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Orbital stability of the black soliton for the quintic Gross-Pitaevskii equation

In this work, a rigorous proof of the orbital stability of the black soliton solution of the quintic Gross-Pitaevskii equation in one spatial dimension is obtained. We first build and show explicitly black and dark soliton solutions and we prove that the corresponding Ginzburg-Landau energy is coercive around them by using some orthogonality conditions related to perturbations of the black and dark solitons. The existence of suitable perturbations around black and dark solitons satisfying the required orthogonality conditions is deduced from an Implicit Function Theorem. In fact, these perturbations involve dark solitons with sufficiently small speeds and some proportionality factors ari\-sing from the explicit expression of their spatial derivative. We are also able to control the evolution of the modulation parameters along the quintic Gross-Pitaevskii flow by estimating their growth in time. As a direct consequence, we also prove the orbital stability of the dark soliton in a small speed interval.


Introduction
In this work we consider the one-dimensional quintic Gross-Pitaevskii equation (quintic GP) iu t + u xx = (|u| 4 − 1)u, (t, x) ∈ R 2 , u(0, x) = u 0 (x), (1.1) where u is a complex-value function and the initial data u 0 satisfies the boundary condition lim |x|→+∞ |u 0 (x)| 2 = 1. (1.2) From the physical point of view is interesting to look for solutions u(t, x) of (1.1) satisfying the boundary condition (1.2) for all t ≥ 0. This is a defocusing nonlinear Schrödinger equation modeling for example ultra-cold dilute Bose gases in highly elongated traps. More specifically, it describes dynamics of weak density modulations of one dimensional bosonic clouds (Tonks-Girardeau gases) when the tight transverse confinement potential is turned off. In fact (1.1), in the case of one dimensional atomic strings, allows to explain many fermionic properties arising in one dimensional chains of bosons, phenomena usually named as bosonic fermionization. See [7,18,19,21] and references therein for a complete background on the physical phenomena accounted for by this quintic defocusing model.
The quintic GP equation is phase (also called U (1) invariance) and translation invariant, meaning that if u is a solution of (1.1), then e iθ u(t, x + a), a ∈ R, θ ∈ R, is also a solution of (1.1). The quintic GP (1.1) also bears Galilean invariance, namely e i( c 2 x− c 2 4 t) u(t, x − ct), c ∈ R, but this will not be used in our approach. Note moreover that in (1.2) the asymptotic value 1 can be changed to any number ζ > 0 without loss of generality by rescaling the values of u with a rotation in time u = e i(1−ζ 2 )t v. Under this change (1.1) recasts as iv t + v xx = (|v| 4 − ζ 2 )v, (t, x) ∈ R 2 .
Some previous results on the Cauchy problem of (1.1) are well known in the literature. For example, the global well-posedness of (1.1)-(1.2) was established by Gallo in [9], where it was considered the general model iu t + u xx + f (|u| 2 )u = 0, (1.6) with regular nonlinearity f : R + → R satisfying f (1) = 0 and f (1) < 0. (1.6) includes, as particular cases, other important equations such as • Pure powers: f (r) = 1 − r p , p ∈ Z + .
Concerning solutions, complex constants with modulus one are the simplest solutions contained in (1.1). Moreover, and with respect to particular soliton solutions, and specifically to the stability of solitonic waves for (1.6), the situation is well understood in the case of the cubic GP equation, proffiting its integrable character ( [24]). Indeed, it is well known that the black soliton of the cubic GP is which is a stationary, i.e. time independent, wave solution. Furthermore, the study of orbital and asymptotic stability for ν 0 (x) was considered in several works [2,8,12,15]. Beside that, for some cases of the cubic-quintic model (f (r) = (r − 1)(2a + 1 − 3r)) Lin showed the stability of traveling solitonic bubbles [20]. See [1,2,22] for more details on these models. Finally, Chiron [4] dealt with stability and instability problems for stationary and subsonic traveling waves giving an explicit condition on a general C 2 nonlinearity f for stability (as well as for instability) in the NLS model. Such a general approach also encompasses the quintic GP (1.1) but with a different method, which as far as we know, does not give clues on how to proceed with the asymptotic stability of stationary kinks in this specific model.
In comparison with the cubic GP, the non-integrability of the quintic GP equation makes the search of solutions even harder as well as the rigorous study of the analytical properties related to them, as we will have the opportunity to check in the following sections of this work. Actually, and as far as we know, the black soliton solution for the quintic GP was discovered by Kolomeisky et al. (see [18,Eq.(12)]) Beside that, we present in this work the explicit expression of this solution as well as its formal derivation (see Section 2). Namely, the black soliton of (1.1) is given by which is a solution of φ + (1 − φ 4 )φ = 0, (1.8) the corresponding differential equation describing stationary real waves of (1.1) with u(0, x) = φ(x) (see Section 2 for further details). Therefore, it is natural to question whether, in the case of the quintic GP, the stability of φ 0 is preserved in some sense. In fact, the main result of this work is the following (see Section 5 for a more detailed version and proof of this result) Theorem 1.1. The black soliton solution φ 0 (1.7) of the quintic GP equation (1.1) is orbitally stable in a subspace of the energy space Σ (1.4).
The black soliton (1.7), stationary by nature, belongs to a greater family of traveling waves. As far as we know, an explicit and correct expression of a traveling wave family of solutions for the quintic GP (1.1) was missed. In fact, we show in this work that the quintic GP (1.1)-(1.2) also contains explicit traveling-wave solutions. These waves, with the form are currently known as dark solitons, a reminiscent terminology coming from nonlinear optics (see [17]). The function φ c satisfies the complex nonlinear ordinary differential equation (1.9) Indeed, for |c| < 2 we are able to obtain the following explicit family of dark solitons: and µ ≡ µ(c) verifying the constraint relation for all |c| < 2 and which comes from (1.2). Therefore, µ is explicitly . (1.14) Note that Also notice that, as a consequence of the above limits, we get As a direct consequence of Theorem 1.1, we obtain the orbital stability of dark solitons (1.10) in a small speed region c ∈ [0, c) or also for c ∈ (−c, 0] for c 1, due to the discontinuity in (1.15).
Remark 1.2. The main ingredient is the use of the associated family of complex dark profiles φ c whose real parts are odd functions with respect to the speed c and are laterally approximated to the stationary black solution when c → 0 + , as shown in (1.15). The same arguments can be applied to prove the orbital stability of −φ 0 . Note that as we said before, the orbital stability of black solitons (stationary kinks) for defocusing NLS with a general C 2 nonlinearity is covered in the pioneer work of Chiron [4,Theorem 24]. Besides that, our motivation to deal with the orbital stability of black solitons with this particular quintic nonlinearity comes firstly from the physical relevance that this model has in quantum gases as we said above. We were also motivated to prove this orbital stability result without resorting to a hydrodynamical formulation because, this approach only describes non-vanishing solutions, and therefore excluding the black soliton solution. From a specific mathematical point of view, we avoid technical issues coming from the non integrable character of the model and therefore not being allowed to use classical integrability methods. Therefore, the main goal of this paper is to revisit the orbital stability of black solitons (i.e. stationary kinks) getting rid of the hydrodynamical formulation. In particular and as a wished consequence, this makes possible to pave the technical approach to study the asymptotic stability of the black soliton, in the same way as it was previously done by Gravejat-Smets [15] for the simplest case of the cubic GP.
