslant (K \nu)^n \|
\varphi \|^2_{H_Q^{\exp}}
\end{equation}

by (<ref>). For the last remaining term, since

\[ \mathfrak{I}\mathfrak{m} \left( \frac{F (\psi)}{Q} \right)
=\mathfrak{I}\mathfrak{m} (- \eta \nabla \psi . \nabla \psi), \]

we have $\int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) \mathfrak{I}\mathfrak{m} \left( \frac{F (\psi)}{Q} \right)
\mathfrak{I}\mathfrak{m} (\psi) = - 2 \int_{\mathbb{R}^2} | Q |^2 \eta^{n +
1} \mathfrak{Re}^n (\psi) \mathfrak{I}\mathfrak{m} (\nabla \psi)
.\mathfrak{Re} (\nabla \psi) \mathfrak{I}\mathfrak{m} (\psi)$. In

\begin{eqnarray}
&  & \left| \int_{\mathbb{R}^2} | Q |^2 \eta^n \mathfrak{Re}^n
(\psi) \mathfrak{I}\mathfrak{m} \left( \frac{F (\psi)}{Q} \right)
\mathfrak{I}\mathfrak{m} (\psi) \right| \nonumber\\
& \leqslant & (K \nu)^n \| \eta \mathfrak{I}\mathfrak{m} (\psi)
\|_{L^{\infty} (\mathbb{R}^2)} \int_{\mathbb{R}^2} \eta | \nabla \psi |^2
\nonumber\\
& \leqslant & (K \nu)^n \| \eta \mathfrak{I}\mathfrak{m} (\psi)
\|_{L^{\infty} (\mathbb{R}^2)} \| \varphi \|^2_{H_Q^{\exp}} .
\label{329ziz}
\end{eqnarray}

Combining this result with the previous estimates, this implies that

\begin{equation}
| A_n | \leqslant (C_6 \nu)^n (1 + \| \eta \mathfrak{I}\mathfrak{m} (\psi)
\|_{L^{\infty} (\mathbb{R}^2)}) \| \varphi \|^2_{H_Q^{\exp}}
\label{CP32188}
\end{equation}

for some universal constant $C_6 > 0$, but that is not enough to show that
we have $\left| \int_{\mathbb{R}^2} | Q |^2 \eta^n
\mathfrak{Re}^n (\psi) \mathfrak{I}\mathfrak{m} \left( \frac{F
(\psi)}{Q} \right) \mathfrak{I}\mathfrak{m} (\psi) \right| \leqslant (K
\nu)^n \| \varphi \|^2_{H_Q^{\exp}}$, since we have no control on $\| \eta
\mathfrak{I}\mathfrak{m} (\psi) \|_{L^{\infty} (\mathbb{R}^2)}$ other than
the fact that it is a finite quantity. By integration by parts (integrating
$\mathfrak{Re} (\nabla \psi)$), with computations similar as for
the proof of (<ref>), we infer that

\begin{eqnarray*}
&  & \left| 2 \int_{\mathbb{R}^2} | Q |^2 \eta^{n + 1}
\mathfrak{Re}^n (\psi) \mathfrak{I}\mathfrak{m} (\nabla \psi)
.\mathfrak{Re} (\nabla \psi) \mathfrak{I}\mathfrak{m} (\psi)
\right|\\
& \leqslant & \left| 2 \int_{\mathbb{R}^2} | Q |^2 \eta^{n + 1}
\mathfrak{Re}^n (\psi) \mathfrak{I}\mathfrak{m} (\Delta \psi)
\mathfrak{Re} (\psi) \mathfrak{I}\mathfrak{m} (\psi) \right| +
(K \nu)^n \| \varphi \|^2_{H_Q^{\exp}}\\
& \leqslant & 2 | A_{n + 1} | + (K \nu)^n \| \varphi \|^2_{H_Q^{\exp}} .
\end{eqnarray*}

Combining this result with estimates (<ref>) to (<ref>), we
deduce that for some universal constant $C_7 > 0$,

\[ | A_n | \leqslant 2 | A_{n + 1} | + (C_7 \nu)^n \| \varphi
\|^2_{H_Q^{\exp}}, \]

therefore, by induction,

\[ | A_1 | \leqslant 2^n | A_n | + \sum_{k = 1}^{n - 1} (2 C_7 \nu)^k \|
\varphi \|^2_{H_Q^{\exp}}, \]

hence, with (<ref>),

\[ | A_1 | \leqslant \left( (2 C_6 \nu)^n (1 + \| \eta
\mathfrak{I}\mathfrak{m} (\psi) \|_{L^{\infty} (\mathbb{R}^2)}) + \sum_{k
= 1}^{n - 1} \nobracket (2 C_7 \nu \nobracket)^k \right) \| \varphi
\|^2_{H_Q^{\exp}} . \]

Taking $\nu > 0$ such that $2 C_6 \nu < 1 / 2$ and $2 C_7 \nu < 1 / 2$, then
$n \geqslant 1$ large enough (depending on $\| \eta \mathfrak{I}\mathfrak{m}
(\psi) \|_{L^{\infty} (\mathbb{R}^2)}$) such that

\[ \frac{1}{2^{n - 1}} (1 + \| \eta \mathfrak{I}\mathfrak{m} (\psi)
\|_{L^{\infty} (\mathbb{R}^2)}) \leqslant 1, \]

we conclude that

\[ | A_1 | \leqslant \left( 2 C_6 + 2 C_7 \sum_{k = 0}^{n - 2} \frac{1}{2^k}
\right) \nu \| \varphi \|^2_{H_Q^{\exp}} \leqslant 2 (C_6 + 2 C_7) \nu \|
\varphi \|^2_{H_Q^{\exp}} . \]

This concludes the proof of equation (<ref>).

Combining estimates (<ref>) to (<ref>) in equation
(<ref>), we deduce that

\[ (1 - C_8 \nu) \| \varphi \|_{H_Q^{\exp}}^2 \leqslant 0 \]

for some universal constant $C_8 > 0$, therefore, taking $\nu > 0$ such that
$C_8 \nu < 1 / 2$, we have $\| \varphi \|_{H_Q^{\exp}} = 0$. From Lemma
<ref>, we deduce $c'' = c$. The proof is complete.

