Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.


The problem
We consider the one-dimensional nonlocal Gross-Pitaevskii equation for Ψ : R × R → C introduced by Gross [38] and Pitaevskii [55] to describe a Bose gas with the boundary condition at infinity Here * denotes the convolution in R, and W is a real-valued even distribution that describes the interaction between particles. The nonzero boundary condition (1) arises as a background density. This model appears naturally in several areas of quantum physics, for instance in the description of superfluids [53,1] and in optics when dealing with thermo-optic materials because the thermal nonlinearity is usually highly nonlocal [58]. An important property of equation (NGP) with the boundary condition at infinity (1), is that it allows to study dark solitons, i.e. localized density notches that propagate without spreading [41], that have been observed for example in Bose-Einstein condensates [31,6].
There have been extensive studies concerning the dynamics of equation (NGP), and the existence and stability of traveling waves in the case of the contact interaction W = δ 0 (see [15,10,14,13,24,23,34,49,26,40,39,42] and the references therein). However, there are very few mathematical results concerning general nonlocal interactions with nonzero conditions at infinity. In [26,54] the authors gave conditions on W to get global well-posedness of the equation and in [28] conditions were established for the nonexistence of traveling waves (in higher dimensions). Nevertheless, to our knowledge, there is no result concerning the existence of localized solutions to (NGP) when W is not given by a Dirac delta. The aim of this paper is to provide conditions on W in order to have stable finite energy traveling wave solutions, more commonly refereed to as dark solitons due to the nonzero boundary condition (1). More precisely, we look for a solution of the form representing a traveling wave propagating at speed c. Hence, the profile u satisfies the nonlocal ODE icu + u + u(W * (1 − |u| 2 )) = 0 in R.
(TW W,c ) By taking the conjugate of the function, we assume without loss of generality that c ≥ 0.
Let us remark that when considering vanishing boundary conditions at infinity, this kind of equation has been studied extensively [35,20,52] and long-range dipolar interactions in condensates have received recently much attention [43,19,4,7,48]. However, the techniques used in these works cannot be adapted to include solutions satisfying (1).
When W is given by a Dirac delta function, equation (TW δ 0 ,c ) corresponds to the classical Gross-Pitaevskii equation, which can solved explicitly. As explained in [9], if c ≥ √ 2 the only solutions in E(R) are the trivial ones (i.e. the constant functions of modulus one) and if 0 ≤ c < √ 2, the nontrivial solutions are given, up to invariances (translations and a multiplications by constants of modulus one), by Thus there is a family of dark solitons belonging to N E(R) for c ∈ (0, √ 2) and there is one stationary black soliton associated with the speed c = 0. Notice also that the values of u c (∞) and u c (−∞) are different, and thus we cannot relax the condition (1) to lim |x|→∞ Ψ = 1, as is usually done in higher dimensions.
The study of equation (TW δ 0 ,c ) can be generalized to other types of local nonlinearities such as the cubic-quintic nonlinearity and some cubic-quintic-septic nonlinearities as shown in [22,51]. The techniques used by the authors rely on the analysis of a second-order ODE of Newton type, so that the Cauchy-Lipschitz theorem can be invoked and some explicit formulas can be deduced. These arguments cannot be applied to (TW W,c ) due to the nonlocal interaction. For this reason, our approach to show existence of traveling waves relies on a priori energy estimates and a concentration-compactness argument, that allow us to prove that there are functions that minimize the energy at fixed momentum. These minimizers are solutions to (TW W,c ) and we can also establish that they are orbitally stable (see Theorem 4). These kinds of arguments have been used by several authors to establish existence of solitons for the (local) Gross-Pitaevskii equation in higher dimensions and for some related equations with zero conditions at infinity (see e.g. [10,49,24,47,50,5,44]). The main difficulty in our case is to handle the nonvanishing conditions at infinity, the fact that the constraint given by the momentum is not a homogeneous function along with the nonlocal interactions.

The critical speed and assumptions on W
Linearizing equation (NGP) around the constant solution equal to 1 and imposing e i(ξx−wt) as a solution of the resulting equation, we obtain the dispersion relation where W denotes the Fourier transform of W. Supposing that W is positive and continuous at the origin, we get the so-called speed of sound c * (W) = lim ξ→0 w(ξ) ξ = 2 W(0).
The dispersion relation (3) was first observed by Bogoliubov [17] in the study of a Bose-Einstein gas. He then argued that the gas should move with a speed less than c * (W) to preserve its superfluid properties. This leads to the conjecture that there is no nontrivial solution of (TW W,c ) with finite energy when c > c * (W). Actually, one of the authors proved this conjecture in [28] in dimensions greater than one, under some conditions on W.
In order to simplify our computations, we can normalize the equation so that the critical speed is fixed. Indeed, it is easy to verify that the rescaling x → x/ W(0) 1/2 and t → t/ W(0) allows us to replace W(ξ) by W(ξ)/ W(0) in (NGP). Therefore, we assume from now on that W(0) = 1 and hence that the critical speed is Before going any further, let us state the assumptions that we need on W.
(H1) W is an even tempered distribution with W ∈ L ∞ (R), and W ≥ 0 a.e. on R. Moreover W is continuous at the origin and W(0) = 1.
(H3) W admits a meromorphic extension to the upper half-plane H := {z ∈ C : Im(z) > 0}, and the only possible singularities of W on H are simple isolated poles belonging to the imaginary axis, i.e. they are given by {iν j : j ∈ J}, with ν j > 0, for all j ∈ J, 0 ≤ Card J ≤ ∞, and their residues satisfy i Res( W, iν j ) ≤ 0, for all j ∈ J.
Also, there exists a sequence of rectifiable curves (Γ k ) k∈N * ⊂ H, parametrized by γ k : [a k , b k ] → C, such that Γ k ∪ [−k, k] is a closed positively oriented simple curve that does not pass through any poles. Moreover, for all t ∈ [a k , b k ], W(γ k (t)) |γ k (t)| 4 = 0.
Here C k b (R) denotes the bounded functions of class C k whose first k derivatives are bounded. We have also used the convention that the Fourier transform of (an integrable) function is In particular, the Fourier transform of the Dirac delta isδ 0 = 1 and thus assumptions (H1)-(H3) are trivially fulfilled by W = δ 0 . Let us make some further remarks about these hypotheses. Assumption (H1) ensures that the critical speed exists and that the energy functional is nonnegative and well defined in E(R). Indeed, let us consider v ∈ E(R), set η = 1 − |v| 2 and write the energy in terms of the kinetic and potential energy as By hypothesis (H1) and the Plancherel theorem, we deduce that so that the functions in E(R) have indeed finite energy and their potential energy is nonnegative. Moreover, using the space M p (R) (see e.g. [36]), i.e. the set of tempered distributions W such that the linear operator f → W * f is bounded from L p (R N ) to itself, and denoting by · Mp its norm, we see that (H1) implies that W ∈ M 2 (R), with Hypothesis (H2), combined with (H1), implied that W(ξ) ≥ (1 − ξ 2 /2) + a.e., that can be seen as a coercivity property for the energy. In particular, it will allow us to establish the key energy estimates in Lemmas 2.1 and 2.3. The condition ( W) (0) > −1 will be crucial to show that the behavior of a solution of (TW W,c ) can be formally described in terms of the solution of the Korteweg-de Vries equation at least for c close to √ 2 (see Section 3).
