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 [7] (see also [13]). 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 [15], [14], 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.

The (NLS) equation is associated with the Ginzburg-Landau energy which is formally conserved by the (NLS) flow.We denote by E the set of functions with finite energy, that is ), E(u) < +∞}.
Besides the energy, the momentum is another quantity formally conserved by the (NLS) flow, and is associated with the invariance by translation of (NLS).Formally, the momentum of u is 1 2 Ê 2 Re(i∇uū) ∈ Ê 2 , but its precise definition requires some care in the energy space due to the condition at infinity (see [36] in dimension larger than two and [13] in dimension two).If u ∈ 1 + C ∞ c (Ê 2 ) for instance, or if u is a travelling wave tending to 1 at infinity, then the expression of the momentum reduces to Re i∇u(ū − 1) .
In addition to the translation invariance, the (NLS) equation is also phase shift invariant, that is invariant by multiplication by a complex of modulus one, and rotation invariant.

Travelling waves for (NLS)
Following the works in the physical literature of Jones and Roberts (see [31], [30]), there has been a large amount of mathematical works on the question of existence and properties of travelling wave solutions in the (NLS) equation, that are solutions of 0 = (TW c )(u for some c > 0, corresponding to particular solutions of (NLS) of the form Ψ(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 − − → e 2 ).We refer to [6] for an overview on these problems in several dimensions.A natural approach is to look at the minimizing problem for p > 0 E min (p) := inf u∈E {E(u), P 2 (u) = p}.
It was shown by Bethuel-Gravejat-Saut in [7] that there exists a minimizer to this problem.
Theorem 1.2 ( [7]) For any p > 0, there exists a non constant function u p ∈ E and c(u p ) > 0 such that P 2 (u p ) = p, u p is a solution to (TW c(up) )(u p ) = 0 and E(u p ) = E min (p).
Furthermore, any minimizer for E min (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 (p), the compactness of minimizing sequences has been shown later on in [13] for the natural semi-distance on E D 0 (u, v) := ∇u − ∇v L 2 (Ê 2 ) + |u| − |v| L 2 (Ê 2 ) .Theorem 1.3 ( [13]) For any p > 0, and any minimizing sequence (u n ) n∈AE for E min (p), there exists a subsequence (u nj ) j∈AE , a sequence of translations (y j ) j∈AE and a non constant function u p ∈ E such that D 0 (u nj , u p ) → 0, P 2 (u nj ) → P 2 (u p ) = p and E(u nj ) → E(u p ) = E min (p) as j → +∞.In particular, there exists c(u p ) > 0 such that P 2 (u p ) = p, u p is a solution to (TW c(up) )(u p ) = 0 and Furthermore, the set S p := {v ∈ E, P 2 (v) = p and E(v) = E min (p)} of minimizers for E min (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 S p consists of a single orbit under phase shift and space translation, that is: do we have, for some minimizer U p , S p = {U p (. − X)e iγ , γ ∈ Ê, X ∈ Ê 2 }?
The main consequence of our work is to solve this open problem of uniqueness for large momentum.
In fact, we will be able to show slightly stronger results than Theorem 1.4, see Theorem 1.11 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 (p) for any p > 0. However, the numerical results given in [16] suggest that this may no longer be the case for more general nonlinearities.
In the analysis of the minimization problem in [7] (and also [13]), the following properties of E min play a key role.

A smooth branch of travelling waves for large momentum
There have been several ways of constructing travelling waves of the (NLS) equation, with different approaches.For instance, we may use variational methods, such as a mountain pass argument in [10] and in [3], or by minimizing the energy at fixed kinetic energy ( [7], [13]).Also, we have constructed in [15] a travelling wave by perturbative methods, taking for ansatz a pair of vortices, by following the Lyapounov-Schmidt reduction method as initiated in [20].Vortices are stationary solutions of (NLS) of degrees n ∈ Z * (see [25], [39], [45], [28], [12]): V n (x) = ρ n (r)e inθ , where x = re iθ , solving In the previous paper [15], we constructed solutions of (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.
Theorem 1.6 ([15], Theorem 1.1 and [14], Proposition 1.2) There exists c 0 > 0 a small constant such that for any 0 < c c 0 , there exists a solution of (TW c ) of the form where d c = 1+oc→0 (1)   c is a C 1 function of c.This solution has finite energy, that is Q c ∈ E, and Q c → 1 at infinity.Furthermore, for all 2 < p +∞, there exists c 0 (p) > 0 such that, if 0 < c c 0 (p), for the norm and the space X p := {f ∈ L p (R 2 ), ∇f ∈ L p−1 (R 2 )}, one has (1).
with the estimate Finally, we have hence the C 1 mapping is a strictly decreasing diffeomorphism from ]0, c 0 ] onto [P 2 (Q c0 ), +∞[.

Remark 1.7
With the same kind of approach, [35] also provides an existence result of travelling waves for (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 1.6, we see that we may also parametrize the branch c → Q c by its momentum 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 [31] and [16] for more general nonlinearities (see also [17]).We will show here that the construction of Theorem 1.6 are the unique, up to the natural translation and phase invariances, constrained energy minimizers.