More specifically, our proof establishes the coercivity of the functional Q c [z] (3.46), when the function z satisfies suitable orthogonality conditions well adapted to this specific quintic nonlinearity. These orthogonality conditions are guaranteed by the introduction of modulation parameters (see Proposition 4.2) and needed to perturb a stationary object as the black soliton of the quintic GP. This approach has the advantage to show a better control on the perturbation with respect to the black soliton. Such a control is very useful in order to tackle the asymptotic stability of the black soliton, problem which is currently in advance.
Besides that, we are able to explicitly obtain (despite the nonintegrable nature of the model) dark solitons (traveling wave kinks) of the quintic GP, being one of the special cases where it is still possible to get these explicit solutions (the other one is the cubic GP). The explicit knowledge about dark solitons of quintic GP allowed us to perform precise perturbations on the black soliton. More specifically, our proof is able to capture the discontinuity of dark solitons of the quintic GP equation with respect to the speed parameter around zero, being that we have to consider a bilateral proof for each case.
Summarizing, we highlight here the main results involved in the proof of the orbital stability of the black soliton of the quintic GP (1.1). Our proof introduces new theoretical and technical tools, with respect to the integrable cubic GP equation [15] or more recently with respect to systems of cubic GP equations [5]. These tools are specially suited to deal with the associated nonlinear solutions of (1.1), namely the black soliton φ 0 (1.7) and the dark soliton φ c profile (1.10). Specifically we introduced • a new family of traveling wave solutions φ c (1.10), close to the stationary black soliton φ 0 , for the quintic GP equation (1.1). Obtaining non-constant solutions of this non-integrable equation is not a simple task, and even more in the case when one has to solve a coupled nonlinear ODE system (1.9). Only by proposing a suitable ansatz and a careful tuning of the free parameters allowed us to obtain them. Just, compare these solutions of the quintic GP equation with the corresponding ones of the cubic GP equation, where a simple complex constant translation gives the traveling family. See Section 2.2 for further reading. • a new metric d c . This is a weighted metric with nonlinear weight φ 3 c as it is dictated from the coercivity estimates we need to prove on the Ginzburg-Landau energy on black and dark solitons. See (2.14) for a precise definition of d c and also Propositions 3.2 -3.3.
• new functional spaces, in order to correctly measure the distance between black and dark solitons and their perturbations z. See Section 2.3 for details. • new orthogonality conditions associated to perturbations of the black and dark solitons (3.22) and (4.4) and specially adapted to the spectral properties of the quintic GP equation. • splitting of the stability proof in two different speed intervals (on the left and on the right hand side of c = 0) because the discontinuity of the dark soliton solution (1.10) in terms of the speed variable c.
We think that the technique introduced in this work can be used in the analysis of stability properties of black type solutions in different nonlinear quantum models, because it generalizes the constant factor 1 in the integrable cubic GP case (where its black soliton verifies U 0 = 1 √ 2 (1−U 2 0 )). See Section 5.1 for further details on applications.
In this work, we were able to overcome several technical issues coming from the nonlinear functional structure of the quintic GP and its black and dark solitons, by working in a small speed region c ∈ [0, c) or also for c ∈ (−c, 0] for c 1. Note that as a consequence of the discontinuous character of the dark soliton function in terms of the speed variable c (1.15), we have to consider along the orbital stability proof, not a continuous speed interval around c = 0, but two different speed intervals on the left and on the right hand side of c = 0. For the sake of simplicity we worked on c ∈ [0, c) since it gives +φ 0 in the limit. The proof for the (−c, 0] subinterval is performed in a similar way. Moreover, the apparent structural difference between black solitons in the cubic GP and the quintic GP, is reflected in many identities and related functions around these black (and dark) solitons, e.g. the Ginzburg-Landau energy E 2 (1.3) or the spatial derivative φ 0 .
The strategy we used for the proof of the orbital stability result for the black soliton φ 0 of (1.1) was focused to first show that the Ginzburg-Landau energy E 2 is coercive around the black and dark solitons. This was done by using some orthogonality relations based on perturbations z of the black and dark solitons and arising from the particular spectral problem related to (1.1), suitable nonlinear identities and some proper Gagliardo-Nirenberg estimates on functions of the black and dark solitons of (1.1).
We notice that the orthogonality conditions arising from the coercivity result (Proposition 3.2) on the black soliton φ 0 , do not include a linear term appearing after expansion of E 2 (1.3) around the black and dark solitons, and therefore we must deal with this remaining linear term along the proof, estimating it in a suitable way to obtain the expected bounds, in contrast with previous approaches ( [15]) where their natural orthogonality conditions imposed its cancellation.
After that main step, we continued with Proposition 4.2, proving, through a modulation of parameters, the existence of suitable perturbations z of the dark soliton which satisfy the orthogonality conditions defined in (4.1). Furthermore, we show in Corollary 4.5 and Proposition 4.7 that these modulation parameters and their growth are bounded in time.
Finally note that, related with the orbital stability, is the concept of asymptotic stability which essentially states the convergence of perturbations of the black soliton to a special element in the tubular neighborhood generated by its symmetries, e.g. phase and translation invariances. A detailed study on the asymptotic stability of the black soliton (1.7) of the quintic GP (1.1) is currently being made and it will appear elsewhere.
1.1. Structure of the paper. In Section 1 we introduce the problem and the main result. In Section 2 we obtain the black and dark solitons of (1.1) and describe some properties and nonlinear identities and norms based on them. In Section 3 we present the coercivity properties of the Ginzburg-Landau energy E 2 around black and dark solitons. In Section 4 we study the existence and time growth of some modulation parameters associated to black and dark solitons. Finally in Section 5 we prove the main Theorem on the orbital stability of the black soliton of (1.1), gathering the results obtained in the previous sections.