The more technical and restrictive assumption (H3) is used only to prove that the curve associated with the minimizing problem is concave. Indeed, using a reflexion argument, we are led to show that for all odd functions f ∈ C ∞ c (R), wheref is given byf (x) = f (x) for x ∈ R + , andf (x) = −f (x) for x ∈ R − . Using the sine and cosine transformŝ we show in Section 3 that inequality (6) is equivalent to the following assumption.
Therefore, we can replace (H3) by the weaker (but less explicit) condition (H3'). Finally, let us notice that if W = δ 0 , we can verify that condition (H3') is satisfied by using the Plancherel formula At the end of this section we will give some examples of potentials satisfying (H1)-(H3).

Main results
In the classical minimization problems associated with Schrödinger equations with vanishing conditions at infinity, the constraint in given by the mass. In our case, the momentum is the key quantity that we need to take as a constraint to show the existence of dark solitons. Let us verify that the momentum is well defined in the nonvanishing energy space. Indeed, a function v ∈ N E(R) is continuous and admits a lifting v = ρe iφ , where ρ = |v| and φ are real-valued functions in H 1 loc (R) (see e.g. [33]). Hence, setting η = 1 − |v| 2 , the energy and momentum can be written as Using the classical Cauchy inequality ab ≤ a 2 /2 Thus, using Lemma 2.1, we conclude that the right-hand side of (8) can be bounded in terms of E(v) and therefore, the momentum is well defined in N E(R), under the assumptions (H1)-(H2).
Let us now describe our minimization approach for the existence problem, assuming that W satisfies (H1) and (H2). For q ≥ 0, we consider the minimization curve that is well defined in view of Lemma 3.1. Moreover, this curve is nondecreasing (see Lemma 3.11). We also set If (H3) is also fulfilled and q ∈ (0, q * ), we will show that the minimizers associated with E min (q) are attained and that the corresponding Euler-Lagrange equation is exactly (TW W,c ), where c appears as a Lagrange multiplier (see Section 6 for details). More precisely, our first result establishes the existence of a family of solutions of (TW W,c ) parametrized by the momentum.
It is important to remark that the constant q * is not necessarily small. For instance, in the case W = δ 0 , the explicit solution (2) allows us to compute the momentum of u c , for c ∈ (0, √ 2), and to deduce that q * = π/2. Moreover E min can be determined and its profile is depicted in Figure 1. Notice that E min is constant on (q * , ∞) and that in this interval the minimizers are not attained (see e.g. [9]). Since (H1)-(H3) are satisfied by W = δ 0 , and since there is uniqueness (up to invariances) of the solutions to (TW δ 0 ,c ), we deduce that the branch of solutions given by Theorem 1 corresponds to the dark solitons in (2), for c ∈ (0, √ 2). In the general case, we do not know if the solution given by Theorem 1 is unique (up to invariances). Actually, the uniqueness for nonlocal equations such as (TW W,c ) can be difficult to establish (see e.g. [3,44]) and goes beyond the scope of this work. Concerning the regularity, the solutions given by Theorem 1 are smooth and we refer to Lemma 6.2 for a precise statement.
To establish Theorem 1, we analyze two problems. First, we provide some general properties of the curve E min . Then, we study the compactness of the minimizing sequences associated with E min . The next result summarizes the properties of E min .
(i) The function E min is even and Lipschitz continuous on R, with Moreover, it is nondecreasing and subadditive on R + .
(iv) We have q * > 0.027. If E min is concave on R + , then E min is strictly increasing on [0, q * ), and for all v ∈ E(R) satisfying E(v) < E min (q * ), we have v ∈ N E(R).
(v) Assume that E min is concave on R + . Then E min (q) < √ 2q, for all q > 0, E min is strictly subadditive on R + , and the right and left derivatives of E min , denoted by E + min and E − min respectively, satisfy To prove the existence of solutions we use a concentration-compactness argument. Applying Theorem 2, we show that the minimizers are attained at least for q ∈ (0, q * ), so that the set is nonempty, and thus there are nontrivial solutions to (TW W,c ) (see Theorem 6.3). Hence, we can rely on the Cazenave-Lions [21] argument to show that the solutions are stable. Let us remark that the Cauchy problem for (NGP) was studied in [27]. Precisely, using the distance the energy space E(R) is a complete metric space and for every Ψ 0 ∈ E(R) there is a unique global solution Ψ ∈ C(R, E(R)) with initial condition Ψ 0 , provided that W ∈ M 3 (R) and that W ≥ 0 or that inf R W > 0 (see Theorem 5.1). However, these conditions are not necessarily fulfilled by a distribution satisfying (H1)-(H2). Nevertheless, using the energy estimates in Section 2, we can generalize a result in [27] in the following way.
Theorem 3. Assume that W ∈ M 3 (R) is an even distribution, with W ≥ 0 a.e. on R, and that W of class C 2 in a neighborhood of the origin with W(0) = 1. Then for every Ψ 0 ∈ E(R), there exists a unique Ψ ∈ C(R, E(R)) global solution to (NGP) with the initial condition Ψ 0 . Moreover, the energy is conserved, as well as the momentum as long as inf x∈R |Ψ(x, t)| > 0. Remark 1.1. As explained before, the condition W(0) = 1 in Theorem 3 is due to the normalization, and it can be replaced by W(0) > 0.
We can also endow E(R) with the pseudometric distance or with the distance used in [9] then the solution Ψ(t) of (NGP) associated with the initial condition Ψ 0 satisfies Similarly, the set S q is orbitally stable in (E(R), d A ) if for all Ψ 0 ∈ E(R) and for all ε > 0, there exists δ > 0 such that if d A (Ψ 0 , S q ) ≤ δ, then sup t∈R inf y∈R d A (Ψ(· − y, t), S q ) ≤ ε. Here we need to introduce a translation of the flow, since the d A is not invariant under translations.
Now we can state our main result concerning the existence and stability of traveling waves.
Theorem 4. Suppose that W satisfies (H1) and (H2), and that E min is concave on R + . Then the set S q is nonempty, for all q ∈ (0, q * ). Moreover, every u ∈ S q is a solution of (TW W,c ) for some speed c q ∈ (0, √ 2) satisfying Also, c q → √ 2 as q → 0 + .
In addition, if W ∈ M 3 (R), then S q is orbitally stable in (E(R), d) and in (E(R), d A ), for all q ∈ (0, q * ). Furthermore, for all Ψ 0 ∈ E(R) and for all ε > 0, there exists δ > 0 such that if d(Ψ 0 , S q ) ≤ δ, then the solution Ψ(t) of (NGP) associated with the initial condition Ψ 0 satisfies In this manner, it is clear that Theorem 1 is an immediate corollary of Theorems 2 and 4, and that the branch of solutions given by Theorem 1 is orbitally stable provided that W ∈ M 3 (R). In particular, we recover the orbital stability proved by several authors for the solitons given in (2) (see e.g. [45,15,23] and the references therein).