A uniqueness result for symmetric functions
We have shown in [14] several coercivity results for the travelling waves constructed in Theorem 1.6.This will allow us to show the following uniqueness result for symmetric functions close to the branch constructed in Theorem 1. Proposition 1.8There exists λ * > 1 such that, for any λ λ * , there exists ε(λ) > 0 such that if a function u ∈ E satisfies then, there exist X ∈ R and γ ∈ R such that u = Q c (. − X e 2 )e iγ , where Q c is defined in Theorem 1.6.Remark 1.9 In view of the symmetry assumption, we may replace the second hypothesis by We will discuss the main arguments of the proof of Proposition 1.8 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 1.8 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 ±d − → e 1 only need to satisfy |dc − 1| ε(λ), but for instance both d = 1 c and d = 1 c + 1 √ c satisfy these conditions for c > 0 small enough at fixed λ.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 1.8 will rely on methods used in [14] 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 1.8 is a strong unicity result is that it implies the following local uniqueness result in L ∞ for even functions in x 1 .
Corollary 1.10There exist c 0 , ε > 0 such that, for 0 We may now state our main result.It establishes that any travelling wave solution which is even in x 1 and within O(1) of the minimizing energy must be, for large momentum, the Q c travelling wave constructed in Theorem 1.6, up to the natural translation and phase invariances.
where Q c is defined in Theorem 1.6.In particular, P 2 (u) = P(c) (where P is defined in Theorem 1.6).
Section 3 is devoted to the proof of this result.We show there that a function satisfying the hypothesis of Theorem 1.11 also satisfies the hypothesis of Proposition 1.8.Our result applies in particular to the constraint minimizers for the problem E min (p), for large p.
Proof.By a first translation in x 1 , we may assume, by Theorem 1.2, that this minimizer U is even in x 1 .By Proposition 1.5, the last hypothesis 4 of Theorem 1.11 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 ) = p = P 2 (Q c ), thus c = P −1 (p).✷ Theorem 1.4 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 1.6 coincides with the global energy minimizer at fixed momentum (up to translation and phase shift).Theorem 1.13There exists c * > 0 such that, for 0 < c c * , Q c is a minimizer for E min (P 2 (Q c )).Moreover, there exists p 0 > 0 such that the following statements hold.

3.
For any p p 0 , Q P −1 (p) is orbitally stable for the semi-distance D 0 (or, equivalently, for 0 < c c * , Q c is orbitally stable for the semi-distance D 0 ).

4.
For p p 0 and any minimizer u for E min (p), then, up to a space translation and a phase shift, u enjoys the symmetry in addition to the symmetry in x 1 .
Proof.By Theorems 1.2 and 1.3, we have existence of at least one minimizer U p for E min (p), whatever is p > 0.
For large p, by applying Corollary 1.12, we have U p = Q c (• − X)e iγ for some X ∈ Ê 2 and γ ∈ Ê, thus proving that Q c is a minimizer for E min (p) and that P 2 (Q c ) = P(c) = p.For 1., it suffices to notice that, in view of Corollary 1.12 applied to any minimizer (we have existence by Theorems 1.2 and 1.3) E min (p) = E(Q P −1 (p) ).We then conclude by using that P is a C 1 diffeomorphism and that in view of the Hamilton like relation (formally shown in [31] and rigorously proved for the branch constructed in Theorem 1.6 in [14]) Since 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 [14].Integration would yield E min (p) ∼ 2π ln p, but we may slightly improve this estimate.Indeed, Proposition 1.5 gives E min (p) 2π ln p + O(1), and the lower bound is a straightforward consequence of Theorem 3.4 (i) and the study in subsection 3.2.3.
Statement 2. is a rephrasing of Corollary 1.12 combined with the existence of at least one constrained minimizer.Statement 3. is then a direct consequence of Theorem 1.3.Statement 4. simply follows from the fact that Q c enjoys by construction this symmetry (see [15]).Finally, statement 5. is also a rephrasing of Theorem 1.11.✷ Remark 1.14 Concerning the stability stated in statement 3. in the above theorem, we quote the work [34], where a linear "spectral" stability result is proved (through ad hoc hypotheses that have been checked in [14]), namely that the linearized equation i∂ t v = L Qc (v) does not have exponentially growing solutions (in Ḣ1 (Ê 2 ; ), say).Statement 3. in the above theorem does not rely on the result in [34], and is for the nonlinear (orbital) stability (following the Cazenave-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 1.6, 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 [19].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 1.6 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 [14]), but then the coercivity constant goes to 0 when c → 0. This would pose a problem for the uniqueness result.The last statement in Theorem 1.13 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 [14] 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 [31], [16] and [17] assume the two symmetries.