Acknowledgments
We really thank here to Henrik Kalisch and Didier Pilod for some valid discussions and observations along this work.

Derivation of black and dark solitons for the quintic GP
In this section we explain the derivation of the black and dark solutions given in (1.7) and (1.10). The following basic result will be useful for obtaining the black family (1.7).
See Appendix B for a proof of this identity.
2.1. Derivation of black solitons. Using (1.8), we get after multiplication by φ and then d dx From the boundary conditions at infinity in (1.2) we conclude that K 0 = 2 3 and we obtain the following first order ODE where we consider x 0 = φ −1 (0). Then, making the change s = φ(x) we have Without loss of generality we can assume x 0 = 0. Since |φ(x)| < 1, using Lemma 2.1 it follows that and consequently , which is a stationary real solution of the quintic GP (1.1)-(1.2) and named as black soliton.
An important observation is that the black soliton φ 0 (1.7) has a definite variational structure. More precisely, considering the Ginzburg-Landau energy E 2 [u] defined in (1.3) as the corresponding Lyapunov functional, and considering a small perturbation z of the black soliton φ 0 , namely a z ∈ H 0 (R) with H 0 (R) ⊂ H 1 loc (R) to be defined in (2.9), we get, after a power expansion in z of Because the first variation of E 2 vanishes for (1.8), the black soliton is characterized as critical point of the functional E 2 associated to the quintic GP (1.1). In fact, it is easy to see that Moreover, it is possible to state the following minimality's characterization on the black soliton solution Proof. This result is essentially contained in [2, Lemma 2.6], where the black soliton case for the cubic GP was considered. The extension to the quintic GP case, once we work with the energy E 2 is direct and does not require additional steps. We therefore skip the details.

2.2.
Derivation of dark solitons. Once obtained the black soliton (1.7), the detailed construction of the dark soliton solution (1.10) to (1.9) is presented in the Appendix A. A sketch of the derivation is the following: bearing in mind that (1.9) reduces to (1.8) at c = 0, we proposed a suitable ansatz like with a 1 , a 2 , a 3 and k as free parameters to be determined in order that (2.3) is actually a solution of (1.9), and verifying the asymptotic behavior Hence, substituting (2.3) into (1.9) and after lengthy manipulations, we got (1.10), with a 1 = µ 1 , a 2 = µ 2 and a 3 = µ, satisfying the relation (1.13) and k = κ as in (1.11).
Finally, we introduce the notion of dark profile.