We point out that we have not discussed what happens with the minimizing curve for q ≥ q * . As mentioned before, for all q > q * , the curve E min (q) is constant for W = δ 0 (see Figure 1) and S q is empty. Moreover, the critical case q = q * is associated with the black soliton and its analysis is more involved (see e.g. [11,37]). Numerical simulations lead us to conjecture that similar results hold for a potential satisfying (H1)-(H3), i.e. that E min (q) is constant and that S q is empty on (q * , ∞), and that there is a black soliton when q = q * . In addition, in the performed simulations the value q * is close to π/2 (see Section 7). Furthermore, these simulations also show that (H2) and (H3') are not necessary for the concavity of E min nor the existence of solutions of (TW W,c ). We think that (H2) could be relaxed, but that the condition ( W) (0) > −1. As seen from Theorem 2, we have only used (H3') as a sufficient condition to ensure the concavity of E min . If for some W satisfying (H1) and (H2), one is capable of showing that E min is concave, then the existence and stability of solutions of (TW W,c ) is a consequence of Theorem 4.
In addition to the smoothness of the obtained solutions (see Lemma 6.2), it is possible to study further properties of these solitons such as their decay at infinity and uniqueness (up to invariances). Another related open problem is to show the nonexistence of traveling waves for c > √ 2. We will study these questions in a forthcoming paper.
We give now some examples of potentials satisfying conditions (H1), (H2) and (H3) so that W α,β (0) = 1, and it is simple to check that (H1) and (H2) are satisfied. To verify (H3), it is enough to notice that the only singularity on H of the meromorphic function W α,β is the simple pole ν 1 = iβ and that Since W α,β is bounded on H away from the pole, we conclude that (H3) is fulfilled. We recall that, by the Young inequality, L 1 (R) is a subset of M 3 (R). Therefore W α,β ∈ M 3 (R) and Theorem 4 applies.
In Section 7 we perform some numerical simulations to illustrate the shape of the solitons and the minimization curves associated with these and other examples. The rest of the paper is organized as follows: we give some energy estimates in Section 2. In Section 3, we establish the properties of the minimizing curve and the proof of Theorem 2, and in Section 4 we show the compactness of the sequences associated with the minimization problem. The orbital stability of the solutions and Theorem 3 are proved in Section 5. We finally complete the proof of Theorem 4 in Section 6.

Some a priori estimates
We start by establishing an L ∞ -estimate for the functions in the energy space in terms of their energy. In the sequel, we use the identity that is a consequence of parity of W stated in (H1). for some κ ≥ 0. Let v ∈ E(R) and set η := 1 − |v| 2 . Then and Proof. Let W ∈ M 2 (R) and v ∈ E(R), and set ρ = |v|, η = 1 − ρ 2 and x ∈ R. By Plancherel's identity By (2.2), we have 1 ≤ W(ξ) + κξ 2 a.e. on R, so that the term on the right-hand side of (2.5) can be bounded by On the other hand, the set Ω := {v = 0} coincides with the set {η = 1}, and v = 0 and η = 0 a.e. onΩ. Therefore, we conclude that Combining (2.5), (2.6) and (2.7), we have Bearing in mind that η(±∞) = 0, we deduce that there is some x 0 ∈ R such that a := min Therefore, using (2.8) for x 0 and (2.9), we get Solving the associated quadratic equation and using that By putting together (2.8), (2.9) and (2.10), we obtain (2.3).
To prove (2.4), we use the Plancherel identity and argue as before to get Therefore, using (2.10), inequality (2.4) is established.
Remark 2.2. Let us suppose that W ∈ M 2 (R) is even and that also W is of class C 2 in some interval [−r, r], with r > 0. Then ( W) (0) = 0, and by the Taylor theorem we deduce that for any ξ ∈ (−r, r), there existsξ ∈ (−r, r) such that Assuming also that W ≥ 0 a.e. on R, we conclude that in both cases condition (2.2) is fulfilled.
From now on we assume that (H1) and (H2) are satisfied. A key point to obtain the compactness of the sequences in Section 4 is that the momentum can be controlled by the energy. This kind of inequality is crucial in the arguments when proving the existence of solitons by variational techniques in the case W = δ 0 (see [10,24]). Moreover, for an open set Ω ⊂ R and u = ρe iθ ∈ N E(R), we need to be able to control the localized momentum by some localized version of the energy. By the Cauchy inequality, setting as usual but it is not clear how to define a localized version of energy, due the to the nonlocal interactions. We propose to introduce the localized energy Notice that if Ω = R, then E Ω (u) = E(u) and p Ω (u) = p(u). Since η Ω can be discontinuous (and thus not weakly differentiable) when Ω is bounded, we also need to introduce a smooth cut-off function as follows: for Ω 0 an open set compactly contained in Ω, i.e. Ω 0 ⊂⊂ Ω, we set a function χ Ω,Ω 0 ∈ C ∞ (R) taking values in [0, 1] and satisfying In the case Ω = Ω 0 = R, we simply set χ Ω,Ω 0 ≡ 1.
Let Ω, Ω 0 ⊂ R be two smooth open sets with Ω 0 ⊂⊂ Ω and let χ Ω,Ω 0 ∈ C ∞ (R) as above. Let u ∈ E(R) and assume that there are some ε 1 ∈ (0, 1) and ε 2 ≥ 0 such that where the remainder term ∆ Ω (u) satisfies the estimate ) is a constant depending on E(u), ε 1 and ε 2 , but not on Ω nor Ω 0 . In particular, in the case Ω = Ω 0 = R, we have Proof. As usual, we write u = ρe iθ on Ω. Using (2.11) and that 1 − and also whereη Ω = ηχ Ω,Ω 0 . Using the condition (2.13), the Plancherel theorem and (H2), we have Therefore which combined with (2.17) gives us (2.14). It remains to show the estimate in (2.15). For the first term in the right-hand side of (2.18), we see that For the second term, using (2.1), we have (2.20) Concerning the last integral, we havẽ From now on, we set for q > 0, In this manner, the condition E min (q) < √ 2q is equivalent to Σ q > 0. We also define for q > 0 and δ > 0, the set (2.23) Proof. We argue by contradiction and suppose that the statement is false. Hence, for all δ 0 > 0, there exists δ ∈ [0, δ 0 ] and v ∈ X q,δ such that Then, taking δ 0 = 1/n, there is δ n ∈ [0, 1 n ] and v n ∈ X q,δn such that Since Σ q ∈ (0, 1], considering ε 1 = ε 2 = Σ q /L, we have ε 1 ∈ (0, 1) and the condition (2.13) is fulfilled. Therefore we can apply Lemma 2.3 to conclude that and letting n → ∞, we get which is equivalent to Σ q ≤ Σ q /L, contradicting the fact that L > 1.
Lemma 2.5. Let E > 0 and 0 < m 0 < 1 be two constants. There is l 0 ∈ N, depending on E and m 0 , such that for any function v ∈ E(R) satisfying E(v) ≤ E, one of the following holds: Proof. The proof is a rather standard consequence of the energy estimates. For the sake of completeness, we give a proof similar to the one given in [9].
Let us suppose that (i) does not hold. Then the set is nonempty, where η = 1 − |v| 2 as usual. Setting I j = [j − 1/2, j + 1/2], for j ∈ Z, the assertion in (ii) will follow if we show that l := Card{j ∈ Z, I j ∩ C = ∅} can be bounded by some l 0 , depending only on E and m 0 .
