The Gardner equation and the stability of multi-kink solutions of the mKdV equation

Multi-kink solutions of the defocusing, modified Korteweg-de Vries equation (mKdV) found by Grosse are shown to be globally $H^1$-stable. Stability in the one-kink case was previously established by Zhidkov, and Merle-Vega. The proof uses transformations linking the mKdV equation with focusing, Gardner-like equations, where stability and asymptotic stability in the energy space are known. We generalize our results by considering the existence, uniqueness and the dynamics of generalized multi-kinks of defocusing, non-integrable gKdV equations, showing the inelastic character of the 4-kink collision in some regimes.


Introduction and main results
In this paper we continue our work on stability of multi-soliton solutions for some well-known, dispersive equations, started in a joint work with M.A. Alejo and L. Vega [3]. In this opportunity, we consider the nonlinear H 1 -stability, and asymptotic stability, of the multi-kink solution of the defocusing, modified Korteweg-de Vries (KdV) equation u t + (u xx − u 3 ) x = 0. (1.1) Here u = u(t, x) is a real valued function, and (t, x) ∈ R 2 . Solutions u = u(t, x) of (1.1) are invariant under space and time translations, and under suitable scaling properties. Indeed, for any t 0 , x 0 ∈ R, and c > 0, both u(t − t 0 , x − x 0 ) and c 1/2 u(c 3/2 t, c 1/2 x) are solutions of (1.1). Finally, u(−t, −x) and −u(t, x) are also solutions.
From a mathematical point of view, equation (1.1) is an integrable model [2], with a Lax pair structure and infinitely many conservation laws. Moreover, equation On the one hand, since the Cauchy problem associated to (1.1) is locally well posed in ϕ c (· + ct) + H 1 (R) (cf. Merle-Vega [37,Prop. 3.1]), each solution is indeed global in time thanks to the conservation of Energy: A simple inspection reveals that this is a non-negative quantity.
On the other hand, the standard Cauchy problem for initial data in the Sobolev space H s (R) is locally well-posed for s ≥ 1 4 (Kenig-Ponce-Vega [24]), and globally well-posed for s > 1 4 (Colliander et al. [9]). This result is almost sharp since for s < 1 4 the solution map has been shown to not be uniformly continuous, see Christ-Colliander-Tao [8] (see also Kenig-Ponce-Vega [25] for an early result in the focusing case).
It is also important to stress that (1.1) has in addition another less regular conserved quantity, called mass: Of course this quantity is well-defined for solutions u(t) such that (u 2 (t) − c) has enough decay at infinity. In particular, one has M [ϕ c ] < +∞.
Now we focus on the study of suitable perturbations of kinks solutions of the form (1.2). This question leads to the introduction of the concepts of orbital and asymptotic stability. In particular, since the energy (1.5) is a conserved quantity -in other words, it is a Lyapunov functional-, well defined for solutions at the H 1 -level, it is natural to expect that kinks are (orbitally) stable under small perturbations in the energy space. Indeed, H 1 -stability of mKdV kinks has been considered initially by Zhidkov [47], see also Merle-Vega [37] for a complete proof, including an adapted well-posedness theory. We recall that their proof is strongly based in the non-negative character of the energy (1.5) around a kink solution ϕ c , which balances the bad behavior of the mass (1.6) under general H 1 -perturbations of a kink solution.
For additional purposes, to be explained later, we recall that in [37], the main objective of Merle and Vega was to prove that solitons of the KdV equation were L 2 -stable, by using the Miura transform . (1.8) This nonlinear H 1 − L 2 transformation links solutions of (1.1) with solutions of the KdV equation (1.7). In particular, the image of the family of kink solutions (1.2) under the transformation (1.8) is the well-known soliton of KdV, with scaling 2c (cf. [37]): Therefore, by proving the H 1 -stability of single kinks -a question previously considered by Zhidkov [47]-, and (1.8), they obtained a form of L 2 -stability for the KdV soliton. Additionally, a simple form of asymptotic stability for the kink solution was proved. Related asymptotic results for soliton-like solutions can also be found e.g. in [12,43,41,26,30,31].
Kinks are also present in other nonlinear models, such as the sine-Gordon (SG) equation, the φ 4 -model, and the Gross-Pitaevskii (GP) equation [1,13]. In each case, it has been proved that their are stable for small perturbations in a suitable space, cf. [23,18,47,15,6]. Let us also recall that the SG and GP equations are integrable models in one dimension [1,13].