§.§ Proof of Corollary <ref>

Take a function $u$ satisfying the hypotheses of Corollary <ref>.
Then, $u$ is even in $x_1$ and it has finite energy. Furthermore, by Theorem
<ref> (for $p = + \infty$),

\begin{eqnarray*}
&  & \| u - V_1 (x - d_c \vec{e}_1) V_{- 1} (x + d_c \vec{e}_1)
\|_{L^{\infty} (\mathbb{R}^2)}\\
& \leqslant & \| u - Q_c \|_{L^{\infty} (\mathbb{R}^2)} + \| Q_c - V_1 (x
- d_c \vec{e}_1) V_{- 1} (x + d_c \vec{e}_1) \|_{L^{\infty}
& \leqslant & \varepsilon + o_{c \rightarrow 0} (1),
\end{eqnarray*}

hence $u$ satisfies the third hypothesis of Proposition <ref> for $d =
d_c$ for $c, \varepsilon > 0$ small enough, and $d_c$ is large if $c > 0$ is
small (by Theorem <ref>). Now,

\[ \| | u | - 1 \|_{L^{\infty} (\{ \tilde{r}_d \geqslant \lambda \})}
\leqslant \| u - Q_c \|_{L^{\infty} (\{ \tilde{r}_d \geqslant \lambda
\})} + \| | Q_c | - 1 \|_{L^{\infty} (\{ \tilde{r}_d \geqslant \lambda
\})} \leqslant \varepsilon + \frac{K}{\lambda} \]

by equation (2.6) of <cit.>, therefore $u$ satisfies the fourth
hypothesis of Proposition <ref> by taking $\varepsilon > 0$ small
enough and $\lambda > 0$ large enough (independently of any parameters).
Finally, $(\tmop{TW}_c) (u) = 0$ with $| d_c c - 1 | = o_{c \rightarrow 0}
(1)$ by Theorem <ref>, which concludes the proof of the corollary by
taking $c > 0$ small enough.

§ PROPERTIES OF QUASI-MINIMIZERS OF THE ENERGY AND PROOF OF THEOREM <REF>

§.§ Tools for the vortex analysis

We list in this section some results useful for the analysis of travelling waves
for small speeds or, equivalently, large momentum, with vorticity. We shall denote
$ \langle u| v \rangle = \textrm{Re} ( u \bar{v} ) $ the real scalar product of the complex
numbers $u$, $v $. The jacobian (or vorticity)

\[
J v \assign \langle i \p_1 v | \p_2 v \rangle
= \frac{1}{2} \p_1 \langle i v | \p_2 v \rangle - \frac{1}{2} \p_2 \langle i v | \p_1 v \rangle
\]

is then relevant, and we shall use the following concentration property of the jacobian. We denote

\[
E_\e ( u , \Omega )
\assign \frac{1}{2} \int_{ \Omega } \lvert \nabla u \rvert^2 + \frac{1}{2\e^2} ( 1 - |u|^2 )^2 \, dx .
\]

Let $ M_0 > 0 $, $ R > 0 $ and $ \beta \in ] 0,1 ] $. Then, for every $ \delta >0 $, there exists $ \e_0 > 0 $
(depending only on $ \beta $, $ \delta $, $ R $ and $ M_0 $) such that for any $ 0 < \e < \e_0 $, and for
any $ u \in H^1( B(0,R) ) $ such that $ E_\e ( u , B(0, R) ) \ls M_0 \lvert \ln \e \rvert $,
there exist $ N \in \N $, $ y_1, \dotsc , y_N \in B(0, R ) $, $ d_1, \dotsc , d_N \in \Z $ such that

\[
\Big\| Ju - \pi \sum_{k=1}^N d_k \delta_{y_k} \Big\|_{ [ \BC^{0, \beta}_c( B(0, R) ) ]^*} \ls \delta
\]

\[
\pi \sum_{k=1}^N \lvert d_k \rvert \ls \frac{E_\e (u, B(0,R )) }{ \lvert \ln \e \rvert } + \delta .
\]

Finally, we may choose the points $ y_k $, $1 \ls k \ls N $, in $ \{ |u | \ls 1/2 \} $.

Here, we recall that the space $ [ \BC^{0, \beta}_c( B(0, R) ) ]^* $ is endowed with the dual norm
associated with $ \| \zeta \|_{ \BC^{0, \beta}_c( B(0, R) )} = \sup_{x \neq y \in B(0,R) } \frac{ \lvert \zeta( x) - \zeta( y) \rvert}{|x-y|^\beta} $,
for $ \zeta \in \BC^{0, \beta} ( B(0, R) ) $ compactly supported.

Let $ M_0 > 0 $ and $ \sigma > 0 $ be given. Then there exist $ \ep_0 > 0 $ and $ \eta > 0 $, depending
only on $ M_0 $ and $ \sigma $, such that, if $ R_0 = 1/ (1 + M_0 ) $, if $ U : B (0, R_0) \to \C $ solves

\begin{equation}
\label{GL_transpo}
\Delta U + i \mathfrak{c} \p_2 U + \frac{1}{\ep^2} U ( 1 - |U|^2 ) = 0
\end{equation}

in $ B (0, R_0) \subset \R^2 $, with $ \ep < \ep_0 $, $ \lvert \mathfrak{c} \rvert \ls M_0 \lvert \ln \ep \rvert $, and

\[
E_\ep (U, B (0, R_0) )
% \frac{1}{2} \int_{D (0,R_0)} \lvert \nabla U \rvert^2 \, d\hat{x} + \frac{1}{4 \ep^2} \int_{D (0,R_0)} (1 - \lvert U \rvert^2 )^2 \, d\hat{x}
\ls \eta \lvert \ln \ep \rvert ,
\]

\[
\lvert U(0) \rvert \gs 1 - \sigma .
\]

For the elliptic PDE

\begin{equation}
\label{GL}
\Delta \BU + \frac{1}{\e^2} \BU ( 1 - |\BU|^2 ) = 0 ,
\end{equation}

that is without the transport term $  i \p_2 U $, this result has been shown in 2d in <cit.> for
minimizing maps, then in <cit.> for the Ginzburg-Landau equation with magnetic field.
In higher dimension, see <cit.> and <cit.> for (<ref>)
and <cit.> for an equation including the Ginzburg-Landau equation with magnetic field and (<ref>).