Using that ||v| | = |v |, the Cauchy-Schwarz inequality and (2.3), we deduce that there exists a constant C, depending on E, such that for all x, y ∈ R, Thus, setting r = m 2 0 /(4C 2 ), we deduce that for any z ∈ C and for any y ∈ [z − r, z + r], Taking r 0 = min(r, 1/2) and integrating this inequality, we get, for any z ∈ C,

Properties of the minimizing curve
For the study of the minimizing curve, it will be useful to use finite energy smooth functions that are constant far away from the origin. For this purpose we introduce the set The next result shows that E min is well defined and that its graph lies under the line y = √ 2x on R + .
In particular the function E min : R → R is well defined, and for all q ≥ 0 Proof. The case q = 0 is trivial since it is enough to take v ≡ 1. Let us assume that q > 0 and consider Then it is enough to consider v n = ρ n e iθn , where ρ n (x) = 1 − α n χ (β n x) and θ n (x) = √ 2 α n β n χ(β n x).
We can assume that v n does not vanish since |v n | = |ρ n | ≥ 1−|α n | χ L ∞ (R) . Thus the momentum of v n is well defined and we have For the kinetic part, we have since α n , β n → 0 and α 2 n /β n → 1. For the potential energy, using Plancherel's theorem, the dominated convergence theorem and the continuity of W at 0, we get Therefore we conclude that (3.1) holds true for q ≥ 0. In the situation q < 0, it is enough to proceed as above taking This concludes the proof of (3.1). By the definition of E min , we also have E min (q) ≤ E(v n ).
Lemma 3.2. The curve E min is even on R.
Proof. Let q ∈ R and u n = ρ n e iφn ∈ N E(R) be such that E(u n ) → E min (q) and p(u n ) = q.
and letting n → ∞ we conclude that E min (q) ≥ E min (−q). Replacing q by −q, we deduce that E min (−q) = E min (q), i.e. that E min is even.
. Since the right-hand is an increasing function of q, and since the solution of the equation the conclusion follows from the definition of q * .
In view of Lemma 3.2, it is enough to study E min on R + . Concerning the density of the space we have the following result.
In particular Then g ∈ L 2 (R) and since g = v , v /|v|, we conclude that g ∈ H 1 (R). Therefore, there exists we conclude that θ n −θ → 0 in L 2 (R) and that v n := ρ n e iθn belongs to E ∞ 0 (R). The convergences in (3.4) are a direct consequence of the convergences in (3.3) and the Sobolev injection Hence the function θ is constant outside supp(θ ) and without loss of generality we can assume that there is R > 0 such that θ(x) ≡ 0 for all x ≤ −R, or that θ(x) ≡ 0 for all x ≥ R (but we cannot assume that θ(x) ≡ 0 for all |x| ≥ R). Therefore, w.l.o.g. we can To handle the nonlocal interaction term in the energy in the construction of comparison sequences, we use introduce the functional The following elementary lemma will be useful.
Assume further that g ∈ C ∞ c (R) and that there is a sequence of numbers (y n ) such that y n → ∞, as n → ∞. Then, setting set g n (x) = g(x − y n ), we have Proof. The identity (3.6) is a direct consequence of (2.1). The convergence in (3.7) follows from the fact that g n 0 in L 2 (R).
We finally conclude that we can modify a function with energy close to E min (q) such that it is constant far away, but the momentum remains unchanged.
Proof. Let v n = ρ n e iθ ∈ E ∞ 0 (R) be the sequence given by Lemma 3.4 such that If p(u) = 0, we set α n = p(u)/p(v n ). Therefore α n → 1 and it is straightforward to verify that the sequence u n = ρ n e iαnθn satisfies (3.8).
The case p(u) = 0 is more involved. In this instance, we may assume that δ n := p(v n ) = 0 for n sufficiently large. Otherwise, up to a subsequence, the conclusion holds with u n = v n . By Lemma 3.1, we get the existence of a sequence w n ∈ E ∞ 0 (R) such that Let R n , r n > 0 be such that the functions f n := 1 − |v n | 2 and g n := 1 − |w n | 2 are supported in the balls B(0, R n ) and B(0, r n ), respectively. Taking into account Remark 3.5, without loss of generality, we can assume that the following function is continuous and belongs to E ∞ 0 (R) on [R n , −r n + y n ], w n (· − y n ), on (−r n + y n , ∞), (3.11) where y n is a sequence of points such that R n < −r n + y n . For simplicity, we setw n = w n (· − y n ) andg n := 1 − |w n | 2 . It follows that p(u n ) = p(v n ) + p(w n ) = 0 and E k (u n ) = E k (v n ) + E k (w n ). (3.12) In particular, combining with (3.9) and (3.10), we infer that E k (u n ) → E k (u). In addition, 1 − |u n | 2 = f n +g n , so that (3.6) leads to Using the estimate (2.4), (3.9) and (3.10), we conclude that f n L 2 is bounded and that g n L 2 → 0, so that E p (u n ) → E p (u), which completes the proof of the corollary.
In particular Proof. Let q ≥ 0 and ε > 0. By definition of E min , there is a sequence v m ∈ N E(R) such that p(v m ) = q and E(v m ) → E min (q), as m → ∞. Hence there is m 0 such that (3.14) By Corollary 3.7, we deduce the existence of v ∈ E ∞ 0 (R) such that p(v) = p(v m 0 ) = q and |E(v m 0 ) − E(v)| ≤ ε/2. Combining with (3.14), the conclusion follows. Proposition 3.9. E min is continuous and Proof. We assume without loss of generality that q ≥ p ≥ 0. It is enough to show that Let δ > 0. By Corollary 3.8 and Remark 3.
Now, setting s = q − p and invoking Lemma 3.1, we deduce that there is w δ ∈ E ∞ 0 (R) such that for some r δ > 0, 1 − |w δ | 2 is supported on B(0, r δ ), w δ = 1 on (−∞, r δ ], Then f δ and g δ have compact supports and applying Lemma 3.6 we can choose y δ ∈ R, large enough, such that their supports do not intersect. Finally, we infer that the function applying Lemma 3.6 and increasing y δ if necessary, we conclude that Letting δ → 0, we obtain (3.16).
As noticed by Lions [46], the properties established above are usually sufficient to check that the minimizing curve is subadditive, as stated in the following result. Proof. Let p, q ≥ 0 and δ > 0. By using Corollary 3.8 and arguing as in the proof of Proposition 3.9, we get the existence of v, w ∈ E ∞ 0 (R) such that with v and w constant on B(0, R) c and B(0, r) c , respectively, for some R, r > 0. As in previous proofs, we define with y large enough such that Letting δ → 0, inequality (3.22) is established.
In some minimization problems, there is some kind of homogeneity in the functionals that allows to obtain the strict subadditive property. In our case, the homogeneity give us only the monotonicity of the curve.
Lemma 3.11. E min is nondecreasing on R + .
As mentioned in the introduction, hypothesis (H3) provides a sufficient condition to ensure the concavity of the function E min . The proof relies on a reflexion argument and some identities developed by Lopes and Mariş in [47].
