Full family of flattening solitary waves for the mass critical generalized KdV equation

For the mass critical generalized KdV equation $\partial_t u + \partial_x (\partial_x^2 u + u^5)=0$ on $\mathbb R$, we construct a full family of flattening solitary wave solutions. Let $Q$ be the unique even positive solution of $Q''+Q^5=Q$. For any $\nu\in (0,\frac 13)$, there exist global (for $t\geq 0$) solutions of the equation with the asymptotic behavior \begin{equation*} u(t,x)= t^{-\frac{\nu}2} Q\left(t^{-\nu} (x-x(t))\right)+w(t,x) \end{equation*} where, for some $c>0$, \begin{equation*} x(t)\sim c t^{1-2\nu} \quad \mbox{and}\quad \|w(t)\|_{H^1(x>\frac 12 x(t))} \to 0\quad \mbox{as $t\to +\infty$.} \end{equation*} Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

Then, the function , for any (λ 0 , x 0 ) ∈ (0, +∞) × R, is a solution of (1.1). It is well-known that E(Q) = 0 and that Q is related to the following sharp Gagliardo-Nirenberg inequality (see [34]) It follows from (1.4) and the conservation of the mass and the energy that any initial data u 0 ∈ H 1 (R) satisfying u 0 L 2 < Q L 2 generates a global in time solution of (1.1) that is also bounded in H 1 (R). Now, we summarize available results on blow-up solutions for (1.1) in the case of initial data with mass equal or slightly above the threshold mass, i.e. satisfying • At the threshold mass u 0 L 2 = Q L 2 , there exists a unique (up to the invariances of the equation) blow-up solution S(t) of the equation, which blows up in finite time (denoted by T > 0) with the rate S(t) H 1 ∼ C(T − t) −1 as t → T . See [2,24]. • For mass slightly above the threshold, there exists a large set (including negative and zero energy solutions, and open in some topology) of blow-up solutions, with the blow-up rate u(t) H 1 ∼ C(T − t) −1 as t → T . See [23,28] and other references therein. • In the neighborhood of the soliton for the same topology (H 1 solutions with suitable decay on the right), there exists a C 1 co-dimension one threshold manifold which separates the above stable blow-up behavior from solutions that eventually exit the soliton neighborhood by vanishing. Solutions on the manifold are global and locally converge to the ground state Q up to the invariances of the equation. In this class of initial data, one thus obtains the following trichotomy: stable finite time blowup, soliton behavior or exit. See [22,23,24]. • There also exists a large class of exotic finite time blow-up solutions, close to the family of solitons, enjoying blow-up rates of the form u(t) H 1 ∼ C(T − t) −ν for any ν > 11 13 . Note that the exponent 11 13 does not seem sharp and it is an open question to determine the lowest finite time blow-up exponent for H 1 initial data. Global solutions blowing up in infinite time with u(t) H 1 ∼ Ct ν as t → ∞, were also constructed for any positive power ν > 0. See [25].
Such exotic behaviors are generated by the interaction of the soliton with explicit slowly decaying tails added to the initial data. Because of the tail, these H 1 solutions do not belong to the class where the trichotomy (blowup, soliton, exit) occurs. We refer to the above mentioned articles and to the references therein for detailed results and previous references on the subject.
Recall that for the mass critical nonlinear Schrödinger equation (NLS), there exists a large class (stable in H 1 ) of blow-up solutions enjoying the so-called log log blow-up rate (see [29] and references therein), whereas (unstable) blow-up solutions with the conformal blow-up rate u(t) H 1 ∼ C(T − t) −1 were also constructed by perturbation of the explicit minimal mass blow-up solution ( [1,13,30]). Moreover, in the vicinity of the soliton, it is proved in [32] that solutions cannot have a blow-up rate strictly between the log log rate and the conformal rate. It is an open question to build solutions with a blow-up rate higher than the conformal one (see however [26] in the case of several solitons). The only available results concerning flattening solitons are deduced from the pseudo-conformal transformation applied to the solutions discussed above. For the mass critical (NLS), the question of the existence of exotic behaviors is thus widely open.
The systematic study of exotic blow-up behaviors was initiated by the articles [15,16] for energy critical dispersive models, followed by [5,8,9,14]. (We also refer to [7] for the construction of exotic solutions in other contexts.) The article [5], where a class of flattening bubbles is constructed for the energy critical wave equation on R 3 , is particularly related to our work. More precisely, W being the unique radial positive solution of ∆W + W 5 = 0 on R 3 , it is proved in [5] that for any |ν| ≪ 1, there exist global (for positive time) solutions of ∂ 2 t u = ∆u + |u| 4 u such that u(t, x) ∼ t ν/2 W (t ν x) as t → +∞; the case 0 < ν ≪ 1 corresponds to blow-up in infinite time, while 0 < −ν ≪ 1 corresponds to flattening solitons.