The travelling wave Q c and two other variational characterizations
Before providing other variational characterizations of Q c , we have to define a distance on the energy space E. One can use (see [24]) which is adapted to the Cauchy problem.Actually, we may also use the pseudo-distance Is it shown in [13], Corollary 4.13 there, that both the energy E and the momentum P 2 are continuous for the distance D E , and actually even for the pseudo-distance D 0 .
The travelling wave Q c as a mountain pass solution.Thanks to the results in Theorem 1.13, it is easy to show that we have locally, near Q c , a mountain-pass geometry.Indeed, let c * > 0 be small, and define (1.2) Indeed, let υ ∈ Υ c * .By the intermediate value theorem, there exists Moreover, considering the particular in view of the Hamilton group relation [14]).Since d dc P 2 (Q c ) < 0, we deduce that (E − c * P 2 )(υ * (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 [14] and the above mentioned Hamilton group relation As a consequence, we have implying that u is a minimizer for E min (P 2 (Q c * )), hence there exist γ ∈ Ê and X ∈ Ê 2 such that u = Q c * (•− X)e iγ , 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 [10], the mountain pass is implemented in the space 1 + H 1 (Ê 2 ) whereas we know (by the result in [27]) that the nontrivial traveling wave do not belong to this affine space; in [3], the solution is constructed by working first on [−N, +N ] × Ê and then passing to the limit, and it is then not immediate to compute the functional E − cP 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 − cP 2 for fixed kinetic energy.In [13], for κ 0, the following variational problem is investigated: Any minimizer v for I min (κ) is such that there exists c > 0 satisfying (TW c )(v(•/c)) = 0.In 2d and for the Ginzburg-Landau nonlinearity, existence of minimizers for κ > 0 is established in Theorem 1.2 there.Furthermore, it is shown in [13] (see Proposition 8.4 there) that if p > 0 and if U is a minimizer for E min (p) with speed c, then U (c •) is a minimizer for I min (κ) with κ = 1 2 Ê 2 |∇U | 2 dx (this last quantity is scale-invariant in 2d) and I min is differentiable at this κ, with we shall conclude that I min is of class C 1 on [κ 0 , +∞[, and that (by the arguments in [13]), the only minimizer for κ = 1 2 Ê 2 |∇Q c | 2 dx (for some suitable c ∈]0, c 0 ]) is Q c (c •) up to the natural translation and phase invariances and, in addition, I ′ min (κ) = −1/c 2 .In order to prove that statement, it suffices to use the Pohozaev identity (2.2) and deduce Therefore, by using the Hamilton like relation and then the asymptotics of c → P 2 (Q c ) obtained in [14], we arrive at and this concludes.
The paper is organized as follows.In section 2, we give the proof of the uniqueness result given in Proposition 1.8.Section 3 is devoted to the vortex analysis of travelling waves with energy E min (p) + O(1) and even in x 2 , in order to show that they satisfy the hypotheses of Proposition 1.8.Subsection 3.4 contains a few remarks on the nonsymmetrical case.Finally, in subsection 3.3, we provide some decay estimates slightly away from the vortices.For the Ginzburg-Landau (stationary) model, such estimates have been first given in [37] for minimizing solutions and later generalized in [18] to non-minimizing solutions.They improve some estimates in [15] and are not specific to the way we construct the solutions.

Proof of the local uniqueness result (Proposition 1.8)
This section is devoted to the proof of Proposition 1.8 and Corollary 1.10.The proof of Proposition 1.8 uses arguments from the proof of Theorem 1.14 of [14], another local uniqueness result for this problem, but in different spaces.We explain here the core ideas of the proof.
Let us explain schematically the proof of Proposition 1.8.We first pick c ′ , X, γ ′ in such a way that Q = Q ′ c (. − X)e iγ has the same vortices as u.This is possible because c → d c , the position of the vortices, is smooth.We then decompose u = Qe ψ , where ψ 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 ψ is then , where we regroup the linear terms in L and the nonlinear terms in NL, and (TW c )(Q) = 0 because c = c ′ .We then take the scalar product of this equation with ψ, and we get 0 = (TW c )(Q), ψ + B Q (ψ) + NL(ψ), ψ .Now, the coercivity of B Q has been studied in [14].It holds (for even functions in x 1 ) up to three orthogonality conditions, that can be satisfied by changing slightly the modulation parameters c ′ , X, γ.We deduce that B Q (ψ) K ψ 2  1 for some norm • 1 .There are two main difficulties at this point.First, since the hypothesis on u in Proposition 1.8 are weak, we simply have ||ψ|| 1 < +∞, but not the fact that it is small.Therefore, an estimate of the form would not be enough to conclude.Secondly, the norm • 1 is rather weak, and in fact NL(ψ), ψ can not be controlled by powers of ψ 1 .
Concerning the term (T W c )(Q), ψ , we may show that we always have Then, even if ψ 1 is not small, by the hypothesis of Proposition 1.8, ψ will be small in other (non equivalent) norms.Let us write one of them • 2 .Our goal is then to show an estimate of the form 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 ∆ψ, which also appears in the equation 0 . We thus replace it using the equation, which leads to another term containing two derivatives (from NL(ψ)), and other terms that can be successfully estimated.After n such integration by parts, we have an estimate of the form , where • 3 is another (semi-)norm in which ψ is not necessarily small.Now, taking n large enough (depending on ψ), since , concluding the proof.The problem is a lot simpler near the vortices.There, we write u = Q + φ and the coercivity norm is equivalent to the H 1 norm, and the hypothesis of Proposition 1.8 gives us φ L ∞ = 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 2.6 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 1.6
We recall here properties on the branch c → Q c from Theorem 1.6, coming mainly from [14] and [15].In this section, we will use the notation Re(f ḡ).

Properties of vortices
We start with some estimates on vortices, that compose the travelling wave (see Theorem 1.6).

Toolbox
We list in this section some results useful for the analysis of travelling waves for not necessarily small speeds. Then, The following Pohozaev identity (see [7] for instance) will be useful in our analysis.If c ∈ Ê and U ∈ E satisfies We shall also make use of the algebraic decay of the travelling waves conjectured in [31] and shown in [26].
Theorem 2.4 (Algebraic decay of the travelling waves - [26]) Up to a phase shift, we may assume U (x) → 1 for |x| → +∞.Then, there exists M , depending on U and c such that, for x ∈ Ê 2 ,

Symmetries of the travelling waves from Theorem 1.6
We recall from [15] that the travelling wave Q c constructed in Theorem 1.6 satisfies for all

This implies that for all
where [14].Remark that these quantities all have different symmetries.