Proposition 3.12. Assume that (H3') holds. Then for all p, q ≥ 0, In particular E min is concave on R + .
Proof. Let p, q > 0 and δ > 0. By Corollary 3.8, there is u = ρe iθ ∈ E ∞ 0 (R) such that For notational simplicity, we continue to write u, ρ and θ forũ,ρ andθ. Now we introduce the reflexion operators Since ρ and θ are continuous, the functions (T ± ρ) and (S ± ρ) are continuous and therefore, they belong to H 1 loc (R). Then it is simple to verify that the functions belong to E ∞ 0 (R). Bearing in mind (3.25), we obtain p(u + ) = p and p(u − ) = q, which implies that E min (p) ≤ E(u + ) and E min (q) ≤ E(u − ). (3.26) In addition We claim that which combined with (3.27), allows us to conclude that E(u + ) + E(u − ) ≤ 2E(u). By putting together this inequality, (3.24) and (3.26), we get so that (3.23) is proved. Since E min is a continuous function by Proposition 3.9, we conclude that E is concave on R + .

Thus, introducing the complex-valued function
(3.29) Then, using thath(ξ) = h(−ξ) and that W is even, we conclude that We will compute the integral in the right-hand side of (3.30) by using Cauchy's residue theorem. First we notice that h is real-valued and nonnegative on the imaginary line since Also, since f ∈ C ∞ c (R), h is a holomorphic function on C. To establish the decay of h on the upper half-plane, we use that h(z) = H(z) 2 , where Using the fact that e ixz = 1 iz d dx e ixz and integrating by parts, we get for z = 0, Since f is odd, f (0) = 0, so that integrating by parts once more, we have Therefore, where C = (|f (0)| + f L 1 ) 2 . Using the curves γ k , Cauchy's residue theorem yields where J k refers to the poles enclosed by Γ k . Taking into account (3.31), we see that so that the decay in (5) gives that the integral goes to 0 as k → ∞. Therefore, using the dominated convergence theorem, we can pass to the limit in (3.32), and using (3.30), we conclude that condition (H3') is satisfied.
The following propositions provide estimates for the curve E min near the origin.
Proposition 3.15. There exist constants q 1 , K 1 , K 2 > 0, depending on W C 3 , such that As an immediate consequence of Propositions 3.14 and 3.15, is that E min is right differentiable at the origin, with E + min (0) = √ 2. Moreover, if E min is concave we also deduce that E min is strictly subadditive as a consequence of the following elementary lemma (see e.g. [10,24]).  In particular, if E min is concave on R + , then E min is strictly subadditive on R + .
The proof of Proposition 3.15 is inspired on the fact that the Korteweg-de Vries (KdV) equation provides a good approximation of solutions of the Gross-Pitaveskii equation when W = δ 0 in the long-wave regime [59,12,25]. Our aim is to extend this idea to the nonlocal equation (NGP). Let us explain how this works in the case of solitons, performing first some formal computations. We are looking to describe a solution of (TW W,c ) with c ∼ √ 2, so we consider c = 2 − ε 2 , and use the ansatz u ε (x) = (1 + ε 2 A ε (εx))e iεϕε(εx) .
Therefore, setting W ε (ξ) := W(εξ), (3.36) i.e. W ε (x) = W(x/ε)/ε in the sense of distributions, we deduce that u ε is a solution to (TW W,c ) if (A ε , ϕ ε ) satisfies To handle the nonlocal term, we use the following lemma. where Proof. Let us set Now, by Taylor's theorem and the fact that ( W) (0) = 0, we deduce that for all ξ ∈ R and ε > 0, there exists z ε,ξ ∈ R such that Replacing this equality into (3.40), we conclude that which completes the proof of the lemma.

Proof. Let us first compute the momentum. Bearing in mind that
so using that R sech 4 (x)dx = 4/3 and that R sech 6 (x)dx = 16/15, we obtain the expression for p(v ε ) in (3.47). For the kinetic energy we can proceed in the same manner. Indeed, using that we get Now, for the potential energy, invoking Lemma 3.18 and (3.44), we have where we have also used that W (0) = ω 2 − 1. Adding the expressions for E k and E p , we obtain the estimate for the energy in (3.47).
Proof of Proposition 3.15. For q small, we can parametrize q as a function of ε as so q ε is a strictly increasing function of ε ∈ [0, 1]. The idea is to express ε in terms of q ε in order to obtain E(v ε ) in (3.47) as a function of q ε . Then (3.35) will follow from the facts that p(v ε ) = q ε and that E min (q ε ) ≤ E(v ε ). For notational simplicity, we set Applying Taylor's theorem and noticing that ε 5 /10 ≤ s ε , we infer that there is some p ε ∈ (s ε , 2s ε ) such that Using again (3.49), we conclude that Combining this asymptotics with (3.47), (3.48) and (3.49), we get , we conclude that (3.35) holds true.
We are now in position to prove Theorem 2.
By Corollary 3.3, q * > 0.027. Let us proof now the rest of the statement in (iv). Since E min is nondecreasing on [0, q * ), if we suppose that E min is not strictly increasing, then E min is constant in some interval [a, b], with 0 ≤ a < b < q * . Since E min is concave, this implies that E min is constant on [a, ∞) and therefore E min (a) = E min (q * ), which contradicts the definition of q * in (9). Finally, we remark that if E(v) < E min (q * ), for some v ∈ E(R), using the fact that E min (0) = 0, the intermediate value theorem gives us the existence of someq ∈ [0, q * ) such that E(v) = E min (q). Sinceq < q * , the definition of q * implies that |v| does not vanish.
We now establish (v). Arguing by contradiction, we show that E min (q) < √ 2q, for all q > 0. Indeed, in view of (3.2), let us suppose that for some p > 0 we have E min (p) = √ 2p. Since E min is concave, the function q → E min (q)/q nonincreasing, thus Therefore E min (q) = √ 2q, for all q ∈ (0, p), which contradicts (ii).
At this point, we recall that the concavity of E min implies that E + min is right-continuous, so that, by Corollary 3.17, we have E + min (q) → E + min (0) = √ 2, as q → 0 + . Using also that E min is nondecreasing, (3.2) and Corollary 3.17, we deduce the other statements in (v).

Compactness of the minimizing sequences
We start now the study of the minimizing sequences associated with the curve E min . The following result shows that the set S q in Theorem 4 is nonempty, and also allows us to establish the orbital stability in the next section.
Theorem 4.1. Assume that W satisfies (H1) and (H2), and that E min is concave on R + . Let q ∈ (0, q * ) and (u n ) in N E(R) be a sequence satisfying as n → ∞. Then there exists v ∈ N E(R), a sequence of points (x n ) such that, up to a subsequence that we still denote by u n , the following convergences hold
In the rest of the section we will assume that the hypotheses in Theorem 4.1 are satisfied and therefore the conclusion in Theorem 2-(v) holds. Thus, in the sequel, E min is strictly subadditive and E min (q) < √ 2q, for all q > 0.
For the sake of clarity, we state first the following elementary lemma.