One may use the change of unknown

\[
\BU(x) \assign ( 1+ \mathfrak{c}^2 \ep^2 /4 )^{-1/2} \ex^{i \mathfrak{c} x_2/2} U(x) ,
\quad \quad \quad
\e = \ep ( 1+ \mathfrak{c}^2 \ep^2/4 )^{-1/2}
\]

to transform the equation (<ref>) without the transport term into the equation
(<ref>) with the transport term. However, the assumptions
$ E_\ep (U, B(0, R_0) ) \ls \eta \lvert \ln \ep \rvert $ and $ E_\e ( \BU, B (0, R_0) ) \ls \eta \lvert \ln \e \rvert $
are not equivalent (due to the extra phase term).

§.§ Vortex structure for quasi-minimizers of $E$ at fixed $P$

In this section, some $ \Lambda_0 > 0 $ is fixed and we consider a large momentum
$ \mathfrak{p} $ and $ u_\mathfrak{p} $ such that

\begin{equation}
\label{upper_bound}
E( u_\mathfrak{p} ) \ls 2\pi \ln \mathfrak{p} + \Lambda_0
\end{equation}

and such that there exists $ c_\mathfrak{p} > 0 $ (depending on $ u_\mathfrak{p} $) such that

\[
0 = (\tmop{TW}_{c_\mathfrak{p}} ) (u_\mathfrak{p})
= - i c_\mathfrak{p} \partial_{x_2} u_\mathfrak{p} - \Delta u_\mathfrak{p} - (1 - | u_\mathfrak{p} |^2) u_\mathfrak{p} .
\]

It then follows from <cit.> (see Theorem <ref>) that we may assume, using the phase shift invariance,
that $ u_\mathfrak{p} \to 1 $ at spatial infinity. In particular, we have

\[
\mathfrak{p}
= P_2 ( u_\mathfrak{p} )
= \frac{1}{2} \int_{\R^2} \langle i \p_2 u_\mathfrak{p} | u_\mathfrak{p} - 1 \rangle \, dx .
\]

Our goal is to show that $ u_\mathfrak{p}$ satisfies the hypothesis of Proposition <ref>.
We shall follow <cit.> and <cit.> in order to analyze the vortex structure of $ u_\mathfrak{p} $.

§.§.§ Localizing the vorticity set at scale $ x / \mathfrak{p} $

We define the following rescaling $ \hat{u}_\mathfrak{p} $ of $ u_\mathfrak{p} $:

\begin{equation}
\label{defuhat}
\hat{u}_\mathfrak{p} ( \hat{x} ) = u_\mathfrak{p} ( \mathfrak{p} \hat{x} ) .
\end{equation}

Therefore, $ \hat{u}_\mathfrak{p} $ solves

\begin{equation}
\label{TW_resc}
\Delta \hat{u}_\mathfrak{p} + i c_\mathfrak{p} \mathfrak{p} \p_2 \hat{u}_\mathfrak{p}
+ \mathfrak{p}^2 \hat{u}_\mathfrak{p} ( 1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 ) = 0
\end{equation}

which is a particular case of (<ref>) with

\[
\ep = 1 / \mathfrak{p},
\quad \quad \quad
\mathfrak{c} = c_\mathfrak{p} \mathfrak{p} .
\]

The universal $ L^\ii $ bound on the gradient of Corollary <ref> reads now

\begin{equation}
\label{LinfGradResc}
\| \nabla \hat{u}_\mathfrak{p} \|_{L^\ii (\R^2) } \ls K \mathfrak{p} .
\end{equation}

We shall have, in the end, $ c_\mathfrak{p} \sim 1/ \mathfrak{p} $.
The first step provides a rough upper bound for the speed $ c_\mathfrak{p} $ (the Lagrange
multiplier for the minimisation problem $ E_{\rm min} (\mathfrak{p} ) $).

Step 1: there exists $ \mathfrak{p}_1 = \mathfrak{p}_1 ( \Lambda_0) $ such that, for
$ \mathfrak{p} \gs \mathfrak{p}_1 $, we have

\[
0 \ls c_\mathfrak{p} \ls \frac{2E ( u_\mathfrak{p} ) }{ \mathfrak{p} }
\ls 13 \frac{ \ln \mathfrak{p} }{ \mathfrak{p} } .
\]

In particular, $ c_\mathfrak{p} \ls 1/2 $ and $ \ln \mathfrak{p} \ls 2 \lvert \ln c_\mathfrak{p} \rvert $.

We shall use the Pohozaev identity (<ref>), that is:

\[
\frac{1}{2} \int_{\R^2} (1 - \lvert u_\mathfrak{p} \rvert^2 )^2 \, dx
= c_\mathfrak{p} \mathfrak{p} .
\]

At this stage, we only have the rough upper bound $ 0 \ls \frac{1}{4} \int_{\R^2} (1 - \lvert u_\mathfrak{p} \rvert^2 )^2 \, dx
\ls E ( u_\mathfrak{p} ) \ls  2\pi \ln \mathfrak{p} + \Lambda_0 $, which concludes.

Another argument we could use for minimizers is that we know from <cit.> (see also <cit.>)
that $ 0 \ls c_\mathfrak{p} \ls d^+ E_{\rm min} (\mathfrak{p} ) \ls E_{\rm min} (\mathfrak{p} ) / \mathfrak{p} $.