Definition 2.3 (Dark profile).
Let c ∈ (−2, 2), and x 0 ∈ R be fixed parameters. We define the complex-valued dark profile φ c with speed c as follows .

(2.4)
Remark 2.4. Note that the profile φ c is the standard profile associated to the dark soliton solution (1.10). Note moreover, that although φ c is not an exact solution of (1.1), it can be interpreted as follows: for each (t, , is an exact dark soliton solution of (1.1) moving with speed c. Hereafter, and if no confusion arises, we will refer to the dark soliton by (2.4), setting x 0 = 0 for the sake of simplicity.

Preliminaries.
First of all, we introduce the following notation for the nonlinear weights and for the real and imaginary parts of the dark soliton . (2.7) To simplify the notation, we shall also denote Moreover, we define the following functional spaces: given c ∈ (−2, 2) we consider the weighted Sobolev space with the norm We will also use H real c (R) to denote the set of real-valued functions in H c (R), that is, Using the exponential decay of η c we can check that the space H c does not depend on the velocity c when |c| ≤ c, for some c small enough. Even more, the norms · Hc are equivalent with · H0 . For further details see Lemma 2.6. Therefore, hereafter we simplify the notation using the identification for all |c| < c. Beside that, we define a proper subset of Z(R) ⊆ H(R), namely which has metric structure with the distance.
for all |c| < c. Also notice that if u ∈ Z(R), from the computations in (1.5) we see that, the energy E 2 (1.3) is well defined for elements in Z(R).

Remark 2.5.
Similarly to the theory developed in [15] in the context of the cubic GP model, here we also have that the unique global solution u of (1.1) with initial data u 0 ∈ Z(R) remains continuous from R to Z(R) endowed with the metric structure induced by d c (2.14).

2.4.
Nonlinear identities and estimates for black and dark solitons. Now we present some nonlinear identities related to the black and dark solitons (1.7) and (1.10), which shall be useful along the work. Firstly we note that from (2.1) we get In comparison, the dark soliton satisfies the following identity: and which shows the localized character of φ c .
Notice that from (1.8), (2.1) and (2.2), the black soliton solution (1.7) satisfies the identity and curiously, by direct calculation we have also η 0 Coming back to (1.10) and using (1.14) and (1.13), we get that the explicit Ginzburg-Landau On the other hand, we get the right convergence of (2.17) to (2.16) when the speed c goes to 0, namely (see Figure 2) In fact, the expansion of (2.17) around φ 0 up to c 2 order is and therefore, for c small enough one gets For the sake of completeness, we also show here the following related amounts: We now have that if |c| < c, with some c 1, and , and therefore, we get that In the following lines, we show some interesting computations, which they are justified in Appendix C. Firstly we have that for all |c| < 2. On the other hand, because −1 < µ < 0 for all |c| < 2, we have with I c defined in (2.7). We also have the next useful estimates: for all |c| ≤ c with some c 1. For more details on these L 2 and L ∞ -norms, see Appendix C.1 and C.2 respectively.
The next estimate will be useful in subsequent technical results on perturbations of the black soliton φ 0 , and therefore we present a brief proof of it. Lemma 2.6 (Equivalent norms). Let φ 0 and φ c be the black and dark solitons (1.7) and (1.10) respectively. Then, there exists c ∈ (0, 2) such that for all |c|, |c * | < c and any z ∈ H c * . Furthermore, we conclude that H c ≡ H 0 for all |c| < c and there exist positive constants σ 1 and σ 2 such that Proof. From (2.31) and using the identity .

(2.34)
Now we show that the last two integrals are uniformly bounded in c, with |c| ≤ 1. To do this first we observe that max 1 + x 2 , (1 + x 2 )|x| cosh κ(c)x , for all x ∈ R and |c| ≤ 1. Furthermore, due the exponential decay of η c (x) one gets .
So, using this control, from (2.34) we conclude that Then, using that the functions are bounded on R (uniformly for |c * | < c), combined with the Sobolev embedding, we have Thus, (2.32) follows by substituting (2.36) into (2.35). Finally, in view of (2.32), and using the relation we check that H c ≡ H 0 for all |c| < c and further we have (2.33).