Lemma 4.2. Let (u n ) be a sequence as in Theorem 4.1. Then there is function u ∈ N E(R) such that, up to a subsequence, In addition, E(u) ≤ E min (q), and writing u = ρe iφ and u n = ρ n e iφn , the following relations hold, up to a subsequence, for all A > 0, Proof. In view of (4.1), E(u n ) is bounded, so that, using also Lemma 2.1, we deduce that u n and that η n := 1 − |u n | 2 are bounded in L 2 (R) and that u n is bounded in L ∞ (R). Therefore, by weak compactness in Hilbert spaces and the Rellich-Kondrachov theorem, there is a function u ∈ H 1 loc (R) such that, up to a subsequence, the convergences in (4.6)-(4.8) hold, as well as (4.9), and also u L 2 (R) ≤ lim inf n→∞ u n L 2 (R) . (4.12) At this point we remark that the function B(f ) = R (W * f )f is continuous and convex in L 2 (R), since W ≥ 0 a.e. Thus it is weakly lower semi-continuous, so that (4.13) Combing with (4.12), we deduce that E(u) ≤ E min (q). Using (4.8) and the fact that W ∈ M 2 (R), we get W * η n W * η in L 2 (R), (4.14) which together with (4.6) lead to (4.10).
Since q ∈ (0, q * ), Theorem 2 and the fact that E(u) ≤ E min (q) < E min (q * ) imply that u ∈ N E(R), so that we can write u = ρe iφ . Then, setting u n = ρ n e iφn and by using that E k (u n ) is bounded and (4.6), we get for A > 0, so that, up to a subsequence, φ n φ in L 2 ([−A, A]). Using again (4.6), we then establish (4.11).
Proof of Theorem 4.1. By hypothesis, we can assume that E(u n ) ≤ 2E min (q). (4.15) Since E min (q) < √ 2q, we have Σ q ∈ (0, 1), so that applying Lemma 2.4 with L = 1 + Σ q , and Lemma 2.5 with E = 2E min (q) and m 0 =Σ q := Σ q /L, we deduce that there exist an integer l q , depending on E and q, but not on n, and points x n 1 , x n 2 , . . . , x n ln , with l n ≤ l q such that and Since the sequence (l n ) is bounded, we can assume that, up to a subsequence, l n does not depend on n and set l * = l n . Passing again to a further subsequence and relabeling the points (x n j ) j if necessary, there exist some integer , with 1 ≤ ≤ l * , and some number R > 0 such that Hence, by (4.17), we deduce that Applying Lemma 4.2 to the translated sequence u n,j (·) = u n (· + x n j ), we infer that there exist functions v j = ρ j e iφ j ∈ N E(R), j ∈ {1, . . . , }, satisfying the following convergences η n,j := 1 − |u n,j | 2 η j := 1 − |v j | 2 , in L 2 (R), (4.22) as n → ∞, and also where u n,j = ρ n,j e iφ n,j and q j = p(v j ). Moreover, using (4.16) and (4.20), we infer that In particular, v j cannot be a constant function of modulus one. Now we focus on proving the following claim.
For this purpose, we fix µ > 0. By the dominated convergence theorem, there exists By (4.18), we can assume that B(x n k , R µ ) ∩ B(x n j , R µ ) = ∅, for all 1 ≤ k = j ≤ . Hence, using (4.24) and (4.31), we deduce that there exists N µ ≥ 1, such that for all n ≥ N µ and for all By adding the inequality (4.32) from j = 1 to j = , we conclude that Similarly, using again the dominated convergence theorem and possibly increasing R µ , we obtain for all 1 ≤ j ≤ , (4.34) By (4.25), and increasing N µ if necessary, we have for n ≥ N µ , Combining (4.34), (4.35) and adding from j = 1 to j = , we deduce that Applying the same argument to η n,j φ n,j and η j φ j instead of (W * η n,j )η n,j and (W * η j )η j , we get Now we handle the integrals on Let us start with the momentum. We split p(u n ) as Rµ −Rµ η n,j φ n,j + p Aµ (u n ), with p Aµ (u n ) := 1 2 Aµ η n φ n . (4.38) By (2.4), (2.11), (4.15) and (4.19), we obtain Hence, p Aµ (u n ) is is uniformly bounded with respect to n and µ, so that, passing possibly to a subsequence (in n and µ), we infer that there existsq ∈ R such that Hence, passing to the limit n → ∞ and then letting µ → 0 in (4.37), and using (4.38), we obtain (4.29). To prove (4.28), we first remark that since E k (u n ) and E p (u n ) are bounded, passing possibly to a subsequence, there are constants Thus, decomposing the kinetic part as Rµ −Rµ |u n,j | 2 , and using (4.33), we deduce as before that To prove (4.28), it remains to study the potential energy. However, E p (u) is more involved because of the nonlocal interactions. To make the decomposition, we introduce the functions so that Using Plancherel's identity, the Cauchy-Schwarz inequality and (2.4), we deduce that R (W * g n,µ )f n,µ ≤ W L ∞ (R) g n,µ L 2 (R) f n,µ L 2 (R) ≤ C(E min (q)), and the same argument shows that R (W * f n,µ )f n,µ can also be bounded in terms of E min (q). Passing possibly to a subsequence, we conclude that there existsẼ p ≥ 0 such that lim µ→0 lim n→∞ R (W * f n,µ )f n,µ = 4Ẽ p . (4.42) We will show that lim Assuming (4.43), we can now establish inequality (4.28). Indeed, letting n → ∞ and then µ → 0 in (4.41), and using (4.42) and (4.43), we obtain Combining with (4.36), we have and bearing in mind that E min (q) = E k +E p and that E(v j ) ≥ E min (q j ), inequality (4.28) follows by adding (4.40) and (4.44).
It remains to show (4.43). By definition of g n,µ , we obtain (W * f n,µ ) (x + x n j )η n,j (x)dx.
Now we establish an inequality betweenq andẼ that will be key to conclude that both quantities are equal to zero.
We suppose first thatq > 0. By definition of Σ q in (2.22), and using thatΣ q = Σ q /L < Σ q , we have In addition, since E min is concave, we obtain for all 0 < p < q, Then, setting s := q −q = j=1 q j , the assumptionq > 0 implies that s < q, and combining with (4.49), (4.56) and (4.57), we also obtain Hence, using (4.28), we get Since E min is even, nondecreasing and subadditive, the inequality s ≤ j=1 |q j | yields which contradicts (4.58). Thusq ≤ 0 and (4.29) gives q ≤ j=1 |q j |. As before, this implies that On the other hand, sinceẼ ≥ 0, we see from (4.28) that In view of (4.28) and (4.49), (4.59) yieldsẼ = 0 andq = 0. Finally, if there are at least two nonzero values q k and q m , with 1 ≤ k = m ≤ , then the strictly subadditivity of E min implies that contradicting (4.59). Therefore we can suppose without loss of generality that = 1, which finishes the proof of Claim 3.

Stability
We start recalling the following result concerning the Cauchy problem.
(i) W ∈ M 1 (R) and W ≥ 0 in a distributional sense.
Then, for every w 0 ∈ H 1 (R N ) there exists a unique solution Ψ ∈ C(R, φ 0 + H 1 (R N )) to (NGP) with the initial condition Ψ 0 = φ 0 + w 0 . Moreover, the energy is conserved, as well as the momentum as long as inf x∈R |Ψ(x, t)| > 0.