Such construction is especially motivated by the soliton resolution conjecture, which states that any global solution should decompose for large time into a certain number of decoupled solitons plus a dispersive part. We refer to [6] and references therein for the proof of the soliton resolution conjecture for the 3D critical wave equation in the radial case. It follows from [5] that some flexibility on the geometric parameters is necessary in the statement of the conjecture.
The above mentioned works are a strong motivation for investigating exotic behaviors related to flattening solitons in the context of mass critical dispersive models. Our main result is the existence of such solutions for the critical generalized KdV equation.
(1.6) Theorem 1.1 states the existence of solutions arbitrarily close to the soliton Q which eventually defocus in large time with scaling ℓ(t) ∼ t −ν where ν = (1 − β)/2 is any value in (0, 1 3 ). The values of the exponents and multiplicative constants in (1.5) are consistent with the formal equation x ′ (t) = ℓ −2 (t) relating the two geometrical parameters x(t) and ℓ(t).
Note that by continuous dependence of the solution of (1.1) with respect to the initial data, the constant T δ in Theorem 1.1 satisfies T δ → ∞ as δ → 0 . The estimates in (1.5) make sense only for t ≫ T δ when the flattening regime appears. Of course, one can use the scaling invariance of the equation to generate solutions with different multiplicative constants in (1.5). In the statement of Theorem 1.1, the scaling is adjusted so that one can compare the initial data with the soliton Q. We refer to Remark 5.1 for details . We also notice that w(t) does not converge to 0 in H 1 (R) as t → +∞; otherwise, it would hold E(u(t)) = 0 and u 2 (t) = Q 2 and by variational arguments, u(t) would be exactly a soliton. However, the residue w is arbitrarily small in H 1 and converges strongly to 0 as t → ∞ in the space-time region x > 1 2 x(t) ≫ ℓ(t) which largely includes the soliton. To complement Theorem 1.1, we prove in Section 5.6 that the solutions do not behave as solutions of the linear Airy equation ∂ t v + ∂ 3 x v = 0 as t → ∞ (non-scattering solutions). We claim that the restriction β ∈ ( 1 3 , 1) in Theorem 1.1 corresponds to the full range of relevant exponents. Indeed, the exponent β = 1 3 is related to self-similarity, and in the region x < t 1/3 , the question of existence or non-existence of coherent nonlinear structures is of different nature. See [31] for several results in this direction.
As mentioned above, infinite time blow-up solutions with any positive power rate were constructed in [25]. Thus Theorem 1.1 essentially settles the question of all possible single soliton behaviors as t → +∞. It also sheds some light on the classification of all possible behaviors in H 1 , while the results in [22,23,24] hold in a stronger topology.
Remark 1.1. We note from the proof that all initial data in Theorem 1.1 have a tail on the right of the soliton of the form c 0 x −θ , for c 0 > 0 and θ = 5β−1 4β ∈ ( 1 2 , 1). Observe that for such value of θ, this tail does not belong to L 1 (R).
Recall from [25] that θ ∈ (1, 5 4 ] corresponds to blowup in infinite time and θ ∈ ( 5 4 , 29 18 ) to exotic blowup in finite time (for negative values of the multiplicative constant c 0 ). This means that, except the remaining question of the largest value of θ leading to exotic blowup, the influence of such tails on the soliton is now well-understood.
Remark 1.2. The more general statement Theorem 5.2 given in Section 5.2 provides a large set of initial data, related to a one-parameter condition to control the scaling instability direction (in particular responsible for blowup in finite time). As in the classification given by [23], a strong topology related to L 2 weighted norm is necessary to avoid destroying the tail leading to the soliton flattening. Therefore, though the phenomenon of flattening solitons may seem exotic, it is rather robust by perturbation in weighted norms, its only instability in such spaces being related to the scaling direction. Moreover, it follows from formal arguments that any small perturbation in that direction should lead to blowup with the blow-up rate C(T − t) −1 or to exit of the soliton neighborhood. This is analogous to the situation described by the construction of the C 1 threshold manifold in [22]. Here, because of weaker decay estimates on the residue, we do not address the question of the regularity of this set. Remark 1.3. Flattening solitary waves were constructed in Theorem 1.5 of [17] for the following double power (gKdV) equations with saturated nonlinearities x u + u 5 − γ|u| q−1 u) = 0 where q > 5 and 0 < γ ≪ 1. The blow-down rate and the position of the soliton are fixed as t → +∞.
Analogous results (construction of minimal mass solutions with exotic blow-up rates) were also established for a double power nonlinear Schrödinger equation in [18].