Step 2: there exists $ \mathfrak{p}_2 > \mathfrak{p}_1 $, $ R_* \gs 1/ 8 $ and $ n_{*} \in \N $,
depending only on $ \Lambda_0 $, such that, if $ \mathfrak{p} > \mathfrak{p}_2 $, there exist $ n_\mathfrak{p} $ points
$ \hat{z}_{ \mathfrak{p} , j } $, $ 1 \ls j \ls n_\mathfrak{p} $ with $ n_\mathfrak{p} \ls n_* $ such that
$ \{ \lvert\hat{ u}_\mathfrak{p}( \hat{x} ) \rvert \ls 1/2 \} \subset \cup_{j=1}^{n_\mathfrak{p}} B( \hat{z}_{\mathfrak{p} ,j} , R_* ) $
and the disks $ \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 4 R_* ) $, $ 1 \ls j \ls n_\mathfrak{p} $, are mutually disjoint.

We apply Theorem <ref> with $ \ep = 1/ \mathfrak{p} $, $ \mathfrak{c} = c_\mathfrak{p} \mathfrak{p} $
and $ \sigma = 1/2 $ to $ \hat{u}_\mathfrak{p} $.
This is possible in view of the upper bound on $ 0 \ls c_\mathfrak{p} \mathfrak{p} \ls 13 \ln \mathfrak{p} $ of Step 1
(that is $ M_0 = 13 $). We then let $ R_0 \assign 1 / (1+ 13 ) = 1/14 $ for $ \mathfrak{p} \gs \mathfrak{p}_1 $
and denote $ \eta_{1/2} $ the positive constant $ \eta $ given by Theorem <ref>.

We now proceed in this way: we choose (if it exists) some $ \hat{z}_{\mathfrak{p},1} \in \R^2 $ such that
$ \lvert \hat{u}_\mathfrak{p} ( \hat{z}_{\mathfrak{p} ,1} ) \rvert < 1/2 $. If $ \{ \lvert \hat{u}_\mathfrak{p} \rvert \ls 1/2 \} \subset
\bar{B} ( \hat{z}_{ \mathfrak{p} ,1}, 2 R_0 ) $, then we stop. If not, we choose
$ \hat{z}_{\mathfrak{p},2} \in \R^2 \setminus \bar{B} ( \hat{z}_{\mathfrak{p}, 1} , 2 R_0 )  $
such that $ \lvert \hat{u}_\mathfrak{p} ( \hat{z}^{\mathfrak{p}, 2 } ) \rvert < 1/2 $.
If $ \{ \lvert \hat{u}_\mathfrak{p} \rvert \ls 1/2 \} \subset \cup_{j=1}^{2} \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 2 R_0 ) $,
then we stop, if not, we continue. This process ends in a finite number of steps (depending only on $K_0 $) since,
by construction, the disks $ \bar{B} ( \hat{z}_{\mathfrak{p}, j} , R_0 ) $, $ 1 \ls j \ls n $, are
pairwise disjoint, hence, by Theorem <ref>, we have

\begin{align*}
2 \pi \ln \mathfrak{p} + K_0
\gs E( u_\mathfrak{p} )
= E_{1/ \mathfrak{p} } ( \hat{u}_\mathfrak{p} )
\gs \sum_{j=1}^n E_{1/ \mathfrak{p} } ( \hat{u}_\mathfrak{p} , B(\hat{z}_{\mathfrak{p}, j} , R_0 ) )
\gs n \times \eta_{1/2} \ln \mathfrak{p} ,
\end{align*}

which implies

\[
n \ls \frac{ 2 \pi \ln \mathfrak{p} + K_0 }{ \eta_{1/2} \ln \mathfrak{p} }
\ls \frac{ 7 }{ \eta_{1/2} }
\]

for $ \mathfrak{p} $ large enough, say $ \mathfrak{p} \gs \mathfrak{p}_2 $.

At this stage, the disks $ B (\hat{z}_{\mathfrak{p}, j} , R_0 ) $, $1 \ls j \ls n_\mathfrak{p} $, cover the vorticity set
$ \{ \lvert \hat{u}_\mathfrak{p} \rvert \ls 1/2 \} $, but the disks $ \bar{B} (\hat{z}_{\mathfrak{p} ,j} , 4 R_0 ) $ may not
be pairwise disjoint. To get this property, we argue as in <cit.> (Theorem IV.1). Let us recall the
idea: if the disks $ \bar{B} (\hat{z}_{\mathfrak{p}, j} , 4 R_0 ) $, $1 \ls j \ls n_\mathfrak{p} $ are pairwise disjoint, then
we are done with $ R_* = R_0 $. If not, then we have, for instance,
$ \lvert \hat{z}_{\mathfrak{p},1} - \hat{z}_{\mathfrak{p},2} \rvert \ls 8 R_0 $. We then remove the disk
$ B (\hat{z}_{\mathfrak{p},1} , 4 R_0 ) $ from the list and set $ R_1 \ddef 9 R_0 $. The disks
$ B (\hat{z}_{\mathfrak{p},j} , R_1 ) $, $2 \ls j \ls n_\mathfrak{p} $ cover
$ \cup_{1 \ls j \ls n_\mathfrak{p} } B (\hat{z}_{\mathfrak{p}, j} , R_0 ) $, hence the vorticity set
$ \{ \lvert \hat{u}_\mathfrak{p} \rvert \ls 1/2 \} $, and their number has decreased. In a finite number of steps
(depending only on $K_0 $), we obtain the conclusion. The radius $ R_* $ is necessarily
$ \ls R_0 \times 9^{n_\mathfrak{p}} \ls R_0 \times 9^{n_* } $.

Similar arguments are given in <cit.>, whereas in <cit.> the vorticity
set is included in some disks of radii of order $ c_\mathfrak{p}^\gamma $, which requires some extra work.