In the case (ii), we also have the growth estimate for any t ∈ R, where C is a positive constant that depends only on E(Ψ 0 ), W L ∞ , φ 0 and σ.
Let us remark that the author in [27] uses a sightly different definition of the momentum to allow a possible vanishing of Ψ(t). However, the proof of the conservation of momentum in [27] also applies to our renormalized momentum as long as Ψ(t) ∈ N E(R). We also notice that other statements for Cauchy problem for the Gross-Pitaevskii equation have been established in different topologies when W = δ 0 (see e.g. [60,34,32,9,30,29] and the reference there in), and these results can probably be adapted to our nonlocal framework.
For the proof of Theorem 5.1, the author proves first a local well-posedness result for W ∈ M 3 (R). Then conditions (i) and (ii) are used to show that the solution is global. In [27], it is also established that the solution is global in dimensions greater that 1, provided that W ≥ σ > 0 a.e. However, the proof given by the author does not apply in the one-dimensional case. Using Lemma 2.1, we can partially fill this gap.
Then we have the same conclusion as in Theorem 5.2, including the growth estimate (5.1), with a constant C depending only on E(Ψ 0 ), W L ∞ , φ 0 and κ.
Proof. In view of the local well-posedness established in Theorem 1.10 in [27], to prove that the solution is global, we only need to show that the solution Ψ(t) = φ 0 + w(t) defined (T min , T max ), satisfies T max = ∞ and T min = −∞. In view of the blow-up alternative in the mentioned theorem, it is sufficient to prove that w(t) L 2 (R) remains bounded in any bounded interval of (T min , T max ). Indeed, from (NGP), we have (see equation (63) in [27]) where η(t) = 1 − |u(t)| 2 . From Lemma 2.1, we deduce from the conservation of energy on (T min , T max ), that there exists a constant K > 0, depending on κ and E(Ψ 0 ), such that η(t) L 2 (R) ≤ K, for all t ∈ (T min , T max ).
As explained in Section 6 in [27], Theorem 5.2 allows us to show that the solutions in the energy space are global. Theorem 5.3. Assume that W ∈ M 3 (R) is an even distribution satisfying (5.2). Then for every Ψ 0 ∈ E(R), there exists a unique Ψ ∈ C(R, E(R)) global solution to (NGP) with the initial condition Ψ 0 . Moreover, the energy is conserved, as well as the momentum as long as inf x∈R |Ψ(x, t)| > 0.
Proof of Theorem 3. In view of Remark 2.2, we deduce if W ∈ M 3 (R) is an even distribution, with W ≥ 0 a.e. on R, and W of class C 2 in a neighborhood of the origin, then W satisfies (5.2), for some κ ≥ 0. Therefore, we can apply Theorem 5.3 and the conclusion follows.
The rest of the section is devoted to prove that the set S q is orbitally stable in the energy space. Using the Cazenave-Lions approach [21] and Theorem 4.1, we obtain the following result.
Theorem 5.4. Assume that W ∈ M 3 (R) satisfies (H1) and (H2). Suppose also that E min is concave on R + . Then, S q is orbitally stable for (E(R), d) and for (E(R), d A ), for all q ∈ (0, q * ). Moreover, for all Ψ 0 ∈ E(R) and for all ε > 0, there exists δ > 0 such that if where Ψ(t) is the solution of (NGP) associated with the initial condition Ψ 0 .
Notice that for u, v ∈ E(R), we have d(u, v) ≤ d A (u, v), and thus Therefore, the implication in (5.3) shows the orbital stability for the distance d and d A .
In order to prove Theorem 5.4, we will use the following lemma.
In particular, we have the continuity of the energy E(v n ) → E(v) (with respect to d). In addition, if v n , v ∈ N E(R), then we also have the continuity of the momentum p(v n ) → p(v).
Proof. First, we remark that since for all n ∈ N. By the sharp Gagliardo-Nirenberg interpolation inequality and using that ||w| | = |w |, for w ∈ H 1 loc (R), we have |v n | − |v| L ∞ (R) ≤ |v n | − |v| L 2 (R) |v n | − |v| L 2 (R) ≤ 2M |v n | − |v| L 2 (R) , so the first convergence in (5.4) follows. Similarly, we deduce the second one noticing that Therefore (5.4) is proved. In particular, we have v n → v in L 2 (R) and η n = 1−|v n | 2 → η = 1−|v| 2 in L 2 (R), so that E(v n ) → E(v). For the momentum, writing v n = |v n |e iθn as usual, we have p(v n ) = 1 2 R η n θ n , so it suffices to prove that θ n θ in L 2 (R) to conclude that p(v n ) → p(v), where v = |v|e iθ . To establish the weak convergence of θ n , we notice that since |v n | → |v| in L ∞ (R), there exists C > 0 such that inf R |v n | ≥ C, for all n ∈ N. Hence, Since E(v n ) is bounded, we conclude as in Lemma 4.2 that for a subsequence, θ n k θ in L 2 (R), as k → ∞. Therefore, we conclude that p(v n k ) → p(v). Since the limit does not depend on the subsequence, we deduce that p(v n ) → p(v).
Proof of Theorem 5.4. Arguing by contradiction, we suppose that there exist ε 0 > 0, (δ n ), (t n ) and (u n 0 ) ⊂ E(R) such that δ n → 0, d(u n 0 , S q ) < δ n (5.5) and inf y∈R d A (u n (· − y, t n ), S q ) ≥ ε 0 , (5.6) where u n denotes the solution to (NGP) with initial data u n 0 . In particular, from (5.5) we deduce that there is v n ∈ S q such that d(u n 0 , v n ) < 2δ n . (5.7) Since E(v n ) = E min (q) and p(v n ) = q, applying Theorem 4.1 to (v n ), we infer that there exists v ∈ S q and points (a n ) such that, up to a subsequence, the functionṽ n (x) = v n (x + a n ) satisfies Using also the estimate (4.5) in Theorem 4.1, we conclude that and also d A (ṽ n , v) → 0. On the other hand, by the triangle inequality and (5.7), d(u n 0 (· + a n ), v) ≤ d(u n 0 (· + a n ),ṽ n ) + d(ṽ n , v) < 2δ n + d(ṽ n , v).
Combining with (5.9), we conclude that d(u n 0 (· + a n ), v) → 0. Applying the conservation of energy in Theorem 5.3 and Lemma 5.5, we thus get, for all t ∈ R, E(u n (t)) = E(u n 0 ) = E(u n 0 (· + a n )) → E(v) = E min (q). Otherwise, there are values s n , with |s n | ≤ |t n |, such that inf R |u n (s n )| = 0. By (5.10), we conclude that E(u n (s n )) → E min (q) and thus, using that E min is strictly increasing on (0, q * ), we can find n 0 such that E(u n (s n )) < E min (q * ), for all n ≥ n 0 . This is a contradiction because, by Theorem 2, this implies that u n (s n ) ∈ N E(R).
In view of (5.11), we can proceed as before invoking the conservation of momentum in Theorem 5.3 and Lemma 5.5, to obtain p(u n (t n )) = p(u n 0 ) = p(u n 0 (· + a n )) → p(v) = q. (5.12) By (5.10) and (5.12), we can apply Theorem 4.1 to (u n (t n )). Then, reasoning as before, we deduce that there exist w ∈ S q and (b n ) such that, up to a subsequence, d A (u n (· + b n , t n ), w(·)) → 0, (5.13) which contradicts (5.6).