Let us come back to the equation (1.1). In addition to the previously mentioned kink solution (1.2), mKdV has multi-kink solutions, as a consequence of the integrability property, and the Inverse Scattering method. This result, due to Grosse [19,20], can be obtained by a different approach using the Miura transform (1.8), see Gesztesy-Schweinger-Simon [16,17], or the monograph by Thaller [44]. According to Gesztesy-Schweinger-Simon [17], there are at least two different forms of multi-kink solutions for (1.1), which we describe below (cf. Definitions 1.1 and 1.4). Moreover, they proved that the Miura transform sends these solutions towards a well defined family of multi-soliton solutions of the KdV equation, provided a criticality property is satisfied (see [18,17] for such an assumption).
In order to present multi-kinks from a different point of view, we need some preliminaries.
In [3] (see e.g. [40] for a short review), Alejo-Muñoz-Vega showed the L 2stability of KdV multi-solitons following the Merle-Vega approach and the Gesztesy-Schweinger-Simon property [17], above described. However, H 1 -stability of multikinks was not known at that moment. Even worse, according to our knowledge, there was no result involving stability of several kink solutions, for any type of dispersive equation with stable kinks. Instead, we avoided this problem and followed a different approach, based in the use of the Gardner transform This nonlinear map links solutions of the KdV equation and the Gardner equation [38,14], In particular, the Gardner transform sends Gardner solitons towards KdV solitons (see [3] for further details). However, we have realized that the existence, uniqueness and stability of multikinks is closely related to the solitons of the Gardner equation, and more generally, dynamical properties of defocusing gKdV equations are closely related to those of suitable focusing counterparts. In particular, as a consequence of our results, we provide the first proof of stability for multi-kinks solutions of the mKdV equation. This result can be also considered a first step towards the understanding of the dynamics of several SG and φ 4 kinks.
In order to explain in more detail this relationship, let us recall that the Gardner equation is also an integrable model [14], with soliton solutions of the form v(t, y) := Q c,β (y − ct), In particular, in the formal limit β → 0, we recover the standard KdV soliton.
On the other hand, the Cauchy problem associated to (1.10) is globally well-posed under initial data in the energy class H 1 (R) (cf. [24]), thanks to the mass and energy conservation laws (see [3] for more details).
The first, striking connection is well-known in the mathematical physics literature, and it was in part used in the recent paper [3]. Indeed, let v = v(t, y) ∈ C(R, H 1 (R)) be a solution of (1.10). Then Miura, [38,37] < Gardner Note that, for t fixed, (1.12) is a diffeomorphism which preserves regularity, a key difference with respect to the Miura and Gardner transforms. Note in addition that u in (1.12) is an L ∞ -function with nonzero limits at infinity. This will allow to consider the first class of multi-kink solutions of (1.1), characterized by the same positive limit (= b) as x → ±∞. Moreover, since −u(t, x) is also a solution of (1.1), one can easily construct a solution with negative limits at infinity. This analysis motivates the following alternative approach for the multi-kink solution: Definition 1.1 (Even multi-kink solutions, see also [19,20,17,44]).
and (c 0 k ), and Q c,β being solitons of the Gardner equation (1.10).

Remarks.
1. Let us emphasize thatc N <c N −1 < . . . <c 1 , which means that bigger Gardner solitons are actually slower than the smaller ones. Note also that they move from the right to the left, as time evolves. In conclusion, as time goes to ±∞, the Gardner components of the multi-kink solution are ordered in the inverse sense compared with the usual solitons of the Gardner equation, or any focusing gKdV equation.
2. The denomination multi-kink above comes from the fact the these solutions can be seen asymptotically as the sum of several kinks ±ϕ c of the form (1.2). For instance, with our notation, given β > 0 and 0 < c < 2 9β , an expression for the 2-kink solution is given by [17, p. 505] (see also [44, p. 273 and ϕ c as in (1.2). Note that both kinks ±ϕ c/2 have the same velocityc, a key difference with the SG and φ 4 models. After a quick computation, using the identity one can see that (1.16) can be written, as in (1.14)-(1.15), using the Gardner soliton (1.11): U e (t, x) = b − βQ c,β (x +ct + x 0 ), which will be helpful for our purposes. From this fact one can say that in general, the function U e represents a 2N -kink solution. In terms of our point of view, it will represent N different Gardner solitons attached to the non-zero constant b.