For a given small positive constant 0 < α ⋆ ≪ 1, δ(α ⋆ ) will denote a small constant with We will denote by c a positive constant that may change from line to line. The notation a b (respectively, a b) means that a ≤ cb (respectively, a ≥ cb) for some positive constant c. For 1 ≤ p ≤ +∞, L p (R) denote the classical Lebesgue spaces. We define the weighted spaces L 2 loc = L 2 (R; e − |y| 10 dy) and L 2 B (R) = L 2 (R; e y B dy), for B ≥ 100 to be fixed later in the proof, through the norms . For f , g ∈ L 2 (R) two real-valued functions, we denote the scalar product We introduce the generator of the scaling symmetry We also define the linearized operator L around the ground state by From now on, for simplicity of notation, we write instead of R and omit dx in integrals.

1.2.
Strategy of the proof. The overall strategy of the proof, based on the construction of a suitable ansatz and energy estimates, follows the one developed in [19,23,24,25,27,33] in similar contexts. The originality of the present work lies mainly in the prior preparation of suitable tails and the rigorous justification of all relevant flattening regimes.
(i) Definition of the slowly decaying tail. Given c 0 > 0, x 0 ≫ 1 and 1 2 < θ < 1, we introduce a smooth function f 0 corresponding to a slowly decaying tail on the right: In the present case, a special care has to be taken in the preparatory step of understanding the evolution of such slowly decaying tails under the (gKdV) flow. Not only the decay rate is slower than the one in [25] but also the control of the solution is needed close to the larger space-time region x t β , for β > 1 3 . Note that the proof uses the mass criticality of the exponent (it extends to super-critical exponents). See Section 2.
(ii) Emergence of the flattening regime. For t 0 ≫ 1, we consider the rescaled time variable dτ. (1.10) In the variable s, the equations governing the parameters (λ, σ, b) ∈ (0, +∞) × R 2 write By using the first equation (1.11), we also compute To simplify constants, we choose (1.14) To come back to the original time variable, we first need to solve (1.10). We set Then, by choosing Last, we deduce from (1.14) that for some positive constants c λ and c σ (see (5.13)).
(iii) Energy estimates. In order to construct an exact solution of (1.1) satisfying the formal regime (1.15), we use a variant of the mixed energy-virial functional first introduced for (gKdV) in [23] (the introduction of the virial argument in the neighborhood of the soliton for critical (gKdV) goes back to [20]). Considering a defocusing regime induces a simplification (see also the energy estimates in [2]) that allows us to treat the whole range β ∈ ( 1 3 , 1) in spite of a basic ansatz and relatively large error terms. See Section 4.

Persistence properties of slowly decaying tails on the right
In this section, we present a general result concerning the persistence of a class of slowly decaying tails for the critical gKdV equation in a suitable space-time region.
Let θ ∈ ( 1 2 , 1] and define For c 0 > 0 and x 0 ≫ 1, we consider f 0 any smooth nonnegative function such that Now, for t 0 ≫ 1 to be fixed, let f be a solution of the IVP The main result of this section states that the special asymptotic behavior of f 0 (x) on the right persists for f (t, x) in regions of the form x t β .
Proposition 2.1. Let θ ∈ 1 2 , 1 , β = 1 5−4θ and c 0 > 0. For x 0 > 0 large enough, for any κ 0 > 0, setting t 0 := (x 0 /κ 0 ) 1/β , the solution f of (2.4) is global, smooth and bounded in H 1 . Moreover, it holds for all t ≥ t 0 and x > κ 0 t β ≥ x 0 , The rest of this section is devoted to the proof of Proposition 2.1, which requires preparatory monotonicity lemmas based on variants of the so-called Kato identity (see [10,20,21]). This result is a substantial generalization of Lemma 2.3 in [25], where only the case θ = 1 is treated. Our proof allows regions x t β for any β > 1 3 . Complementary results are obtained in [31], where large regions close to x = 0 are investigated by similar functionals.
Remark 2.1. Without loss of generality and for simplicity of notation, we reduce ourselves to prove estimates (2.5) and (2.6) for the special value κ 0 = 2. Indeed, consider the function f (s, y) = λ 1 2 f (λ 3 s, λy). Then f is a solution to (2.4) where . First, note that if x 0 is chosen large enough, it follows directly from the Cauchy theory developed in [11] (see Corollary 2.9) and (2.3) that f ∈ C(R : H s (R)) for all s ≥ 0 and Moreover, by using the sharp Gagliardo-Nirenberg inequality (1.4) and the conservations of the mass and the energy (1.2), we deduce, for x 0 large enough, that . For anyr ≥ 0, we define a smooth function ωr such that (2.10) Observe that |ω ′′ r | + |ω ′′′ r | ≤ Cω ′r on R, (2.11) for some constant C = C(ωr) > 0.