Euler-Lagrange equations and proof of Theorem 4
In this section we establish the Euler-Lagrange equations associated with the minimization problem, which will allow us to complete the proof of Theorem 4. Since the energy and momentum functional are not defined on a vector space, the notion of differential is not trivial. For our purposes, it suffices consider the directional derivatives using only smooth functions with compact support. More precisely, for u ∈ E(R) we define for all h ∈ C ∞ c (R), where we also suppose that u ∈ N E(R) for the definition of dp(u) so that p(u + th) is actually well defined for t small enough. Lemma 6.1. Assume that W satisfies (H1). Then for all h ∈ C ∞ c (R), we have In particular, for all c ∈ R, dE(u) = c dp(u) if and only if u satisfies (TW W,c ).
Notice that the elliptic regularity theory shows that if u is a solution of (TW W,c ), then u is smooth. More precisely, the following result stated in higher dimensions in [28] applies without changes in dimension 1.
Proof of Lemma 6.1 . Using (2.1), the differential in (6.1) is a straightforward consequence of the definition of dE. To show (6.2), let us fix u ∈ N E(R) and h ∈ C ∞ c (R). Then Therefore we obtain (6.2) noticing that − ih, u u, u + iu , u u, h = iu , h |u| 2 .
The last assertion in the statement follows from the fact that if for some v ∈ E(R) we have Theorem 6.3. Suppose that E min is concave on R + and that u ∈ S q , with q > 0. Then there exists c q satisfying

3)
such that u is a solution of (TW W,c ) with of speed c = c q .
Proof. Let u ∈ S q , so that p(u) = q and E(u) = E min (q). Notice that since q > 0, u is not a constant function. Let h ∈ C ∞ c (R). From the definition of E min we have, for all t > 0, If dp(u)[h] > 0, then p(u + th) ≥ p(u) = q for t > 0 small enough, so that letting t → 0 + , we obtain Replacing h by −h, we obtain the following inequalities and Since the functionals dp(u), dE(u) : C ∞ c (R) → R are linear, to establish the Euler-Lagrange equations, it is enough to show that Ker dp(u) ⊂ Ker dE(u). (6.6) Indeed, by Lemma 3.2 in [18], this implies that there exists some c q ∈ R such that dE(u) = c q dp(u), (6.7) and therefore, by Lemma 6.1, u is a solution of (TW W,c ) with c = c q To prove (6.6), let us consider φ ∈ Ker dp(u). Since u is nonconstant, there exists some function ψ ∈ C ∞ c (R) such that dp(u)[ψ] = 0. Thus, for all n ∈ N, we have dp(u)[ψ + nφ] = dp(u)[ψ] = 0. It remains to show (6.3). Let h 0 ∈ C ∞ c (R) such that dp(u)[h 0 ] = 1. Then (6.7) implies that dE(u)[h 0 ] = c q . It follows from (6.4) that E + min (q) ≤ c q ≤ E − min (q), (6.8) which finishes the proof.
Remark 6.4. It is possible to establish the Euler-Lagrange equations using an argument based on the implicit function theorem, without invoking the concavity of E min . Even thought the former argument is more general, we gave the proof using the concavity because it is simpler.
Proof of Theorem 4. Combining Theorems 4.1, 5.4 and 6.3, we obtain that the set S q is nonempty, orbitally stable and that any u ∈ S q is a solution of (TW W,c ). Using (6.3) and Theorem 2-(v), we get the properties for c q , except that c q > 0. Arguing by contradiction, we suppose that there exists p ∈ (0, q * ) such that c p = 0. Thus, by (10) and (11), we get E + min (p) = 0. Since E min is concave, we have for all r < s, E − min (r) ≥ E + min (r) ≥ E − min (s) ≥ E + min (s) ≥ 0, which implies that E − min = E + min = 0 on [p, ∞), so that E min is constant on [p, ∞), which contradicts that E min is strictly increasing on [p, q * ). This completes the proof of the theorem.

Some numerical simulations
In this section, we numerically illustrate the properties of the minimizing curve through some simulations. The numerical method is based on the projected gradient descent and the convolution is computed by the fast Fourier transform algorithm. Given W (or W) and some q > 0 close to 0, we compute the corresponding soliton u q (i.e. p(u q ) = q) and its energy E(u q ). We then increase the value of q > 0 until we obtain enough points to plot E min .
First, we show our results for the examples (i) and (ii) in Section 1. In Figures 2 and 3, we can see E min and the modulus of the solitons associated with q = 0.05, q = 0.55, q = 1.1 and q = 1.5, for the potentials with α = 0.05, β = 0.15, and with α = 0.8. In both cases, we observe that E min is concave and that the line √ 2q is a tangent to the curve. We notice that the shapes of the solitons in Figure 3 and the solitons in Figure 1 are quite similar. On the other hand, the solitons in Figure 2 are very different, they have values greater than 1 and exhibit a bump on R + . Notice also that the curves E min for both potentials seem to be constant for q > 1.55.
We end this section showing some numerical simulations for two interesting potentials. The first one has been proposed in [57] as simple model for interactions in a Bose-Einstein condensate. It is given by a contact interaction δ 0 and two Dirac delta functions centered at ±σ, Noticing that W σ (ξ) = 2 − cos(σξ), we see that for σ > 0, W σ fulfills (H1), (H2), and that W σ is analytic in C, but is exponentially growing on H. Thus, W σ does not satisfy the assumption (5) in (H3). We can also check that (H3') is not fulfilled. Nevertheless, the results of the simulation depicted in Figure 4 show that E min is concave, and in that case Theorem 4 gives the orbital stability of the solitons illustrated in Figure 4. Finally, we consider the potential W a,b,c (ξ) = (1 + aξ 2 + bξ 4 )e −cξ 2 , (7.4) that it has been proposed in [8,56] to describe a quantum fluid exhibiting a roton-maxon spectrum such as Helium 4. Indeed, as predicted by the Landau theory, in such a fluid, the dispersion curve (3) cannot be monotone and it should have a local maximum and a local minimum, that are the so-called maxon and roton, respectively. In Figure 5, we see the dispersion curve associated with potential (7.4), with a = −36, b = 2687, c = 30. In this case, there is a maxon at ξ m ∼ 0.33 and a roton at ξ r ∼ 0.53. For these values, (H1) is satisfied, but not (H2) nor (H3'). However, we observe in Figure 6 that the energy curve is still concave, and that the straight line √ 2q is still a tangent to the curve. Moreover, we found the same critical value as before for the momentum, i.e. q * ∼ 1.55.
It remains to prove the bound in (A.1). Using that and that |f (x)| ≤ exp(−1/x) .
Combining with (A.2), we conclude that |χ (x)| ≤ 4e −2 e In particular, there exists a sequence of integers (n l ) l , with n l → ∞, as l → ∞, such that b km a km |f n l | ≥ δ 2 , for all m ≥ m 0 , for all l ≥ l 0 , for some l 0 ∈ N. Therefore, we deduce that which is a contradiction.