The existence of a solution U e satisfying (1.14) is a simple consequence of (1.12) and the behavior of the N -soliton solution of the Gardner equation (see also [19,20,17] and [44, pp. 272-273] for the standard deduction). Indeed, from the integrable character of this last equation, given parameters β > 0, 0 < c 0 1 < . . . < c 0 N < 2 9β , and x 0 1 , . . . , x 0 N ∈ R, it is well-known that there exists a N -soliton solution of the form (see e.g. Maddocks-Sachs [28] for a similar structure) of (1.10), and which satisfies for some x ± j ∈ R, uniquely depending on the set of parameters (c 0 k , x 0 k ). Moreover, note that V (N ) (t) is unique in the sense described by Martel in [29]: Asymptotic uniqueness: Given β > 0, 0 < c 0 1 < . . . < c 0 N < 2 9β , and x − 1 , . . . , x − N ∈ R, the corresponding multi-soliton V (N ) given in (1.17) is the unique C(R, H 1 (R))solution of (1.10) satisfying (1.18). (1.20) Note that we do not need the criticality assumption required in [17,44]. Following the notation of Maddocks and Sachs [28] and (1.17), we may think U e as a function of three independent set of variables: U e (t, x) := U e (x; c 0 j , x 0 j +c j t), withc j given in Definition 1.1. Therefore, as a conclusion of the preceding analysis, and using (1.12), we get the uniqueness of the corresponding solution U e .
The second problem that we want to consider is the stability of the multi-kink U e . First of all, we recall some important literature.
In [36,35], Martel, Merle and Tsai have showed the stability and asymptotic stability of the sum of N solitons of some generalized KdV equations, well decoupled at the initial time, in the energy space H 1 (R). We say that such an initial data is well-prepared. Their approach is based on the construction of N almost conserved quantities, related to the mass of each solitary wave, plus the total energy of the solution. Although the proof for general nonlinearities is not present in the literature, it is a direct consequence of [36] (see also Section 5 in [35].). An important remark to stress is that their proof applies even for non-integrable cases, provided they have stable solitons, in the sense of Weinstein [45]. In the particular case of the Gardner equation, this condition reads This inequality is directly verifiable in the case of Gardner solitons, see (B.5). From this result and Definition 1.1 we claim the following In Section 2 we give a precise, ε-δ formulation of this result. See Theorem 2.3.
There is a second type of multi-kink solutions for (1.1), which is actually the best known one. Here, the standard kink ϕ c in (1.2) and the Gardner equation play once again a crucial and surprising rôle. Indeed, let β > 0 be a fixed parameter and suppose that one has a solution of (1.1) of the form (the reader may compare with (1.12) and (1.13)) andũ(t) ∈ H 1 (R). Thenũ(t, y) satisfies the equatioñ with ϕ c = ϕ c (y − 2ct). In particular, if the support ofũ(t) is mainly localized in the region where ϕ c ∼ − √ c, namely y 2ct, then the right hand side above is a small perturbation of the left hand side, a Gardner equation with parameter β > 0. As an admissible functionũ, we can take e.g. a sum of Gardner solitons: with support localized in the region c 1 t y c N −1 t, for t 1. In particular, one has c N −1 t 2ct for t 1, which is a necessary condition for the existence of a solution of the form (1.22). Note in addition that Figure 1 can be adapted in the following way: The transformation linking (1.23) and KdV is nothing but (compare with the Gardner transform (1.9)). Finally, the same argument can be done in the case of a solution of the form , and the equation forû(t, y), providedû is supported mainly in the region {ϕ c ∼ √ c}. These two new ideas allow us to consider the following definition of a multi-kink solution, from the point of view of the Gardner equation: Definition 1.4 (Odd multi-kink solutions, [17,44]).
One can also say that U o is composed by (2N − 1) single kinks, in other words, it is a (2N − 1)-kink solution. Additionally, as above mentioned, one may think this solution as function composed of three different class of parameters:  . Below, the behavior as t → −∞; above, the behavior as t → +∞. Each part is ordered according to their respective velocity −c j := c 0 j − 3c 0 3 < 0, j = 1, 2. Note that −c 1 < −c 2 < −c 0 3 , which means that the smallest soliton Q c 0 1 ,β is actually the fastest one.
The proof of the existence of this family is not direct, although it can be explicitly obtained from the solutions found by Grosse [19,20], using the Inverse Scattering method (see also [17], or [44, pp. 270-272] for an alternative procedure involving the inverse Miura transform). In this paper we present a third proof, which gives in addition a uniqueness property, uniform estimates and does not require the criticality property considered in [17,44]. The uniqueness is, of course, modulo the 2N -parameter family (c 0 j , x − j ). Theorem 1.5 (Existence and uniqueness of odd multi-kink solutions).