Then, for x 0 > 1 large enough, and any t ≥ t 0 = x 0 Proof. To prove (2.12), we differentiate M r with respect to time, use (2.9) and integrate by parts in the x variable to obtain By using (2.11), for t 0 large enough, we have and so Next, we estimate M j for j = 4, · · · , 7 separately. For future use, observe that by the assumption 0 < ǫ < 3β−1 20 |r − 5| and 0 < r < 6, we also have 0 < ǫ < 3β−1 4 . We denote (2.14) Estimate for M 4 . It is clear that M + 4 (t) ≤ 0. Next, for t 0 large enough, Then, it follows from the definition of ω r in (2.10) and (2.7) that, for t 0 large enough, it holds We observe that, for t 0 large enough, Thus, we deduce from (2.2), and then 4θβ > 2β > 2ν (since θ > 1 2 and β > 1 3 > ν), that, for t 0 large enough, As before for M − 4 , we have for t 0 large enough, To deal with M 5,2 , we follow an argument in Lemma 6 of [28]. We have by using the fundamental theorem of calculus , and so, by Cauchy Schwarz inequality and then (2.11), Therefore, using also (2.7), for x 0 large enough, Estimate for M 6 . By using interpolation, (2.2) and then the inequality |x| −θ q 5 x −4θ q 2 + q 6 , we observe that By (2.10) and (2.15), and choosing ǫ > 0 such that 0 < ǫ < β − ν = 3β−1 2 , Estimate for M 7 . We get from the Cauchy-Schwarz inequality that .
We prove a similar estimate for a quantity related to the energy. Lemma 2.3. Let 0 < r < 2θ + 4, r = 5 and 0 < ǫ < 3β−1 20 |r − 5|. Define wherex = x−t β t ν+ǫ . Then, for x 0 > 1 large enough, and any t ≥ t 0 = x 0 Proof. We differentiate E r with respect to time and integrate by parts to obtain First, observe that E 1 ≤ 0, E 2 ≤ 0 and E 4 ≤ 0. As in the proof of (2.13), we have for t 0 large, |E 3 | ≤ − 1 2 E 4 . Next, we use the same notation as in (2.14) for E j , j = 5, · · · , 10. We observe that E + 5 ≤ 0. Moreover, as for the estimate of M 4 in the proof of (2.12), it holds E 0 Estimate for E 6 . First, we note We estimate E − 6,1 t −10 and since ǫ < 3β−1 2 . Arguing as for M 5,2 in the proof of (2.2), ; and thus, as before, for x 0 large, Estimate for E 7 . We estimate E − 7 ≤ t −10 and As before, we have Last, by arguing similarly as in (2.19), Estimate for E 8 . We compute First, we have as before E − 8,1 t −10 . Now, arguing as for M 5,1 , we get for t 0 large enough that To handle E 8,2 , we use a similar argument as in (2.19). Observe from the fundamental theorem of calculus that .
Thus, by the Cauchy-Schwarz inequality, To deal with the second term on the right-hand side of the above inequality, we integrate by parts to get .
Using again the Cauchy-Schwarz inequality and then (2.11) and using (2.7), we deduce Therefore, for x 0 large enough, we obtain (2.15) and the Cauchy-Schwarz inequality that whenx > 2. Thus, we get by using again the Cauchy-Schwarz inequality that by taking t 0 large enough, since 4βθ − 2ν = 2(3β − 1). Finally we estimate E 8,4 . First, we have from Young's inequality Hence, we deduce using the estimates for M 0 5,1 , M + 5,1 and M 5,2 that Estimate for E 9 . We have by assumption, and using (2.21).
Last, as before, Estimate for E 10 . We have First, . Then, Now, we deal with E 10,2 . On the one hand, we estimate as before E − 10,2 t −10 . On the other hand, we deduce by using (2.15) and (2.20) that Thus, it follows arguing as in (2.19) that for t 0 large enough, Then, it follows that for t 0 large enough, and 0 < ǫ < 3β−1 4 . Estimate for E 12 . On the one hand, it holds E − 12 t −10 . On the other, we observe arguing as for E 7 and using (2.25) that |E 0 12 | + |E + 12 | t −ǫ E 6,1 + E 6,2 + E + 7,1 + E + 7,2 , so that those terms are estimated similarly.
Gathering all those estimates, we obtain in conclusion, for some c > 0, Therefore, we conclude the proof of (2.23) by using (2.12), integrating the previous estimate over [t 0 , t] and using (2.22).
Proof of Proposition 2.1 in the case k = 0. First, we look for an estimate on (∂ x q) 2 ω r+2 from the energy estimate. Arguing as in (2.16), (2.17) and (2.19), we get that Thus, it follows that for x 0 large enough Hence, we deduce by using (2.12) and (2.23) that, for 0 < r < 2θ + 4, Now, we give the proof of estimate (2.5) in the case k = 0. By the fundamental theorem of calculus and the properties of ω r it holds, for any x, Hence, we obtain from estimates (2.12) and (2.26) that For x > 2t β , we have thatx > t β−ν > 2, for t ≥ t 0 large enough. Then, we deduce from the properties of ω r , estimate (2.27) and the identity (2.21) that and thus, for such t ≥ t 0 and x > 2t β , we have Taking r close enough to 2θ + 4 so that 2 − β − β(2θ − r + 4) > 0, we conclude the proof of (2.5) in the case k = 0 and κ 0 = 2 (see Remark 2.1) using t β < x.