A coercivity result
From Proposition 1.2 of [14], we recall that Q c defined in Theorem 1.6 has two zeros, at ± dc e 1 , with We define (as in [14]) the symmetric expended energy space by where, with ϕ = Q c ψ, r = r dc = min(r 1 , r−1 ), r±1 being the distances to the zeros of Q c (we use r instead of r dc to simplify the notations here), we define By using (2.1), we deduce, for any We take a smooth cutoff function η such that η , where − → 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 [14]) We recall from [14] (or by integration by parts

and that B exp
Qc (ϕ) is well defined for ϕ ∈ H exp,s Qc .This last point is the reason why we write the quadratic form as (2.4), which is equal, up to some integration by parts, to the more natural definition Qc .See [14] for more details on this point.We now quote a coercivity result from [14].
Theorem 2.5 ([14], Theorem 1.13)There exists R, K, c 0 > 0 such that, for 0 < c c 0 , Q c defined in Theorem 1.6, if a function ϕ ∈ H exp,s Qc satisfies the three orthogonality conditions: We will use a slight variation of this result, given in the next corollary.
Corollary 2.6 There exists R, K, c 0 > 0 such that, for 0 < c c 0 , Q c defined in Theorem 1.6, if a function ϕ ∈ H exp,s Qc satisfies the three orthogonality conditions: Remark, with Theorem 1.6 (for ), and that (with Lemma 2.1) they both have the same symmetries.We need to change the quantity Re B( dc e1,R)∪B(− dc e1,R) ∂ c Q c φ in the orthogonality conditions because we will differentiate it with respect to c, and )) |d=dc can be found in Lemma 2.6 of [15].Furthermore, we changed, in the area of the integrals, dc by d c (they are close when c → 0, see (2.3)).Proof.
Step 1: changing the integrand but not the integration domain.
Qc .We want to choose µ ∈ R such that ϕ * satisfied the hypothesis of Theorem 2.5.By the symmetries of subsection 2.1.3and the hypotheses on ϕ, we have that By Theorem 1.6 (for p = +∞) and Lemma 2.6 of [15], we have and also that there exists a universal constant K > 0 (we recall that R > 0 is a universal constant) such that we have Qc by Lemma 2.8 of [14], we deduce that ϕ * satisfies all the hypotheses of Theorem 2.5, therefore
Step 2: changing the integration domain.
To change the conditions we use similar arguments, using |d c − dc | = o c→0 (1) by (2.3).We check for instance that and Notice that the integration domain remains symmetric with respect to the x 2 -axis.✷

Proof of Proposition 1.8
In this subsection, we take ν ∈]0, 1[ a small but universal constant, that will be fixed at the end of the proof.We take ) in the statement of Proposition 1.8 (where R > 0 is defined in Corollary 2.6).Then, for any λ λ * , we take in the statement of Proposition 1.8.The condition ε(λ) 1 10λ 2 +100 is required only to make sure that the two balls B(d − → e 1 , 2λ) and B(−d − → e 1 , 2λ) are disjoint and at a distance at least 1 from one another.This will be used only in the proof of Lemma 2.8.
We take u a function satisfying the hypotheses of Proposition 1.8 for these values of λ * , λ and ε(λ).In the rest of the subsection, K, K ′ > 0 denote universal constants, independent of any parameters of the problem (in particular, λ, λ * , ε(λ) and ν).

Modulation on the parameters of the branch
From Theorem 1.1 and the end of section 4.6 of [15], we have that with ∂ c d c ∼ −1/c 2 for c → 0 (see section 4.6 of [15]).In particular, c → d c is a smooth decreasing diffeomorphism from ]0, c 0 ] onto [d 0 , +∞[, and thus, given d > 1 ν > d 0 (for ν small enough), there exists a unique c ′ > 0 such that From the hypotheses on Γ, and the fact that 2ν, we deduce that (we denote r = rd = rd c ′ to simplify the notations) (2.5) We now claim that, for a universal constant K > 0, That is, u is close to Q c ′ near the vortices (in the region {r λ}) in the C 1 norm and not only in L ∞ .In order to show this, we use the elliptic equation satisfied by u − Q c ′ , that is Let us fix x ∈ {r λ}.We have u − Q c ′ L ∞ ({r 2λ}) K ′ ν by hypothesis, thus the right-hand side of the equation is small in H −1 (B(x, 4)).By a standard H 1 − H −1 estimate, we deduce Then, the right-hand side is small in L 2 , and standard L 2 elliptic regularity yields first and we conclude by Sobolev imbedding.Outside of this domain, u and Q c ′ are close only in modulus.Indeed, by equation (2.6) of [14] (for σ = 1/2) and the hypotheses on u, we have for a universal constant K > 0 that on {r λ}, Now, we modulate on the parameters of the family of travelling waves to get the orthogonality conditions of Corollary 2.6.For c ′′ > 0 close enough to c ′ and X, γ ∈ R, we define Lemma 2.7 There exists K > 0, ν 0 > 0 universal constants such that, for u satisfying the hypotheses of Proposition 1.8 for values of λ * , λ, ε(λ), ν described above, if ν ν 0 , then there exists c ′′ > 0, X, γ ∈ R such that, for R > 0 defined in Corollary 2.6, and d ± := ±d c ′′ e 1 + X e 2 ; Proof.To simplify the notations, in this proof, we define We will keep the notation r for the minimum of the distance to the zeros of Q.First, from equation (7.5) of [14], there exists a universal constant K > 0 such that, for Now, we follow closely the proof of Lemma 7.6 of [14], which is done in Appendix C.3 there.We define Remark that Q, ∂ d V and d ± all depend on X and c ′′ , and Q depends also on γ.From equation (2.6) and the fact that λ λ * > 2R, we have u − Q c ′ L ∞ ({r R}) Kν, and from Theorem 1.6 with p = +∞ as well as Lemma 2.6 of [15], for some universal constant K > 0. Therefore, since We want to show that G is invertible in a vicinity of With equations (2.6) and (2.8), we check that (we and as in Lemma 7.1 of [14], this implies Now, we compute therefore, with (2.1) and (2.10), we check that With similar computations, using Lemma 2.6 of [15], equations (2.1) and (2.10), we infer that By the symmetries of Q(. + X e 2 )e −iγ and ∂ d V (. + X e 2 )e −iγ , we have that and from Theorem 1.6 (with p = +∞), with the symmetries of Q c and V 1 (see subsections 2.1.1 and 2.1.3),we have By decomposition in harmonics and Lemma 2.1, we check easily that Re B(0,R) Similarly, we check that (using c 2 from section 4.6 of [15], and Lemma 2.6 of [15]) (we use here the fact that c → ∂ d V and c → d ± are differentiable) and From (2.1) and Theorem 1.6 (for p = +∞) as well as Lemma 2.6 of [15], there exists a universal constant K > 0 such that provided |X| + c ′′ is small enough.We deduce that there exists K 1 , K 2 , ν 0 > 0 such that, for 0 < ν ν 0 and u satisfying the hypotheses of Proposition 1.8 with the parameters λ, ν, dG is invertible in the ball