Coercivity of the Ginzburg-Landau energy
In this section we establish that the Ginzburg-Landau energy E 2 (1.3) is coercive and pseudocoercive around black and dark solitons respectively. First of all, we establish some preliminary notation and results.
We first expand the energy and N 0 [z] is the nonlinear term In the case of z = f is a real-valued function belonging to the space H real (R) (see (2.12)), Then, considering now H real (R) endowed with the inner product we have that H real (R), ·, · 0 is a Hilbert space with the induced norm Therefore, by the Riesz Theorem, there exists a bounded and self-adjoint operator T 0 such that and also for all f ∈ H real (R).
In particular, from (3.12) we conclude that 16) and also we can assume that i.e. we get uniform convergence on compact subsets of R.
Claim 2: It holds that η To prove this estimate we first observe that which implies that Then, from (3.11), (3.16) and using the exponential decay of η 0 we get the estimate (3.18) in Claim 2. Now given > 0, due to the exponential decay of η 0 , we can take a > 0 such that Then, from (3.17) one gets Therefore, by using (3.18), we get for all n n , so lim n→∞ η , and the proof is finished. Proposition 3.2 (Coercivity of E 2 around the black soliton). Let z ∈ H(R) be such that the perturbation φ 0 + z ∈ Z(R) and set ρ 0 = 2 Re(φ 0z ) + |z| 2 as in (3.1). Then there exists a universal positive constant Λ 0 > 0 such that as soon as Proof. The proof will be divided into 3 steps.
Step 1: There exists a constant Λ 1 > 0 such that for any function f ∈ H real (R) such that Then, using the Spectral Theorem, there exists a sequence{λ n } of eigenvalues for Q 0 with lim n→+∞ λ n = 1 2 and a Hilbert basis {e n } of H real (R) such that Q 0 e n = λ n e n , n ∈ N. Notice that 1 2 ] and λ n 1 2 . Now, let λ ∈ (−∞, 1 2 ] be an eigenvalue with f as the corresponding eigenfunction. Then for all g ∈ H real (R) we have Q 0 f, g 0 = λ f, g 0 . So, from (3.9), it holds that for all g ∈ H real (R). Thus, Note that, since φ 0 has exactly one zero, the Sturm-Liouville theory guarantees that λ = − 1 2 is the only negative eigenvalue of Q 0 with kernel given by the span(φ 0 ), more precisely λ 0 = − 1 2 < 0 = λ 1 < λ 2 < · · · · · · < 1 2 , and Ker( Q 0 + 1 Before presenting the Step 2 we remind that in the case of cubic GP treated in [15], the corresponding black soliton (denoted as U 0 ) satisfies the relation U 0 = 1 √ 2 (1 − U 2 0 ), and hence, the orthogonality condition but in the case of the quintic GP (1.1) this relation is not satisfied anymore.
Step 2: Let z ∈ H(R) fulfilling the orthogonality conditions (3.22). Then, it follows that with Λ 1 as in Step 1.
Proof of Step 2. Recall that from the expansion of E 2 in (3.2) we have . Applying Young's inequality, we obtain the estimate On the other hand, from Step 1, the first two orthogonality conditions in (3.22) imply that while, in addition, the last orthogonality condition in (3.22) ensures that Since Q 0 [z 1 ] ≥ 0, in order to complete the proof of Proposition 3.2, we remark that, bearing in mind the estimate in the Step 2, we only need to show the coercivity property for the operator Q 0 on the full variable z. We will explain this in the next step.
Step 3: Now we proceed with the proof of (3.21).
Proof of Step 3. We begin by estimating the term 2 3 φ 0 ρ 0 2 L 2 which appears in the lower estimate of the Step 2
Finally note that we can define the following tubular subset of U 0 + (α), (3.40) Coming back to the main question on the orbital stability, we use the pseudo-coercivity of E 2 around the black soliton φ 0 to small perturbations around the dark soliton φ c (see Proposition 3.3) . In fact, the idea to introduce a dark soliton family in this argument is to give an extra degree of freedom which allows us to obtain a non-degenerated matrix in the classical modulation theory, as it was explained in Section 4. In that case, the situation is different with respect to the cubic GP equation, because we can not assume the cancellation of the linear term iφ c , z C in our approach, given the orthogonality conditions arising naturally, from the particular structure of the associated spectral problem as we already saw in Proposition 3.2, e.g. (3.24). In fact, this extra technical difficulty introduced by the linear term iφ c , z C is overcome in Proposition 3.3, by using a previously computed L 2 norm (2.25).