Let The family of multi-kink solutions We prove this result in Section 4. In particular, we give a more precise statement in Theorem 4.1.
General nonlinearities. We point out that our results, starting from transformations (1.12)-(1.22), and the Zhidkov theory developed in [47], can be made even more general and include a wide range of non-integrable, defocusing gKdV equations.
Indeed, in the next paragraphs we first introduce the notion of generalized, even and odd multi-kink solutions. Of course these objects have to match with those considered in Definitions 1.1 and 1.4, for the special case of the integrable mKdV model. Second, we study the existence, uniqueness and stability of these new solutions in the case of well-prepared initial data. Next, we consider some particular collision problems, in the spirit of [32,33,39] (note that the collision problem makes sense since we consider non-integrable equations). The method used is the same as in the previous results, so we will skip most of the proofs. We emphasize that the main idea is to exploit the properties contained in the following figure: u ∈ defocusing gKdV (even) < (u ∼ b +ũ) >ũ ∈ focusing gKdV u ∈ defocusing gKdV (odd) < (u ∼ ϕc +ũ) >ũ ∈ focusing gKdV Let us consider the generalized, defocusing KdV equation (1.27) Here f : R → R is a non-linear term, with enough regularity, to be specified below. From now on, we will assume the following hypotheses: Let k 0 ∈ {2, 3, 4} be the first integer k satisfying this property. We assume f is of class C k0+1 (R).
The reader may compare e.g. with the integrable case f (s) = s 3 , where for b 0 = 0 one has f (b 0 ) = 6b 0 = 0. Another important example is the cubic-quintic nonlinearity f (s) := s 3 + µs 5 , µ ∈ R − {0} fixed, for which in the case µ < 0. Otherwise, we are in a degenerate case and f (b 0 ) = 0. Computing the third derivative, one has . For any µ = 0 one has that b 0 = 0 does not satisfy (b) above and therefore is not allowed. In the case Under these two assumptions and using [29, Remark 2], we prove existence and uniqueness of generalized even multi-kinks for (1.27), satisfying the equivalent of (1.15) (cf. Definition 1.1). Theorem 1.7 (Existence and uniqueness of generalized, even multi-kinks).
Let N ≥ 2 and b 0 ∈ R be such that (a)-(b) above are satisfied. There exists is a soliton solution of the focusing gKdV equatioñ Proof. See Section 3. Remarks.
Proof. The proof is similar to that of Theorem 1.3. We skip the details.
Another striking consequence of (3.1) is the fact that we can describe the interaction among even kinks in some regimes, in the spirit of [32,33,39]. Indeed, one has the following Theorem 1.9 (Inelastic interaction of even 4-kinks).
Let b 0 ∈ R such that (a)-(b) are satisfied, and suppose in addition that f is of Remarks. 1. The condition (1.33) allows us to rule out the integrable cases f (s) = αs 2 + βs 3 , α, β ∈ R. 2. By not pure as t → −∞ in Theorem 1.9 we mean that (1.14) cannot happen: 3. The collision problem has been recently considered in the case of the NLS equation: see e.g. Holmer-Marzuola-Zworski [21,22], Perelman [42], and references therein.
Proof of Theorem 1.9. We prove Theorem 1.9 in Section 3. The main idea is that condition (1.33) is the key point to invoke [32]  Another collision result is the following remarkable consequence of the recent Martel-Merle's papers describing the interaction of (i) two very different [32], and (ii) two nearly equal solitons of the quartic gKdV equation [34]. Let b 0 ∈ R, and f b0 (s) := s 4 − 4b 0 s 3 + 6b 2 0 s 2 . Then f b0 satisfies (a)-(b) above, one has c * (f b0 , b 0 ) = +∞ and for any 0 < c 0 1 c 0 2 , the corresponding 4-kink U e (t) constructed in Theorem 1.7, and pure as t → +∞, is globally H 1 -stable, but it is not pure as t → −∞. The same result is valid in the regime 0 < c 0 Remark. The last result is a consequence of the fact that from (1.30) and (1.31), one hasf b0 (s) = s 4 , for which solitons Q c exist for any c > 0. Note in addition that for any b 0 = 0, the corresponding nonlinearity f b0 does not allow to perform the standard transformation u → −u, which links the defocusing and focusing quartic equations. Moreover, the quadratic term in f b0 is always of defocusing nature. Therefore, the above result is completely new for b 0 = 0.