Construction and properties of the perturbation term
We define η a smooth cutoff function with η(x) = 0 for x ∈ B( d ± , 2R) and η(x) = 1 for x ∈ R 2 \B(± d ± , 2R + 1) even in x 1 .We infer the following result, where the space H exp,s Q is simply defined by with, for r the minimum of the distances to the zeros of

and B exp
Q has the same definition than B exp Qc , replacing η by η and Q c by Q.
Lemma 2.8 There exists K 1 , K 2 > 0, ν 0 > ν 1 > 0 universal constants such that, for u satisfying the hypotheses of Proposition 1.8 for values of λ * , λ, ε(λ), ν described above, if ν ν 1 , then there exists a function The goal of this lemma is to decompose the error u − Q in a particular form.In the area {η = 1}, that is far from the zeros of Q, the error is written in an exponential form: u = Qe ψ .This form was already used in [14], [15], and is useful to have a particular form on the cubic error terms.Furthermore, we fix the parameters of Q such that ϕ satisfies the orthogonality conditions of Corollary 2.6, yielding the coercivity.
Remark that we have no smallness on Im(ψ) in {r λ}, where ϕ = Qψ.We will simply be able to show that it is bounded (see equation (2.11) below), with no a priori bound on it.This lack of smallness is one of the main difficulties in the proof of Proposition 1.8.Analogously, we show that ϕ ∈ H exp,s Q , but we have no good control on ϕ H exp Q : this quantity might be a priori very large at this point.Proof.This proof follows some ideas of the proofs of Lemmas 7.2 and 7.3 of [14].First, in the area {r λ}, the proof is identical to that of Lemma 7.2 of [14] for the existence of ϕ Kν (this is a consequence of the estimate u − Q C 1 ({r λ}) Kν, obtained using Lemma 2.7).The main idea is that u − Q is small there (in C 1 ({r λ}, C)), and the equation on ϕ is a perturbation of the identity for functions ϕ that are small in C 1 ({r λ}, C).In particular, since u and Q are symmetric with respect to the x 2 -axis, ϕ and ψ are also symmetric with respect to the x 2 -axis.
We then focus our attention in the area {r λ}, where η ≡ 1, so that the problem reduces to the equation By Theorem 1.6 and the hypotheses of Proposition 1.8, there exists ν 1 > 0 such that, if ν ν 1 , then, as a consequence of ), the domain {r λ} consists in the complement of the two disjointed disks B( d ± , λ), with , does not vanish and has zero degree on the two circles ∂B( d ± , λ).It then follows from standard lifting theorems (even though {r λ} is not simply connected), that there exists ψ † ∈ C 1 ({r λ}) such that e ψ † = u/Q, as wished.We then notice that u and Q are symmetric with respect to the x 2 -axis, thus x → ψ † (−x 1 , x 2 ) is also a lifting of u/Q in the connected set {r λ}, which implies that there exists q ∈ such that ψ † (−x 1 , x 2 ) = ψ † (x 1 , x 2 ) + 2iqπ in {r λ}.Letting x 1 = 0, we obtain q = 0: ψ † is also symmetric with respect to the x 2 -axis.