Proof of Proposition 2.1 in the case k ≥ 1. We will prove estimate (2.5) in the case where k ≥ 1 by an induction on k.
Definition 2.4. Let l ∈ N, 0 < r < 2θ + 4, r = 5 and 0 < ǫ < 3β−1 20 |r − 5|. We say that the induction hypothesis H l holds true if First, it is clear arguing as in (2.27) that if H l and H l−1 hold true for some l ∈ N, l ≥ 1, then Notice in particular that (2.30) would imply (2.5) in the case k = l − 1 arguing as in (2.28). Thus, it suffices to prove that (2.29) hold for any l ∈ N to conclude the proof of Proposition 2.1. Observe from (2.12) and (2.23) that H 0 and H 1 hold true.
Assume that (2.29) holds true for l = 0, 1, · · · , k − 1. The next lemma will prove that (2.29) is true for l = k, which will conclude the proof of Proposition 2.1.
Then, for x 0 > 1 large enough, and any Proof. We differentiate F k,r with respect to time and integrate by parts to obtain First observe that F 1 ≤ 0 and F 3 ≤ 0. Moreover, arguing as in the proof of (2.13), we have that, for t 0 large enough, |F 2 | ≤ − 1 2 F 3 . Next, we use the same notation as in (2.14) for F j , j = 4, · · · , 6. We have F + 4 ≤ 0. Moreover, as for the estimate of M 4 in the proof of (2.12), it holds F 0 We will only explain how to estimate the terms and since the other ones are estimated interpolating between these estimates.
3. Decomposition around the soliton 3.1. Linearized operator. Here, we recall some properties of the linearized operator L around the soliton Q defined in (1.9). We first introduce the function space Y: (3.1) (iv) Invertibility: there exists a unique R ∈ Y, even, such that Moreover, Proof. The properties (i), (ii) and (iii) are standard and we refer to Lemma 2.1 of [23] and the references therein for their proof. Property (iv) is proved in Lemma 2.1 in [25], while property (v) is proved in Proposition 2.2 in [23].

3.2.
Refined profile. We follow [23] to define the one parameter family of approximate self similar profiles : b → Q b , |b| ≪ 1 which will provide the leading order deformation of the ground state profile Q = Q b=0 in our construction. More precisely, we need to localize P on the left hand side. Let χ ∈ C ∞ (R) be such that where P b (y) = χ b (y)P (y) and χ b (y) = χ(|b| γ y). (3.6) The properties of Q b are stated in the next lemma. [23]). There exists b ⋆ > 0 such that for |b| < b ⋆ , the following properties hold. Moreover,

10)
and (recalling the definition of L 2 B in (1.7)), Ψ Note that the implicit constant in (3.11) depends on the constant B ≥ 100. (iii) Projection of Ψ b in the direction Q: (iv) Mass and energy properties of Q b : (3.14) Proof. The proof of Lemma 3.3 can be found in [23]. Actually, the properties (i), (ii) and (iv) are proved in Lemma 2.4 of [23]. The estimate (3.10) follow directly from (3.8) and (3.9). Now, we explain how to prove (3.11) in the case k = 0. It follows from (1.7), (3.8) and the fact that B ≥ 100 that for |b| small enough. The proof of (3.11) in the case k ≥ 1 follows in a similar way by using (3.9) instead of (3.8).
Note that we have added the term −2b 2 ∂Q b ∂b to the definition of Ψ b compared to the definition in [23] in order to get a better estimate for the projection of Ψ b on Q. The property (iii) follows from the computation in the first formula of page 80 in [23] together with (3.3).
Remark 3.1. For future reference, we also observe that 3.3. Definition of the approximate solution. Let any 1 2 < θ < 1. Following (1.13), set For such c 0 , for x 0 large enough and for (our intention is to use Proposition 2.1 with the value We renormalize the flow using C 1 functions λ(t) and σ(t), defining V , F 0 and F as follows We introduce the rescaled time variable (3.18) Note that from (3.17) relating t 0 and x 0 , s 0 can be taken arbitrarily large provided x 0 is large. From now, any time-dependent function can be seen as a function of t ∈ I or s ∈ J , where I is an interval of the form [s 0 , s ⋆ ] and J = s(I). In view of the resolution of the ODE system in (1.14), we will work under the following assumptions on the parameters (λ, σ, b): where ρ is a positive number satisfying . (3.20) We set  and
From the definition of F and r, we have Thus, it follows applying the mean value theorem, splitting into the two cases λ(s)y > − 1 4 σ(s) and λ(s)y < − 1 4 σ(s) as above, and then using (2.2) and (2.5) that, for s large enough, which implies (3.27) by using (3.19).