Step 3: we have

\[
% \sup_{ \hat{X} \in \R^2 } \mathfrak{p}^2 \int_{D(\hat{X}, 1 ) } ( 1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
%= o_{ \mathfrak{p} \to+\ii } (\ln \mathfrak{p} )
\mathfrak{p}^2 \int_{\R^2 } ( 1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
= o_{ \mathfrak{p} \to+\ii } (\ln \mathfrak{p} ) .
\]

This follows exactly as in <cit.> (see Proposition A.1 in the Appendix there). Notice that
the result in <cit.> is stated for the potential on a compact set in a domain $ \Omega $,
but it holds as well in the entire plane.

We then define, as in <cit.>, the function $ \hat{u}'_\mathfrak{p} : \R^2 \to \C $ by

\[
\hat{u}'_\mathfrak{p} (\hat{x} ) \ddef
\left\{\begin{array}{ll}
\hat{u}_\mathfrak{p} ( \hat{x} ) & \textrm{if } \hat{x} \in \cup_{j=1}^{n_\mathfrak{p}} \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 2 R_* )
\\
\frac{ \hat{u}_\mathfrak{p} ( \hat{x} ) }{ \lvert \hat{u}_\mathfrak{p} (\hat{x}) \rvert }
& \textrm{if } \hat{x} \not\in \cup_{j=1}^{n_\mathfrak{p}} \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 3 R_*)
\\
( 3 - \lvert \hat{x} - \hat{z}_{\mathfrak{p}, j} \rvert / R_* ) \hat{u}_\mathfrak{p} ( \hat{x} )
+ ( -2 + \lvert \hat{x} - \hat{z}_{\mathfrak{p},j} \rvert / R_* ) \frac{ \hat{u}_\mathfrak{p} ( \hat{x} ) }{ \lvert \hat{u}_\mathfrak{p} (\hat{x} )\rvert}
& \textrm{if } \hat{x} \in \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 3 R_* ) \setminus \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 2 R_* )
\end{array}
\right.
\]

for some $1 \ls j \ls n_\mathfrak{p} $ (this last formula is valid since the disks $ \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 4 R_* ) $,
$ 1 \ls j \ls n_\mathfrak{p} $, are mutually disjoint).

Step 4: we have, as $ \mathfrak{p} \to +\ii $,

\[
E_{1/ \mathfrak{p} } ( \hat{u}'_\mathfrak{p} )
\ls 2 \pi \ln \mathfrak{p} + o ( \ln \mathfrak{p} ) .
\]

Letting $ \Omega_{R} \ddef \cup_{j=1}^{n_\mathfrak{p}} \bar{B} ( \hat{z}_{\mathfrak{p}, j} , R ) $, we have

\begin{align*}
\int_{\R^2} (1 - \lvert \hat{u}'_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
& = \int_{\Omega_{2R_*} } (1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
+ \int_{\Omega_{3R_*} \setminus \Omega_{2R_*} } (1 - \lvert \hat{u}'_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
\end{align*}

We notice that in $ \Omega_{3 R_*} \setminus \Omega_{2R_*} $, say for
$ \hat{x} \in \bar{B} ( \hat{z}_{\mathfrak{p},j} , 3 R_* ) \setminus \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 2 R_* ) $, we have

\[
\lvert \hat{u}'_\mathfrak{p} ( \hat{x} ) \rvert
= ( 3 - \lvert \hat{x} - \hat{z}_{\mathfrak{p}, j} \rvert / R_* ) \lvert \hat{u}_\mathfrak{p} (\hat{x} ) \rvert
+ ( -2 + \lvert \hat{x} - \hat{z}_{ \mathfrak{p}, j} \rvert / R_* )
\in [ \lvert \hat{u}_\mathfrak{p} (\hat{x} ) \rvert , 1 ]
\]

$ \big\lvert 1 - \lvert \hat{u}'_\mathfrak{p} ( \hat{x} ) \rvert^2 \big\rvert \ls \big\lvert 1 - \lvert \hat{u}_\mathfrak{p} ( \hat{x} ) \rvert^2 \big\rvert $ and thus

\begin{align}
\label{potard_1}
\int_{\R^2} (1 - \lvert \hat{u}'_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
& \ls \int_{\Omega_{2R_*} } (1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
+ \int_{\Omega_{3R_*} \setminus \Omega_{2R_*} } (1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2 \, d\hat{x}
\nonumber \\ &
= \int_{\Omega_{ 3R_*} } (1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2 \, d\hat{x} .
\end{align}

For the kinetic term, we have

\[
\lvert \nabla \hat{u}'_\mathfrak{p} (\hat{x} ) \rvert^2
\lvert \nabla \hat{u}_\mathfrak{p} ( \hat{x} ) \rvert^2
\]

if $ \hat{x} \in \Omega_{2R_*} $. Outside $ \cup_{j=1}^{n_\mathfrak{p}} \bar{B} ( \hat{z}_{\mathfrak{p}, j} , R_* ) $, then
$ \lvert \hat{u}_\mathfrak{p} \rvert \gs 1/2 $ and we may then lift, at least locally,
$ \hat{u}_\mathfrak{p} = A \ex^{i \phi } $ and get

\[
\lvert \nabla \hat{u}_\mathfrak{p} \rvert^2
= A^2 \lvert \nabla \phi \rvert^2 + \lvert \nabla A \rvert^2 .
\]

If $ \hat{x} \not\in \Omega_{3 R_*} $, then, by (<ref>),

\begin{align*}
\lvert \nabla \hat{u}'_\mathfrak{p} \rvert^2
= \lvert \nabla \phi \rvert^2
= A^2 \lvert \nabla \phi \rvert^2 + \frac{1 - A^2}{A^2} \times A^2 \lvert \nabla \phi \rvert^2
\ls \lvert \nabla \hat{u}_\mathfrak{p} \rvert^2 + 4 K \mathfrak{p} \lvert 1 - A^2 \rvert \times \lvert \nabla \hat{u}_\mathfrak{p} \rvert
\end{align*}

since $ A = \lvert \hat{u}_\mathfrak{p} \rvert \gs 1/2 $ outside $ \Omega_{R_*} $. Finally, in
$ \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 3 R_* ) \setminus \bar{B} ( \hat{z}_{\mathfrak{p}, j} , 2 R_* ) $ (for some
unique $1 \ls j \ls n_{\mathfrak{p}} $), we have

\begin{align*}
\lvert \nabla \hat{u}'_\mathfrak{p} \rvert^2
& = \lvert \nabla \phi \rvert^2 \Big( ( 3 - \lvert \hat{x} - \hat{z}_{\mathfrak{p}, j} \rvert / R_* ) A
+ ( -2 + \lvert \hat{x} - \hat{z}_{\mathfrak{p}, j} \rvert / R_* )  \Big)^2
\\& \quad
+ \Big\lvert \nabla \Big[ ( 3 - \lvert \hat{x} - \hat{z}_{\mathfrak{p},j} \rvert / R_* ) A
+ ( -2 + | \hat{x} - \hat{z}_{\mathfrak{p},j} | / R_* ) \Big] \Big\rvert^2 .
\end{align*}