|Q|
, since there exists a universal constant |Q| K ′ ν, we deduce that, for ν ν 1 with ν 1 small enough, Since u is a travelling wave and E(u) < +∞, u converges to a constant at infinity (uniformly in all directions) by [26].Therefore, u Q converges to a constant at infinity, and the function ψ converges to a constant, and thus it is bounded near infinity, that is ψ L ∞ ({r λ}) < +∞. (2.11) Now, we want to show that ϕ ∈ H exp,s Q .We already know that ϕ satisfies the symmetry where K(u, c, Q, c ′′ ) > 0 is a constant depending on u, c, c ′′ and Q, hence |Re(ψ)| K(u,c,Q,c ′′ ) (1+r) 2 and {r λ} Re 2 (ψ) We finally compute and with Theorem 11 of [26], in {r λ}, we deduce that This concludes the proof that ϕ = Qψ ∈ H exp,s Lemma 2.9 For u satisfying the hypotheses of Proposition 1.8 for values of λ * , λ, ε(λ), ν described above, if ν ν 1 (where ν 1 is defined in Lemma 2.8) , then the function ϕ = Qψ defined in Lemma 2.8 satisfies the equation and NL loc (ψ) is a sum of terms at least quadratic in ψ, localized in the area where η = 1.Furthermore, Notice that F (ψ) (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 ϕ is small in the C 1 norm whereas, at infinity, it is small only in a weaker norm.Proof.The proof is identical to the proof of Lemma 7.5 of [14], and it is in the particular case where all the speeds are along e 2 .The proof consists simply in decomposing the equation in the different terms.
The last estimate uses Lemma 2.8 and Lemma 2.7.✷ This result shows in particular that ψ ∈ C 2 ({η = 0}, C), and we can check with it, as in Lemma 7.3 of [14], that ∆ψ 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 [14] (we take the scalar product of the equation of Lemma 2.9 with ∂ c Q), but since we have worse estimates on the error term, we need to be more careful ( ϕ H exp Q is not a priori small at this point).
Lemma 2.10 There exists universal constants K > 0, ν 1 ν 2 > 0 (where ν 1 is defined in Lemma 2.8), such that, for u satisfying the hypotheses of Proposition 1.8 for values of λ * , λ, ε(λ), ν described above, if ν ν 2 , then, with ϕ = Qψ defined in Lemma 2.8, we have First, from equation (2.5) and Lemma 2.7, taking ν > 0 small enough, we have (2.12) We will show the following estimate: This is related to equation (7.13) of [14] (its proof is in step 1 in subsection 7.3.1 of [14]).With both estimates, we can conclude the proof of this lemma.Indeed, either ϕ H exp Q √ c ′′ , and in that case c ′′ , and then with (2.13), therefore, for c ′′ > 0 small enough such that C 2 √ c ′′ < 1/2 (which is implied by taking ν > 0 small enough, independently of λ), we have |c ′′ − c| K √ c ′′ ϕ H exp Q .We now focus on the proof of (2.13).We take the scalar product of the equation We estimate, as in subsection 7.3.1 of [14], that We recall that and we check (estimating the local terms in the area where η = 1 by Cauchy-Schwarz and from Theorem 1.6 for p = +∞ and Lemma 2.6 of [15]) We recall from subsection 7.3.1 of [14] (using decay estimates on c ′′2 ∂ c ′′ Q Q and integrations by parts), that and, from Proposition 1.2 of [14] (we check easily that the translation and phase on Q instead of Q c ′′ do not change the computation), We deduce that, taking ν > 0 small enough (independently of λ), that We take ν 2 > 0 with ν 2 ν 1 such that all the above condition on the smallness of ν are satisfied if ν ν 2 .Since NL loc (ψ) contains terms at least quadratic in ϕ, ϕ C 1 ({η =1}) C 3 ν from Lemma 2.8 and Finally, we estimate, using Similarly, since ηRe(ψ) L ∞ ({r λ}) Kν by Lemma 2.8, diminishing ν 2 if necessary, for ν ν 2 , then ηRe(ψ) L ∞ ({r λ}) 1, hence This concludes the proof of (2.13), and therefore of the lemma.✷

Proof of Proposition 1.8 completed
We take u satisfying the hypotheses of Proposition 1.8 for values of λ * , λ, ε(λ), ν described above, with ν ν 2 , where ν 2 is defined in Lemma 2.10.We want to take the scalar product of the equation of Lemma 2.9 with ϕ.It is however not clear at this point that every term is integrable.In subsection 7.3 of [14], we took the scalar product of the equation with ϕ + iγQ for some γ ∈ R, using a decay estimate Im(ψ + iγ)(1 + r) L ∞ ({r λ}) K(u, Q, c, c ′′ ) to justify that some terms are well defined, and to do some integration by parts.Here, we need to change a little our approach.We first require better decay estimates on ψ.At this stage, we know (see Theorem 11 of [26] and the proof of Lemma 2.8) that Now, let us show the following improvements: The proof of (1 + r) 3 |Re(∇ψ)| L ∞ ({r λ}) K(u, Q, c, c ′′ ) is identical to the one for the same result in Lemma 7.3 of [14] (see the penultimate estimate of its proof).We focus on the estimate on Im(∆ψ).In {r λ}, we have u = Qe ψ , therefore, With the previous estimates and Theorem 11 of [26], we have and since therefore, with [26] (E(Q) < +∞), Similarly, since (TW c )(u) = 0 and E(u) < +∞, We infer, with these two additional estimates on ψ, that we can do the same computations as in the proof of Lemma 7.4 of [14], with γ = 0.The only difference is that, when we used Im(ψ we can use (2.14) instead to get the same decay for these terms, with Im(ψ) L ∞ ({r λ}) K(u, Q).The only two terms where this change is needed are We deduce, taking the scalar product of the equation of Lemma 2.9 with ϕ, that and from Lemma 2.9, (2.17) Let us now show that
We compute, using that therefore, by Lemma 2.9, in {η = 0}, We compute, by integration by parts, with Re n (ψ)Re( We deduce that and with previous estimates, we check easily that and by integration by parts, with computations similar to those for the proof of (2.23), using from (2.9) to (2.11) of [14] (for σ = 1/2) for a universal constant K > 0 and Lemma 2.1, we infer that and we check easily that , for the part of e 2 .H(ψ) related to the cutoff, the estimation can be done as previously, and we are left with the estimation of From equation (2.5) and Lemma 2.7, we have |c − c ′′ | ν (diminishing ν 2 if necessary), and from equation (2.9) of [14], Re and we estimate by (2.23).For the last remaining term, since Re n (ψ)Im(∇ψ).Re(∇ψ)Im(ψ).In particular, Combining this result with the previous estimates, this implies that for some universal constant C 6 > 0, but that is not enough to show that we have other than the fact that it is a finite quantity.By integration by parts (integrating Re(∇ψ)), with computations similar as for the proof of (2.23), we infer that Combining this result with estimates (2.20) to (2.29), we deduce that for some universal constant C 7 > 0, , This concludes the proof of equation (2.19).
Combining estimates (2.16) to (2.19) in equation (2.15), we deduce that for some universal constant C 8 > 0, therefore, taking ν > 0 small enough such that the previous constraints are satisfied and C 8 ν < 1/2 , we have ϕ H exp Q = 0. From Lemma 2.10, we deduce c ′′ = c.The proof is complete.
3 Properties of quasi-minimizers of the energy and proof of Theorem 1.11