Before establishing the main result, and with (2.8), we fix the following notation: there exists Γ c > 0, depending only on c, such that Remark 3.4. We have two important remarks: (a) Note that the quadratic term z 2 ρ c 2 L 2 is not appearing in (3.44) because the lower bound is already guaranteed only with the current terms. Hereafter and for the sake of simplicity we will not include such a quadratic term but note that keeping it, we would recover (3.21) in the limit c −→ 0. Note moreover that in contrast with the coercivity of the black soliton, for the dark soliton we use an extra condition on z, namely z L ∞ = O(1).
(b) Notice that in the process of obtaining the constant Γ c in Proposition 3.3 we see that this coercivity constant is lower bounded by a constantΓ when c → 0. In other words, Γ c ≥Γ for all c in a small interval (0,γ). Hence, we get Proof. First of all, we remind that z = z 1 + iz 2 ∈ H(R) and that defining the quadratic form in z we have Also (see (2.19), (2.20) and (2.21)) we already know that for small c We recall (see (3.2)) that which implies We begin by computing the first term on the r.h.s. of (3.51). Note that subtracting (3.48) and (3.2), we have Substituting (3.52) in the r.h.s. of (3.51) we get Hereinafter, unless otherwise noted, we shall consider the constant c as defined in (2.33). Now we proceed to estimate the last three terms in (3.54) and we begin by the last integral. Using (2.32) we have  we get for all |c| ≤ c and β 2 to be fixed later. Now we split the components of the perturbation z = z 1 +iz 2 in the following way with ω 1 , ω 2 and ω 3 real-valued functions chosen so that z * := z * 1 + iz * 2 satisfies the orthogonality conditions in (3.22). Thus, using (3.43), the functions ω i satisfy the relations (3.60) and the system has a solution, because it has a nonvanishing determinant 2 . Now, using (2.27) and (2.28), the integrals in the left hand of (3.60) can be estimated as follows and therefore there exists a positive constant β 3 such that Notice that we can obtain the following estimates, by using Step 1 of Proposition 3.2 applied to z * 1 and z * 2 , and combined with (3.61), where Q 0 [z * 1 ] ≥ 0, β 4 is a positive constant and |c| < c < 1. Analogously, since z * 2 verifies the H0 , we deduce that 2 Note that for parity reasons R φ 0 η 2 0 dx = 0.
for a larger β 4 if necessary and for all |c| < c < 1.
On the other hand, since z * 1 is orthogonal to η 0 in L 2 , the same arguments used to obtain (3.35) in Step 3 of Proposition 3.2 allow us to conclude the existence of positive Λ ν such that (3.64) and substituting (3.65) in (3.58) it follows that where β 6 = β + 4β 4 + 2β 5 . At this point, to control the effect of J[z 1 ] on the lower bound of (3.66), we shall proceed in a similar way as in (3.28) -(3.29), estimating the following L 2 norm: , (3.67) where R c , I c are defined in (2.6) and (2.7). Now, using that |φ c | ≤ 1, (2.26) and (2.33) one gets On the other hand, since |φ c | ≤ 1 we have (3.69) Note that in a similar way as was treated the integral I 3 in (3.30) we have Now, having in mind the hypothesis (3.42) (z = O(1)) and using (2.29) we get Therefore, collecting estimates J a , J b , J c , we get for some positive numbers γ 1 , γ 2 , β 7 . Finally, we fix β 2 such that β 2 < min(Λ 1 , Λ ν ) and 0 < ν < 1, smaller if necessary, such that Then, substituting (3.73) into (3.66), the second term on the right hand side of (3.73) allows to control the integral J[z 1 ] and consequently we can take a positive constant Γ c such that for all |c| < c.