We then use that, since $ \lvert \hat{u}_\mathfrak{p} (\hat{x} ) \rvert \gs 1/2 $ and letting
$ \theta = 3 - | \hat{x} - \hat{z}_{\mathfrak{p}, j} | / R_* \in [ 0 ,1] $,

\begin{align*}
\lvert \nabla \phi \rvert^2
\Big[ & ( 3 - | \hat{x} - \hat{z}_{\mathfrak{p}, j} | / R_* ) A + ( -2 + | \hat{x} - \hat{z}_{\mathfrak{p}, j} | / R_* ) \Big]^2
\\ &
= A^2 \lvert \nabla \phi \rvert^2 \times \frac{1}{A^2} [ 1 + \theta (A-1) ]^2
\ls A^2 \lvert \nabla \phi \rvert^2 \times ( 1 + K \lvert A^2 -1 \rvert )
\\ &
\ls A^2 \lvert \nabla \phi \rvert^2 + K \mathfrak{p} \lvert \nabla \hat{u}_\mathfrak{p} \rvert \times \lvert A^2 -1 \rvert ,
\end{align*}

by Corollary <ref>. On the other hand, since $ \lvert \cdot \rvert $ is $1$-Lipschitz continuous,

\begin{align*}
\Big\lvert & \nabla \Big[ ( 3 - | \hat{x} - \hat{z}_{\mathfrak{p},j} | / R_* ) A
+ ( -2 + | \hat{x} - \hat{z}_{\mathfrak{p}, j} | / R_* ) \Big] \Big\rvert^2
\\ &
\ls \frac{1}{R_*^2 } \lvert 1- A \rvert^2
+ \lvert \nabla A \rvert^2
+ \frac{2}{R_* } \lvert 1- A \rvert \times \lvert \nabla A \rvert
\\ &
\ls
\lvert \nabla A \rvert^2
+ K  ( A^2 -1 )^2
+ K \lvert \nabla A \rvert \times \lvert A^2 -1 \rvert .
\end{align*}

Therefore, by Cauchy-Schwarz inequality, for some absolute constant $K > 0$,

\begin{align*}
\int_{\R^2} \lvert \nabla \hat{u}'_\mathfrak{p} \rvert^2 \, d\hat{x}
& \ls
\int_{\R^2 } \lvert \nabla \hat{u}_\mathfrak{p} \rvert^2 \, d\hat{x}
+ K \Big( \int_{\R^2} \mathfrak{p}^2 ( 1- \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2  \, d\hat{x} \Big)^{1/2}
\Big( \int_{\R^2} \lvert \nabla \hat{u}_\mathfrak{p} \rvert^2 \, d\hat{x}  \Big)^{1/2}
+ K \int_{\R^2} ( 1- \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2  \, d\hat{x} .
\end{align*}

Combining this with (<ref>) yields

\begin{align*}
E_{ 1/ \mathfrak{p}} (\hat{u}'_\mathfrak{p} )
\ls E_{ \mathfrak{p}} (\hat{u}_\mathfrak{p} ) + K \sqrt{ E_{ \mathfrak{p}} (\hat{u}_\mathfrak{p} )}
\Big( \int_{\R^2} \mathfrak{p}^2 ( 1- \lvert \hat{u}_\mathfrak{p} \rvert^2 )^2  \, d\hat{x}  \Big)^{1/2}
+ K \frac{E_{ \mathfrak{p}} (\hat{u}_\mathfrak{p} ) }{\mathfrak{p}^2}
\ls 2 \pi \ln \mathfrak{p} + o (\ln \mathfrak{p} ) ,
\end{align*}

by the upper bound (<ref>) and the estimate for the potential term of Step 3.

Step 5: we claim that for any $ \delta \in ] 0 , \pi /2 [ $, there exist $ \mathfrak{p}^\dag_\delta > \mathfrak{p}_2 $
such that we are in one of the following cases:

case (I) for any $ 1 \ls j \ls n_\mathfrak{p} $,

\[
\| J \hat{u}'_\mathfrak{p} \|_{ [ \BC^{0,1}_c( B( \hat{z}_{\mathfrak{p}, j} , 4R_*) ) ]^* }
\ls \delta
\]

case (II) there exists (up to a relabelling) two points $ \hat{y}_{\mathfrak{p}, \pm } \in \R^2 $,
depending on $ \hat{u}_\mathfrak{p} $, such that

\[
\max_{1 \ls j \ls n_\mathfrak{p} }
\Big\| J \hat{u}'_\mathfrak{p} - \pi ( \delta_{y_{\mathfrak{p}, + } } - \delta_{y_{\mathfrak{p}, - } })
\Big\|_{ [ \BC^{0,1}_c( B( \hat{z}_{\mathfrak{p}, j} , 4R_*) ) ]^* }
\ls \delta
\]