Finally, by using the identities and Q 5 = Q, we deduce arguing as in (3.27) that which, together with (3.19), concludes the proof of (3.28).
In the next lemma, we derive an estimate for the mass and the energy of W + F . and Proof. Observe by using the decomposition in (3.5) that From the definition of F and the L 2 conservation for (2.4), we have that Moreover, we get from (3.26) that Hence, it follows from (3.5) and (3.27) that which together with (3.2), (3.13), (3.19) and (3.23) imply (3.29). Now, we compute the energy of W : Moreover, it follows from the definition of F , the conservation of the energy for (2.4) and (2.3) that This last estimate combined with (3.23) implies (3.30).
We compute E(W ) in the next lemma.

32)
where, for s large enough,

34)
where the norm L 2 B is defined in (1.7) , and Proof. We compute E(W ) from the definition of W : . Using the definition of Q b and Ψ b , the definitions of m and M and the equation of R, we rewrite the previous identity as follows where Estimates for R 1 . First we deduce from Lemma 3.4 that Q, Second, (brΛR, Q) = −br(R, ΛQ) =: I. Third, we write Moreover, by using the identity (2.52) in [25] − (R, ΛQ) − 20 Q 3 (R + 1)P, Q ′ = 0, we get that I + II 2 = 0. To deal with II 1 , we deduce from (3.28), |σ s − λ| ≤ λ| m| and (3.19) that Next, it is clear from (3.19) and (3.27) that II 3 s −3 . Finally we also claim that II 4 s −3 . Indeed, a direct computation gives Therefore, we deduce gathering those estimates that Estimates for ∂ y R 2 . We claim that We first develop R 2 : arguing as in the proof of Lemma 3.4. Now, we prove the second estimate in (3.39). On the one hand, we get easily from the definition of Q b in (3.5), (3.23) and (3.25) that On the other hand, we see by integration by parts that Observe from the definition of R in (3.2) that 5Q 4 (R + 1) = −∂ 2 y R + R, so that the first term on the right-hand side of the above identity cancels out by symmetry. Hence, it follows from (3.23), (3.25) and (3.27) that which yields the second estimate in (3.39).
Estimates for R 3 . First, we deduce from (3.15) and (3.19) that Arguing similarly, we get from (3.6), (3.19) and (3.23 It follows combining those estimates that  We define Lemma 3.7. It holds Proof. First, observe by a direct computation that Hence, we deduce from (3.19) and (3.32) that where f is defined in Section 3.3. We assume that there exists (λ ♯ (t), σ ♯ (t)) ∈ (0, +∞) 2 and z ∈ C I : for all t ∈ I and where α ⋆ is small positive universal constant. For future use, remark that (3.47) implies (using 1 2 < θ < 1) We collect in the next lemma the standard preliminary estimates on this decomposition related to the choice of suitable orthogonality conditions for the remainder term.
Next, we derive some estimates for ε in H 1 related to the conservation of mass and energy.
for s large enough, where g is defined in Lemma 3.7.
Proof. By using the conservation of the L 2 norm for u, we obtain that We observe by using the third orthogonality condition in (3.51) and then the Cauchy-Schwarz inequality, (3.31) and (3.19) that which combined with (3.29) implies (3.55). We turn to the proof of (3.56). By using the conservation of the energy and the scaling properties, we get that Thus, it follows by using the identity (−Q ′′ − Q 5 )ε = Qε = 0 that We get from the Cauchy-Schwarz inequality, (3.19) and (3.23) that Now, we deal with the term (∂ 2 y F + F 5 )ε. We get from the Cauchy-Schwarz inequality and (2.8) that Similarly, we get from Hölder's inequality, the Gagliardo-Niremberg inequality (1.4) and then (3.52) and (2.8) that Moreover, from interpolation, the Gagliardo-Nirenberg inequality (1.4) and (3.19), (3.23), (3.25), it holds arguing as in (3.57), we estimate the term F 4 ε 2 as follows We conclude the proof of (3.56) by combining those estimates with (3.30) and (3.52).
Next, the equation of the parameters λ, σ and b are deduced from modulation estimates.
Lemma 3.10 (Modulation estimates). Under the bootstrap assumptions (3.19), it holds for s large enough, where the L 2 loc -norm is defined in (1.7). Proof. First, we differentiate the first orthogonality condition in (3.51) with respect to s, use the equation (3.54), follow the computations in the proof of Lemma 2.7 in [23] and use the estimate (3.33) to get that Now, we derive the second orthogonality condition in (3.51) with respect to s. By combining similar estimates with the identity (Q ′ , yΛQ) = ΛQ 2 L 2 , we also get that Next, we derive the third orthogonality condition in (3.51) with respect to s. It follows that We observe the cancellations ∂ y Lε, Q = −(ε, L(Q ′ )) = 0 and Λε, Q = − ε, ΛQ = 0. We also get by using (3.19), (3.23) and (3.25) that for s large enough Moreover, we have from (3.32) that Hence, it follows from the cancellations (ΛQ, Q) = (∂ y Q, Q) = 0 and the estimate (3.35) that We deduce combining those estimates and using (3.19) which conclude the proof of (3.60) by taking s large enough thanks to (3.58).