Modulation of parameters
In order to apply the coercivity property of the Ginzburg-Landau energy E 2 shown in Section 3, we have to ensure that the orthogonality relations hold.
In this section, we prove that there exist small perturbations z ∈ H(R) such that the orthogonality conditions (3.22) for the black soliton are satisfied. In fact, we will prove a more general result, valid for c = 0, and dealing with generalized orthogonality conditions for perturbations z ∈ H(R) around the dark soliton and therefore obtaining the desired orthogonality conditions related with the black soliton (3.22) in the limit c = 0 + .
Firstly, and for the sake of completeness, we will present a preliminary result on the continuous dependence for the shift and phase parameters b, θ on the corresponding dark soliton profile.
Before establishing the next result, we remember the neighborhood (3.39) of the orbit of φ 0 , where we split e −iι w(· + b) = φ c + z. By taking α smaller, if necessary, we can apply a well known standard theory of modulation for the solution u(·) of the Cauchy problem (1.1).

Proposition 4.2 (Modulation).
Let α > 0. Then there existsα ∈ (0, α) such that, if w ∈ U 0 + (α) the following statement holds: there exist functions c, a, θ ∈ C 1 (U 0 + (α), R), such that the perturbation of the dark soliton profile satisfies the generalized orthogonality conditions (4.1) Proof of Proposition 4.2. The proof of this result is a classical application of the Implicit Function Theorem. We begin by considering the functional given by Notice that, similarly to the context of the cubic GP [15], the functional F has C 1 -regularity. Consider now, where 0 := (0, 0, 0). On the other hand, we have that 4 (4.7) Let F(c) be the 3 × 3 matrix F(c) := (∂ σ F, ∂ b F, ∂ ι F )( φ c , c, a, θ), which is a continuously differentiable function on the interval c ∈ [0, c). Here we highlight that F(c) is not continuous in the whole interval (−c, c), since F(c) has different lateral limits when c → ±0.
Note moreover that, from Lemma 4.1, the functions c(w), a(w) and θ(w) (modulo 2π) do not depend on which (b, ι) parameters are chosen. For details, see Step 2 in the proof of [15,Proposition 2].
There exist perturbations z ∈ H around the black soliton such that the orthogonality conditions (3.22) are satisfied Proof. This is a consequence of the above proposition, in the particular case of stationary soliton solutions. Remembering (1.15), we can compute which are the orthogonality conditions for the black soliton (3.22).
Now, from the bound (4.15), we get that Finally, combining (4.15) and (4.19), we conclude for some universal constant A 0 . Now, we will determine the growth in time of the modulation parameters c(t), a(t) and θ(t) for any t ∈ [−T, T ]. We will first show the evolution equation satisfied by the perturbation Lemma 4.6 (Evolution equation for z). Let z(t) = e −iθ(t) u(t, · + a(t)) − φ c(t) (·) the perturbation of the dark soliton profile φ c(t) (·) (2.4). Then we have that with and ρ c(t) = ρ c(t) (z(t)).
Proof. First consider the explicit time derivative of z(t, ·): Now computing the last term ∂ t u(t, · + a(t)), bearing in mind that u fulfills (1.1), and also (2.5), (3.41) and that a direct calculation gives us (4.20).
We now look for an expression for the growth in time of the modulation parameters c(t), a(t) and θ(t). In order to do that, we resort to the continuity of the quintic Gross-Pitaevskii flow in Z(R) ⊂ H(R). Specifically, if an initial data u 0 is chosen such that d 0 (u 0 , φ 0 ) < α, then we get a time T such that the corresponding solution u(t, ·) along the quintic Gross-Pitaevskii flow belongs to V 0 + (α) (3.40), for any t ∈ [−T, T ].
We will see in Section 5, that we can fix the smallness parameter α in such a way that the solution u(t, ·) of the Cauchy problem (1.1) still belongs to V 0 + (α) for all t ∈ R. Proof. We differentiate with respect to time the three generalized orthogonality conditions (4.4) for perturbations around the dark soliton profile φ c(t) .
Since we need to compute the derivatives in time of the orthogonality conditions, we consider initially regular enough initial data, for example ∂ x u 0 ∈ H 2 (R). In fact with this regularity we can justify (4.23), (4.25) and (4.26) below. Also, assuming that the solution u ∈ U 0 + (α) for any t ∈ [−T, T ], using Proposition 4.2 we can set the corresponding modulation parameters as (c(t), a(t), θ(t)) ∈ [0, c) × R 2 for any t ∈ [−T, T ].
Note that c, a and θ belong to C 1 [−T, T ], R by the chain rule theorem and moreover note that z(t) ∈ C 1 ([−T, T ], H(R)), and therefore we can get (4.20). Therefore, derivating the first orthogonality condition in (4.4), and with the notation m ij , i, j = 1, 2, 3, for integrals independent of z(t) and n k , k = 1, . . . , 9 for integrals with terms depending on z(t), we get Now, we derivate the second orthogonality condition in (4.1), and we get  Finally, we derivate the third orthogonality condition in (4.1), and we get Gathering all three previous derivatives, we obtain the following matricial system with Z(t) as in (4.21). See Appendix D for a full expression of the computed matrix elements m i,j , i, j = 1, 2, 3.
Note that, in the case of null perturbation in (4.28), and considering the limit case of c = 0 + , it turns out that M(0 + , 0) has a nonvanishing determinant, namely 5 and therefore M(0 + , 0) is invertible. By using a continuity argument, we can select a small enough parameter α 1 < α such that for small speeds and perturbations (c, z), the matrix M(c, z) is still invertible. In fact, having in mind the Neumann series theorem, it is enough to consider (c, z) verifying . Namely, choosing α 1 < α small enough such that u(t, ·) ∈ U 0 + (α 1 ), for all t ∈ [−T, T ], and therefore that from (4.14) in Proposition 4.2, it holds z(t, ·) H0 + |c(t)| ≤ A 0 α 1 , (4.30) with det M(c, z) = 0 and, consequently, the operator norm of its inverse is bounded by some positive number A 1 (α 1 ). In the same way, the r.h.s. of (4.27) is bounded as follows: for a suitable choice of the constant A 1 (α 1 ). Therefore, from (4.27), we finally get that Finally, we extend the above estimate (4.31) for general initial data u 0 ∈ Z(R). In fact, the flow of (1.1) is continuous with respect to initial data in Z(R) (see [9]). Moreover, from the continuity of the modulation parameters c(t), a(t) and θ(t), we have that the matrices M(c, z) and B(c, z) depend continuously on u ∈ H(R). Therefore, since the matrix M(c, z) is invertible with an operator norm of its inverse depending on α 1 , we can use a standard density argument to extend (4.27) to a general solution. Therefore we get the continuous differentiability property of the modulation parameters c(t), a(t) and θ(t), and we obtain the corresponding estimates (4.22) from (4.27).