We apply Theorem <ref> to $ \hat{u}_\mathfrak{p}' $ on each disk
$ B ( \hat{z}_{\mathfrak{p}, j} , 4 R_* ) $, $ 1 \ls j \ls n_\mathfrak{p} $. This yields
points $ \hat{y}_{\mathfrak{p}, j,k} \in \{ \lvert \hat{u}_\mathfrak{p} \rvert \ls 1/2 \}
\subset B ( \hat{z}_{\mathfrak{p}, j} , R_* ) \subset B ( \hat{z}_{\mathfrak{p}, j} , 4R_* ) $ and integers
$ d_{\mathfrak{p}, j,k} \in \Z $, $ 1 \ls k \ls N_{\mathfrak{p}, j} $ such that

\begin{equation}
\label{step4-1}
\Big\| J \hat{u}'_\mathfrak{p} - \pi \sum_{k=1}^{N_{\mathfrak{p}, j}}
d_{\mathfrak{p}, j,k} \delta_{\hat{y}_{\mathfrak{p}, j, k}} \Big\|_{ [ \BC^{0,1}_c( B ( \hat{z}_{\mathfrak{p}, j} , 4R_* ) ) ]^*}
\ls \delta
\end{equation}

\begin{equation}
\label{step4-2}
\pi \sum_{k=1}^{N_{ \mathfrak{p} , j }}
\lvert d_{\mathfrak{p}, j,k} \rvert
\ls \frac{E_{1/\mathfrak{p}} (\hat{u}'_\mathfrak{p}, B (\hat{z}_{\mathfrak{p},j} , 4R_* )) }{ \ln \mathfrak{p} } + \delta .
\end{equation}

By summing over $ 1 \ls j \ls n_\mathfrak{p} $ the inequalities (<ref>), we infer

\[
\pi \sum_{j=1}^{n_\mathfrak{p}} \sum_{k=1}^{N_{ \mathfrak{p} , j}}
\lvert d_{\mathfrak{p}, j,k} \rvert
\ls \frac{E_{1/\mathfrak{p}} ( \hat{u}'_\mathfrak{p} , \Omega_{4R_*} ) }{ \ln \mathfrak{p} } + \delta
\ls 2.5 \pi
\]

by using $ \delta < \pi /2 $ and Step 3, and for $ \mathfrak{p} $ large enough.

\begin{equation}
\label{Step4-0}
\sum_{j=1}^{n_\mathfrak{p}} \sum_{k=1}^{N_{ \mathfrak{p} , j}}
\lvert d_{\mathfrak{p}, j,k} \rvert
\ls 2
\end{equation}

and two cases may occur: all the integers $ d_{\mathfrak{p}, j,k} $ are zero (this is Case (I))
or at least one of the integers $ d_{\mathfrak{p}, j,k} $ is not zero.

In addition, we have, for $ 1 \ls j \ls n_\mathfrak{p} $,

\begin{equation}
\label{degreoudeforce}
\sum_{k=1}^{N_{ \mathfrak{p}, j } } d_{\mathfrak{p}, j, k}
= {\rm deg} ( \hat{u}_\mathfrak{p} , \p B ( \hat{z}_{\mathfrak{p} , j} , 3 R_*) )  .
\end{equation}

Indeed, since $ \lvert \hat{u}_\mathfrak{p}' \rvert = 1 $ on $ B ( \hat{z}_{\mathfrak{p} , j} , 4 R_*)
\setminus B ( \hat{z}_{\mathfrak{p} , j} , 3 R_*) $, we have $ J\hat{u}_\mathfrak{p}' = 0 $ there.
Therefore, by fixing $ \chi \in \BC^\ii_c ( B (0, 4 R_*) ) $ such that $ \chi \equiv 1 $ on $ \bar{B} (0,3 R_*) $,
we deduce

\begin{align*}
\Big\lvert
\sum_{k=1}^{N_{ \mathfrak{p}, j } } d_{\mathfrak{p}, j, k}
- {\rm deg} ( \hat{u}_\mathfrak{p} , \p B ( \hat{z}_{\mathfrak{p} , j} , 3 R_*) ) \Big\rvert
& = \Big\lvert
\int_{ B (\hat{z}_{ \mathfrak{p}, j}, 3 R_* )} \sum_{k=1}^{N_{ \mathfrak{p}, j } }
d_{\mathfrak{p}, j, k} \delta_{\hat{y}_{\mathfrak{p}, j, k} } \, d \hat{x}
- \frac{1}{\pi} \int_{ B (\hat{z}_{\mathfrak{p}, j}, 4 R_* )} J \hat{u}'_\mathfrak{p} \, d \hat{x}
\Big\rvert
\\ &
= \frac{1}{\pi} \Big\lvert
\int_{ B (\hat{z}_\mathfrak{p}^j, 4 R_* )} \chi ( \hat{x} - \hat{z}_{\mathfrak{p}, j} )
\Big( \sum_{k=1}^{N_{ \mathfrak{p}, j } } d_{\mathfrak{p}, j, k} \delta_{\hat{y}_{\mathfrak{p}, j, k} } - J \hat{u}'_\mathfrak{p}
\Big) \, d \hat{x} \Big\rvert
\\ &
\ls \frac{1}{\pi} \| \chi \| \times
\Big\| J \hat{u}'_\mathfrak{p} - \pi \sum_{k=1}^{N_{\mathfrak{p}, j}}
d_{\mathfrak{p}, j,k} \delta_{\hat{y}_{\mathfrak{p}, j, k}} \Big\|_{ [ \BC^{0,1}_c( \bar{D} ( \hat{z}_{\mathfrak{p}, j} , 4R_* ) ) ]^*}
\end{align*}

by (<ref>).
Since the left-hand side is an integer and the right-hand side is $ \ls 1/2 $ provided
$ \mathfrak{p} \gs \mathfrak{p}_{2,1}(\delta, \Lambda_0) $, (<ref>) follows.