Bootstrap estimates.
Let ψ ∈ C ∞ be a nondecreasing function such that ψ(y) =    e 2y for y < −1, For B > 100 large to be chosen later, we define Note that, directly from the definitions of ψ and ϕ, we have, for all y ∈ R and , for all y < 0. Let 0 < ρ ≪ 1 and B > 100 to be chosen later. In addition of (3.19), we will work under the following bootstrap assumptions.
In particular, from the definition of the L 2 loc -norms in (1.7) and B > 100, it holds For future reference, we state here some consequences of the bootstrap assumptions.
We define the mixed energy-virial functional (1) Time derivative of the energy functional.
We compute using (3.54), Estimate of f 1 . We claim, by choosing α ⋆ small enough, B large enough and then s large enough (possibly depending on B), that where µ 0 is a small positive constant which will be fixed below.
To prove (4.6), we compute following Step 3 of Proposition 3.1 in [23], correspond respectively to integration on the three regions y < − B 2 and y > − B 2 .
To control, the purely nonlinear term in f 1,1 , we recall the following version of the Sobolev embedding (see Lemma 6 of [28] and also (2.18)): Thus, it follows that Note also for future reference that the same proof yields Hence, we deduce gathering those estimates and choosing s and B large enough and α ⋆ small enough that Observe from Young's inequality that so that it follows arguing as for f < 1,1 that, for s and B large enough and α ⋆ small enough, By using again Young's inequality, we have The first two terms on the right-hand side of the above inequality are estimated as before.
For the third one, we deduce arguing as in (2.24) that Thus it follows by taking s and B large enough and α ⋆ small enough that Finally, we get from Young's inequality and (3.63) that Hence, we deduce that Therefore, we conclude gathering all those estimates that (4.10) Estimate of f > 1 . In the region y > − B 2 , one has ϕ B (y) = e y B and ψ B (y) = 1. Thus, where R Vir = R Vir,1 + R Vir,2 + R Vir,3 + R Vir,4 + R Vir,5 and To handle the first term on the right-hand side of (4.11), we rely on the following coercivity property of the virial quadratic form (under the orthogonality conditions (3.51) ) proved in Lemma 3.5 in [2] and which is a variant of Lemma 3.4 in [23] based on Proposition 4 in [20]. (4.12) Now, we turn our attention to R Vir . We begin by explaining how to control |R Vir,1 | and |R Vir,5 |. We rely on the calculus inequality It follows that Hence, |R Vir,1 | and |R Vir,5 | will be controlled by using the contribution coming from the first term on the right-hand side of (4.12) and by taking B large enough.
To estimate R Vir,2 , we write Hence, we get On the one hand, observe from the Sobolev embedding and the bootstrap assumption (3.64) that (4.14) On the other hand, recall that F (s, y) = λ Thus, we deduce by using (3.5), (3.6), (3.23), (4.14) and (4.15) and by taking |s| large enough that To deal with R Vir,3 , we observe Hence, we deduce from (4.14) and (4.15) by taking |s| large enough (possibly depending on B) that To deal with R Vir,4 , we write Thus, we deduce by using (3.5), (3.6), (3.23) and (4.15) and by taking |s| large enough (depending possibly on B) that Then, we deduce gathering all those estimates that The proof of (4.6) follows by combining (4.10), (4.16) and choosing µ 0 = 2 −5 min{1, µ 1 }. Estimate of f 2 . We claim that By using the decomposition in (3.36), we have We first deal with f 2,1 . By using the definition of G B (ε) in (4.5) and integration by parts, we compute and (also using L(∂ y Q) = 0) We estimate each of these terms separately. By using the identity LΛQ = −2Q, the second and third orthogonality identities in (3.51), the localisation properties of ψ B , ψ ′ B and ϕ B , (4.13) and the decay properties of Q and ΛQ, it follows that where in the last line, we have also used the orthogonality condition (from (3.51)) Moreover, it follows from (3.5), (3.6), (3.23) and (3.25) that To deal with the nonlinear term, we recall the Sobolev bound Therefore, we deduce combining those estimates with (3.65) and (3.66), and choosing s and B > 100 large enough that Now, we turn to f 2,2 . We compute from the definition of G B (ε) in (4.5) By the Cauchy-Schwarz inequality and the properties of ϕ B and ψ B in (3.63), it holds To treat the nonlinear term, we observe that On the one hand, we deduce from (4.15), and then (3.63) and (4.7) that On the other hand, (3.63), the Sobolev bound (4.18), and then (3.64), (4.9) yield Then, we deduce combining