Proof of the Main Theorem
In this section we prove a detailed version of Theorem 1.1. and then there exist functions a, θ ∈ C 1 (R, R) such that the solution u of the Cauchy problem for the quintic GP equation (1.1), with initial data u 0 , satisfies d 0 (e −iθ(t) u(t, · + a(t)), φ 0 ) < (5.1) and |a (t)| + |θ (t)| < A * (5.2) for any t ∈ R.
Remark 5.2. An important difference in the proof of the orbital stability Theorem for black solitons of the (1.1), is that we can not achieve a lipschitzian control of the d 0 metric, i.e.
as it is performed in the cubic case [15]. The main reason is that the modified momentum corresponding to (1.1) where u := e −iθ(t) u(t, · + a(t)) (see the definition of χ and ϕ in [15]) gives, after expanding in the perturbation z and once it is computed in the modulated dark solution, the integral term, Now suppose that T * < +∞ and consider where a(t), θ(t) and c(t) ∈ [0, c] are the functions given by Corollary 4.4.
From the global theory developed in [9,Section 3] and Sobolev embedding we have , for all 0 ≤ t ≤ T * . Then, using the pseudo-coercivity of E 2 (1.3) around the dark soliton (3.45) in Proposition 3.3 and Corollary 4.5 with (a, θ) = (0, 0), we obtain Selecting such that A 0 < 1 2Γ 2 and using the conservation of the E 2 energy (1.3) one gets for all t ∈ [0, T * ].
Now, from the expansion (3.2) with z = u 0 − φ 0 there exists a positive constantk such that withk independent of u 0 . Then, putting this estimate in (5.6) we have and hence from (2.24) we have and we also consider u 0 satisfying Then, combining (5.9), (5.10) and (5.11), we get d 0 (e −iθ(t) u(t, · + a(t)), φ 0 ) < 2 , for all t ∈ [0, T * ], (5.12) as claimed in (5.4). Then, the proof of (5.1) is finished. Some explicit models of quantum physics where the present technique might be applied, are • the quintic NLS: The application to this model is rather direct, because it only involves the introduction of a rotation in time transformation u = e it v to connect (1.1) with the quintic NLS. • A cubic-quintic NLS: iu t + u xx + |u| 2 u − |u| 4 u = 0, and • the φ 6 -model: For these last two models, the application of this technique depends on the coercivity behavior of the associated Ginzburg-Landau energies E 2 of the models and the explicit character of their dark and black solitons.
Appendix A. Proof of (1.10) In order to prove that (1.10) is a solution of (1.9), we propose a suitable ansatz (2.3): This ansatz must to reduce to the black solution (1.7) when c = 0, therefore this implies that a 1 has to be dependent on c in some way. We make the following selection for withã 1 to be determined. Hence, we recast (A.1) as follows: whereã 1 , k, a 2 , a 3 are parameters to be determined imposing that (A.2) is a solution of (1.9). Therefore, substituting (A.2) into (1.9) and simplifying (here X = tanh(kξ), D = 1+a 3 tanh(kξ) 2 ), we get where r i , i = 0, . . . , 5, are the following complex coefficients where h(c, k) and g(c, k) are rational functions in c, k variables. Explicitly with h 1 (c, k) = 343c 8 k + 1960c 6 k 3 − 464c 6 k + 2688c 4 k 5 − 1344c 4 k 3 + 576c 4 k + 1408c 2 k 7 + 2304c 2 k 5 + 384c 2 k 3 − 512c 2 k + 256k 9 + 1024k 7 + 1536k 5 + 1024k 3 + 256k and g(c, k) = g 1 (c, k) g 2 (c, k) , Finally, selecting and noting that h(c, k) and g(c, k) are not singular at k = 1 2 √ 4 − c 2 , we have solved system (A.5), and therefore (A.2) is a solution. In order to compare this solution with (1.10), we rewrite it as follows: firstly note that with this value of k, (A.10) and (A.7) reduce tõ and and hence, from (A.6), with the above values ofã 1 , k, a 3 , and simplifying, we get (taking for instance a real + root)   The proof of this identity is made by quadratures. Making the change s = √ 2b tan θ we get the following equalities for the indefinite integrals ds Now, by using the change ρ = √ 3 sin θ we obtain Combining (B.1) and (B.2) the result follows from the Fundamental Theorem of Calculus.
Appendix C. Computation of some L 2 and L ∞ norms We collect some L 2 and L ∞ norms needed along this work, in the following sections. Hereafter, we will consider by K the smallest of the constants that allow us to get the corresponding upper bound.

(C.4)
We now compute the L 2 norm in (2.29). Firstly we write explicitly the integrand In fact, in the same small speed region |c| ≤ c, with c 1, we get, after an expansion of the above expression, that We now compute the L 2 norm in (2.30) proceeding in the same way. In fact after an expansion of the integrand in the small speed region |c| ≤ c, with c 1, we get, C.2. L ∞ norms. We now compute the L ∞ norm in (2.31). Expanding it in the small speed region |c| ≤ c, with c 1, we get |φ c | 2 − φ 2 0 (1 + x 2 )η c ≤ (9 − 4x tanh(x) + tanh 2 (x)) 8(1 + x 2 )(3 + tanh 2 (x)) 3 c 2 , uniformly in x ∈ R, and whose maximum value ( 3 8 ) is attained at x = 0. Therefore we get that Appendix D. Computation of detF(c) and matrix elements of M(c, z) First of all, we remember the expression of detF(c) (4.8): and at c = 0 we have that Re iR c η c ∂ xφc c=0 ± = 0, and we get that R iR c η c , ∂ x φ c C c=0 ± = 0. (D.5) The corresponding third factor is Re iη cφc c=0 ± = 0. Therefore, as above we have, R η c , −iφ c C c=0 ± = 0.
(D.7) 6 For the sake of completeness, we compute the limit at c = 0 through the 0 ± directions, following (1.15).

Integrating, we get
Finally, gathering the five terms above, we have that Now, using a classical continuity argument, we get that, for c ∈ [0, c), with smaller c ≪ 1, if necessary, detF(c) = 0. (D.13) In the same way, we can proceed and conclude that for c ∈ (−c, 0], c ≪ 1, (D.13) also holds.
We now list here the computed matrix elements of M(c, z) in (4.28). By parity reasons some terms vanish. Namely In the limit case when (c = 0 ± , z = 0), the matrix (4.28) has the following simple expression