We finally notice that the degree of $ \hat{u}'_\mathfrak{p} $ on some large circle $ \p D (0, R ) $
(with $ R \gg \max_{1 \ls j \ls n_\mathfrak{p} } \lvert \hat{z}_{\mathfrak{p}, j} \rvert $) is zero, for otherwise
$ \hat{u}'_\mathfrak{p} $ (and $ \hat{u}_\mathfrak{p} $) would have infinite kinetic energy. Therefore,

\[
0 =
\sum_{ j= 1}^{ n_\mathfrak{p}} {\rm deg} ( \hat{u}_\mathfrak{p} , \p D ( \hat{z}_{\mathfrak{p} , j} , 3 R_*) )
= \sum_{ j= 1}^{ n_\mathfrak{p}} \sum_{k=1}^{N_{ \mathfrak{p}, j } } d_{\mathfrak{p}, j, k} .
\]

Combining this with (<ref>), we deduce that if we are not in Case (I), then one of the
$ d_{\mathfrak{p}, j,k} $ must be equal to $+1 $ and another one must be equal to $ -1$, which is Case (II).

Notice that for Case (II), if $ D ( \hat{z}_{\mathfrak{p},j} , 4 R_* ) $ contains neither $ y_{\mathfrak{p}, + } $
nor $ y_{\mathfrak{p}, - } $, then
$ \| J \hat{u}'_\mathfrak{p} \|_{ [ \BC^{0,1}_c( D( \hat{z}_{\mathfrak{p},j} , 4R_*) ) ]^* } \ls \delta $.

As in <cit.>, we now relate the location of the points $ \hat{y}_{\mathfrak{p}, \pm } $ to the
momentum $ P( \hat{u}_\mathfrak{p} ) $.

Step 6: Case (I) does not occur for $ \mathfrak{p} $ sufficiently large, say $ \mathfrak{p} \gs \mathfrak{p}_3 $.
In addition, we have

\[
1 = P ( \hat{u}_\mathfrak{p} )
= \pi \big( ( \hat{y}_{\mathfrak{p}, + } )_1 - ( \hat{y}_{\mathfrak{p}, -} )_1 \big) + o(1) .
\]

First, we have, by computations similar to those of Step 3, $ \hat{u}_\mathfrak{p} = A \ex^{i\vp} $
locally outside $ \Omega_{R_*} $, hence
$ \langle i \hat{u}_\mathfrak{p} | \nabla \hat{u}_\mathfrak{p} \rangle = A^2 \nabla \vp $ and then,
outside $ \Omega_{3 R_*} $,

\[
\langle i \hat{u}_\mathfrak{p} | \nabla \hat{u}_\mathfrak{p} \rangle
- \langle i \hat{u}'_\mathfrak{p} | \nabla \hat{u}'_\mathfrak{p} \rangle
= A^2 \nabla \vp - \nabla \vp
= \frac{A^2 -1}{A} \times A \nabla \vp .
\]

In $ D (\hat{z}_{\mathfrak{p}, j} , 3R_* ) \setminus D (\hat{z}_{\mathfrak{p}, j} , 2R_* ) $, we obtain

\[
\lvert \langle i \hat{u}_\mathfrak{p} | \nabla \hat{u}_\mathfrak{p} \rangle
- \langle i \hat{u}'_\mathfrak{p} | \nabla \hat{u}'_\mathfrak{p} \rangle \rvert
= \lvert A^2 \nabla \vp - \lvert \hat{u}'_\mathfrak{p} \rvert^2 \nabla \vp \rvert
\ls \frac{\lvert A^2 -1 \rvert}{A} \times \lvert A \nabla \vp \rvert ,
\]

since $ \lvert \hat{u}'_\mathfrak{p} \rvert \in [ \lvert \hat{u}_\mathfrak{p} \rvert , 1 ] $. Therefore,

\begin{equation}
\label{pre-jaco}
\| \langle i \hat{u}_\mathfrak{p} | \nabla \hat{u}_\mathfrak{p} \rangle
- \langle i \hat{u}'_\mathfrak{p} | \nabla \hat{u}'_\mathfrak{p} \rangle \|_{L^1(\R^2)}
\ls K \int_{\R^2 \setminus \Omega_{2 R_*} } \big\lvert 1 - \lvert \hat{u}_\mathfrak{p} \rvert^2 \big\rvert
\times \lvert \nabla \hat{u}_\mathfrak{p} \rvert \, d \hat{x}
\ls \frac{K}{ \mathfrak{p} } E_{1/\mathfrak{p}} ( \hat{u}_\mathfrak{p} )
\ls K \frac{\ln \mathfrak{p} }{ \mathfrak{p} } .
\end{equation}

Following <cit.>, <cit.>, we write

\begin{align*}
1 = \frac{ P ( u_\mathfrak{p} ) }{ \mathfrak{p} }
= P ( \hat{u}_\mathfrak{p} )
& = \frac{1}{2} \int_{\R^2} \langle i \p_2 \hat{u}_\mathfrak{p} | \hat{u}_\mathfrak{p} - 1 \rangle \, d \hat{x}
\\
& = \frac{1}{2} \int_{\R^2} \langle i \p_2 \hat{u}'_\mathfrak{p} | \hat{u}'_\mathfrak{p} - 1 \rangle \, d \hat{x}
+ \frac{1}{2} \int_{\R^2} ( \langle i \p_2 \hat{u}_\mathfrak{p} | \hat{u}_\mathfrak{p} - 1 \rangle
- \langle i \p_2 \hat{u}'_\mathfrak{p} | \hat{u}'_\mathfrak{p} - 1 \rangle ) \, d \hat{x}
\end{align*}

For the second integral, we write that, on the one hand,

\begin{align*}
\Big\lvert \int_{\R^2} ( \langle i \hat{u}_\mathfrak{p} | \p_2 \hat{u}_\mathfrak{p} \rangle
- \langle i \hat{u}'_\mathfrak{p} | \p_2 \hat{u}'_\mathfrak{p} \rangle ) \, d \hat{x} \Big\rvert
\ls \| \langle i \hat{u}_\mathfrak{p} | \nabla \hat{u}_\mathfrak{p} \rangle
- \langle i \hat{u}'_\mathfrak{p} | \nabla \hat{u}'_\mathfrak{p} \rangle \|_{L^1(\R^2)}
\ls K \frac{\ln \mathfrak{p} }{ \mathfrak{p} } \to 0