those estimates with (3.34), (3.65), (3.66) and (3.67) that Finally, we conclude the proof of (4.17) gathering (4.19) and (4.20). Estimate of f 3 . We claim that From the definition of G B (ε) in (4.5), we have By using the identities we deduce from (3.66) and (3.63) that To deal with f 3,3 , we compute so that it follows from (3.66), (3.63), and then (4.7), (4.8), (4.9), (4.15), that Therefore, we deduce the proof of (4.21) gathering those estimates and taking s large enough (possibly depending on B). Estimate of f 4 . We claim that Recall from the definition of F in (4.1) that We compute integrating by parts (see also page 97 in [23]) Hence, we deduce gathering those identities that We will control each of these terms separately. Observe that λs λ > 0 since we are in a defocusing regime (see (3.19) and (3.66)). Thus Moreover, we get by using (3.19), (3.66) and (3.63) and, by using Hölder and Young inequalities, On the other hand, the terms (2 + κ) ψ B (∂ y ε) 2 and κ ϕ B ε 2 are positive as well as their product with λs λ . However, we can estimate them as above. It follows from (3.19) and (3. Now, we deal with the nonlinear terms. By using (3.19), (3.66) and (3.63), and then (4.7), (4.9), (4.15), we get that By definition Λ(W + F ) = 1 2 (W + F ) + y∂ y (W + F ). Moreover, we use that, for k = 0 or 1, Thus, we deduce from (3.19), (3.66) and (3.63), and then (4.7), (4.8), (4.9), (4.15), that Therefore, we conclude the proof of (4.22) gathering these estimates. Estimate of f 5 . We claim that We decompose, from the definition of f 5 , and estimate these two terms separately.
Finally, we conclude the proof of (4.3) gathering (4.6), (4.17), (4.21), (4.22) and (4.23). Now, we turn to the proof of (4.4). We decompose F as follows: To bound by below F 1 , we rely on the coercivity of the linearized energy (3.1) with the choice of the orthogonality conditions (3.51) and standard localisation arguments. Proceeding for instance as in the Appendix A of [21] or as in the proof of Lemma 3.5. in [2], we deduce that there existsν 0 > 0 such that, for B large enough, To estimate F 2 , we compute so that We will control each term on the right-hand side separately. First observe from (3.5) and (3.23) and arguing as for (4.7) (but without the restriction y < − B 2 ) that Moreover, we deduce from (4.18) that Therefore, we conclude the proof of (4.4) gathering those estimates.

Construction of flattening solitons
5.1. End of the construction in rescaled variables. In this subsection, we still work with the notation introduced in Sections 3 and 4. We prove that for well adjusted initial data, the decomposition of the solution introduced in (3.49) and the bootstrap estimates (3.19) and such that the solution of (1.1) evolving from has a decomposition λ(s), σ(s), b(s), ε(s) as in Lemma 3.8 satisfying the bootstrap conditions (3.19), (3.64) and if s 0 is chosen large enough, which strictly improves (3.64).

Main result.
We are now in a position to state the main result of this paper, in its full generality. Define the constants Theorem 5.2. There exists ρ 1 > 0 such that for any x 0 large enough, the following holds. Let t 0 = (2x 0 ) 1 β and let σ 0 and λ 0 be such that Let ε 0 ∈ H 1 (R) be such that Then there exists b 0 = b 0 (λ 0 , σ 0 , ε 0 ) with |b 0 | t − 1 2 (3β−1) 0 such that the solution U (t) of (1.1) corresponding to the following initial data at t = t 0 decomposes as where the functions λ(t), σ(t) and η(t) satisfy To prove Theorem 5.2 from Proposition 5.1, it is sufficient to return to the original variables (t, x) (see §5.3) and to prove the additional estimate (5.15) which improves the region where the residue η converges strongly to 0 (see §5.4).

5.3.
Returning to original variables. In the context of Proposition 5.1 and Theorem 5.2, we prove in this subsection the following set of estimates: and where ρ 1 is a small positive number.
Proof of (5.16)-(5.20). First, we relate t and s from (3.18). We claim that for any t ≥ t 0 , s ≥ s 0 , Indeed, from (3.19) and dt = λ 3 (s)ds, one has (ρ is defined in (3.20)) This proves the lower bound in (5.21). The corresponding upper bound is proved similarly.
Remark 5.1. Estimates (5.30)-(5.31) for T δ = T 0 ≫ 1 describe the behavior of the parameters both for large times and for intermediate times. Indeed, by continuous dependence of the solution with respect to the initial data, sinceũ(t, x) = Q(x − t) is a solution, it is clear that T δ → +∞ as δ → 0, and that for 0 < t ≪ T δ , the solution u(t) behaves likeũ(t).