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 u|v = Re(uv) the real scalar product of the complex numbers u, v.
Here, we recall that the space [C 0,β c (B(0, R))] * is endowed with the dual norm associated with The above mentioned theorem is actually Lemma 3.3 in [8].It is related to the works [2], [29], which both correspond to the limit ε → 0, whereas we have here a statement (obtained by compactness) at fixed ε.The hypothesis "|u| 1/2 in B(0, 4R) \ B(0, R)" ensures that the vortices do not approach the boundary ∂B(0, 4R).Theorem 3.3 (Clearing-out Theorem - [8]) Let M 0 > 0 and σ > 0 be given.Then there exist ǫ 0 > 0 and η > 0, depending only on M 0 and σ, such that, if in B(0, R 0 ) ⊂ Ê 2 , with ǫ < ǫ 0 , |c| M 0 |ln ǫ|, and For the elliptic PDE that is without the transport term i∂ 2 U , this result has been shown in 2d in [4] for minimizing maps, then in [9] for the Ginzburg-Landau equation with magnetic field.In higher dimension, see [33] and [5] for (3.2) and [8] for an equation including the Ginzburg-Landau equation with magnetic field and (3.1).One may use the change of unknown U(x to transform the equation (3.2) without the transport term into the equation (3.1) with the transport term.However, the assumptions E ǫ (U, B(0, R 0 )) η|ln ǫ| and E ε (U, B(0, R 0 )) η|ln ε| are not equivalent (due to the extra phase term).

Vortex structure for quasi-minimizers of E at fixed P
In this section, some Λ 0 > 0 is fixed and we consider a large momentum p and u p such that and such that there exists c p > 0 (depending on u p ) such that It then follows from [26] (see Theorem 2.4) that we may assume, using the phase shift invariance, that u p → 1 at spatial infinity.In particular, we have Our goal is to show that u p satisfies the hypothesis of Proposition 1.8.We shall follow [10] and [8] in order to analyze the vortex structure of u p .

Localizing the vorticity set at scale x/p
We define the following rescaling ûp of u p : Therefore, ûp solves which is a particular case of (3.1) with ǫ = 1/p, c = c p p.
The universal L ∞ bound on the gradient of Corollary 2.3 reads now We shall have, in the end, c p ∼ 1/p.The first step provides a rough upper bound for the speed c p (the Lagrange multiplier for the minimisation problem E min (p)).
Step 1: there exists p 1 = p 1 (Λ 0 ) such that, for p p 1 , we have In particular, c p 1/2 and ln p 2|ln c p |.
We shall use the Pohozaev identity (2.2), that is: At this stage, we only have the rough upper bound 0 2 dx E(u p ) 2π ln p+Λ 0 , which concludes.
Another argument we could use for minimizers is that we know from [7] (see also [13]) that 0 c p d + E min (p) E min (p)/p.
We now proceed in this way: we choose (if it exists) some ẑp,1 j=1 B(ẑ p,j , 2R 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 B(ẑ p,j , R 0 ), 1 j n, are pairwise disjoint, hence, by Theorem 3.3, we have At this stage, the disks B(ẑ p,j , 2R 0 ), 1 j n p , cover the vorticity set {|û p | 1/2}, but the disks B(ẑ p,j , 8R 0 ) may not be pairwise disjoint.To get this property, we argue as in [4] (Theorem IV.1).Let us recall the idea: if the disks B(ẑ p,j , 8R 0 ), 1 j n p are pairwise disjoint, then we are done with R * = 2R 0 .If not, then we have, for instance, |ẑ p,1 − ẑp,2 | 16R 0 .We then remove the disk B(ẑ p,1 , 8R 0 ) from the list and set R 1 def = 17R 0 .The disks B(ẑ p,j , R 1 ), 2 j n p cover ∪ 1 j np B(ẑ p,j , 2R 0 ), hence the vorticity set {|û p | 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 R 0 × 17 np R 0 × 17 n * .
Similar arguments are given in [8], whereas in [10] the vorticity set is included in some disks of radii of order c γ p , which requires some extra work.
Step 3: we have This follows exactly as in [8] (see Proposition A.1 in the Appendix there).Notice that the result in [8] is stated for the potential on a compact set in a domain Ω, but it holds as well in the entire plane.
We then define, as in [8], the function û′ for some 1 j n p (this last formula is valid since the disks B(ẑ p,j , 4R * ), 1 j n p , are mutually disjoint).
Step 5: we claim that for any δ ∈]0, π/2[, there exist p † δ > p 2 such that for all p p † δ , we are in one of the following cases: case (I) for any 1 j n p , J û′ The integration by parts is justified by the algebraic decay at infinity given in Theorem 2.4: Then, since J û′ p is supported in Ω R * , we obtain J û′ p dx.
We then fix χ ∈ C ∞ c (B(0, 4R * )) such that χ ≡ 1 on B(0, 3R * ).Next, for any 1 j n p , we write, B(ẑp,j ,3R * ) We now estimate the first integral (actually, a duality bracket) by using Step 5: Since P (û p ) = 1, it follows that for p large enough, we can not be in Case (I), and the conclusion is a recasting of (3.13).

Strong convergence outside the vorticity set at scale x/p
We start with a W 1,p loc bound at scale x, for 1 p < 2.
Step 1: for any 1 p < 2, there exists C p such that, for any X ∈ Ê 2 , we have We shall adapt the proof of [8] (see proof of Theorem 4, Step 3, p. 83) to the two-dimensional case.Actually, the only modification to make in the estimate is to replace (C.26) there by the standard convolution
The proof (relying on Step 1) follows the lines of the proof of Step 7 (p.48) of Theorem 1 in [8] and is omitted.

Lower bound for the energy and upper bound for the potential energy
Step 1: upper bound for the potential.We claim that and that Ê 2 \(B(ŷp,+,2/10)∪B(ŷp,−,2/10)) The proof of this upper bound will be a direct consequence of the lower bounds established in [43] (see Theorems 2 and 3 there).
From the W 1,p loc upper bound of Step 1 in subsection 3.2.2 and by weak compactness, there exists Step 2 of subsection 3.2.2 (for any k ∈ AE).In order to determine Û , we shall pass to the limit in It then follows that Û ∧ ∇ Û = û∞ ∧ ∇û ∞ , hence the existence of Θ ∈ Ê such that Û = e iΘ û∞ .We finally use the x 1 -symmetry to infer Θ = 0.
Step 2: as p → +∞, we have This is claimed in [10] (Proposition VI.7 there), but the proof is not clearly given.One way to prove this point is to use the Hopf differential as in [4] (chapter VII).We shall follow the alternative proof of Theorem VII.2 given in section VII.1 there.The first equality is the Pohozaev identity (2.2).
We shall now compute µ + (the case of µ − is similar).First, we write, for some R 5 2/10, the Pohozaev identity for ûp on B(ŷ ∞,+ , R 5 ) (obtained by multiplying the equation by the conjugate of (x − ŷ∞,+ ) • ∇û p and integrating the real part over B(ŷ ∞,+ , R 5 )), which yields We then pass to the limit p → +∞.For the boundary term, we use the strong convergences outside the vorticity set; for the second term of the first line, we prove that it tends to zero by following the arguments given for Step 6 in subsection 3.2.1.We then get By Step 1, we know that û∞ = exp(iArg(x − ŷ∞,+ ) − iArg(x − ŷ∞,− )) on ∂B(ŷ ∞,+ , R 5 ), and the second term Arg(x − ŷ∞,− ) is smooth and harmonic in D(ŷ ∞,+ , R 5 ).As a consequence, we have the Pohozaev identity for Arg(• − ŷ∞,− ) , and thus by expansion This concludes the proof.

Convergence on the scale x
We shall now focus on the verification of hypothesis 2 of Proposition 1.8.The main tool is the following result.We now work on the scale x.Then, from the uniform bounds of Theorem 2.2 and Corollary 2.3, we may assume, up to a subsequence, that and, by Fatou's lemma, By [11], we know that Ê 2 (1 − |U ∞ | 2 ) 2 dy = 2πd 2 , where d ∈ is the degree of U ∞ at infinity.It follows that |d| 1, and that the case It then follows from [38] that U ∞ = e iβ V d for some β ∈ Ê.
Step 2: conclusion.Applying Proposition 1.8 to e −iβ u p , we infer that there exists γ p ∈ Ê such that (for large p) (no translation is needed in the x 2 direction at this stage since the zeros of ûp are on the x 1 -axis).

Decay slightly away from the vortices
In this section, we provide some estimates for ûp in the region B(ẑ p,+ , 2R 0 )∪B(ẑ p,− , 2R 0 ).For the Ginzburg-Landau (stationary) model, such estimates have been first given in [37] for minimizing solutions and later generalized in [18] to non-minimizing solutions.However, the paper [37] being difficult to find, we give here a proof of these estimates that includes the transport term.They improve some estimates in [15] and are not specific to the way we construct the solutions.and scaling back this yields the conclusion, at least for δ = 2/(p|ŷ|) sufficiently small, say p|ŷ| δ 0 /2, but the estimate is easy to show if p|ŷ| δ 0 /2.✷

Some remarks on the non symmetrical case
In the case where we do not assume the x 1 symmetry for u p , the location of the vortices ŷp,± is more delicate.Indeed, we can no longer assume (3.14), that is Then, the location of the limiting vortices ŷ∞,± = lim p→+∞ ŷp,± can be obtained through the use of the Hopf differential as in [4] (chapter VII), and would lead as before to ŷ∞,± = (±1/(2π), 0).This is of course related to the fact that the only critical point of the action functional The term Θ is somewhat the phase at infinity, even though we do not claim some uniformity at infinity in space.
Next, for the local convergences, there are two phases β ± ∈ Ê such that ûp (ẑ p,± + p•) → e iβ± V ± (3.23) in C k loc (Ê 2 ) for any k ∈ AE.We are then simply able to show that β ± = Θ, but this is not enough for the uniqueness result.This follows from the arguments given in [44], as we explain.
) by(1.2), 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 * ).