Finally, we consider the case of generalized, odd multi-kink solutions. First of all, we have to recall some important facts. For more details, the reader may consult the monograph of Zhidkov [47].
Let ϕ − , ϕ + ∈ R, with ϕ − < ϕ + , and let c > 0, x 0 ∈ R be fixed numbers. Let f be the nonlinearity considered in (1.27). Suppose that the following hypotheses hold: (e) f (ϕ ± ) > c (non degeneracy condition). 4 Then there exists a monotone, generalized kink solution of (1.27), of the form and ϕ c satisfies Moreover, this solution satisfies, for some constants K, γ > 0, the following estimates

Remarks.
1. For the sake of clarity, let us mention that in the integrable case f (s) = s 3 , given ϕ − ∈ R and c > 0, one has that for ϕ + > ϕ − , conditions (c)-(d) and (e) lead to 2. It is important to point out that the multi-kink solution U e constructed in Theorem 1.7 cannot be decomposed as the sum of several kinks of the form (1.34), at least in a general situation (compare e.g. with (1.16)). Therefore, we believe that (1.28) and (1.35) below are the correct ways to define generalized multi-kink solutions, in the case of defocusing gKdV equations.
3. The Cauchy problem associated to (1.27) with initial condition satisfying u(0) − ϕ c ∈ H 1 (R), is locally well-posed in the class ϕ c (· + ct) + H 1 (R). This result is consequence of the analysis carried out by Merle and Vega in [37] and the fact that f is regular enough. In what follows, we will only consider stable solutions, then globally well defined.
Our next objective is to generalize the Zhidkov's results to the case of (2N − 1)kinks, as follows: Theorem 1.11 (Existence and uniqueness of generalized odd multi-kinks).
Finally, we say that an initial configuration u 0 , perturbation of a kink solution ϕ c , is well-prepared if for L, α > 0, 0 < c 0 1 < . . . < c 0 N −1 < c * , and x 0 1 < x 0 2 < . . . < x 0 N , one has (1.29). In addition, by taking c * smaller if necessary, we assume that each soliton Q c 0 j is stable in the sense of Weinstein (1.21). Theorem 1.12 (Stability of the odd multi-kinks).
This result is proved following the lines of the proof of Theorem 1.6, using in addition that single solitons are stable. We skip the details.
Final remarks. 1. We recall that the collision problem in the case of odd multi-kink solutions remains an interesting open question. In addition, we believe that our approach introduces new ideas to deal with the dynamics of kink solutions in the L 2 -critical and supercritical setting, by using a suitable focusing counterpart. 2. Let us mention that a similar transformation to (1.22) can be introduced in the cases of the φ 4 and sine-Gordon models, with different results. For the first equation, with ϕ given in (1.3), is a stationary kink solution. However, the transformation u(t, Looking for an approximate, localized and stationary solution, we arrive to study the elliptic equation associated to the Gardner nonlinearity: This is a Gardner elliptic equation with parameters c := 1 and β := 2 9 (cf. (B.1)), for which the only localized, positive solution is zero. Therefore it is not possible (at least formally) to attach to the kink solution of (1.37) suitable soliton-like structures of the form (1.11). Now we perform the same analysis in the case of the sine-Gordon equation and its kink solution ϕ(x) := 4 arctan e x . Indeed, using the transformation u(t, x) := ϕ(x) +ũ(t, x), we arrive to the following perturbed, sine-Gordon equatioñ where the right hand side is small if we considerũ as a localized solution of (1. 38) in the region where ϕ ∼ 0. We can put for instance, a sum of breather solutions, provided this solution is stable, which is an open problem. We expect to consider some of these problems in a forthcoming publication.
Idea of the proofs. Theorems 1.2, 1.3, 1.7 and 1.8 can be deduced from Martel [29] and Martel-Merle-Tsai [36,35]. We recall that, without using transformation 1.12, these results were unable to be tackled down by using any direct method.
We prove Theorem 1.6 in Section 4. The proof is based in the approach introduced in [36] in order to describe the stability in H 1 (R) of N decoupled solitons. However, in this opportunity we face several new problems since the kink solution and the Gardner solitons are in strong interaction through the dynamics. Moreover, the mass (1.6) cannot be used to control the Gardner solitons, as has been done in [36]. This means that Theorem 1.6 cannot be deduced from the standard Zhidkov [47] and Martel-Merle-Tsai [36] results, and we need new ideas. In that sense, the transformation (1.22) is the first step -and the more important one-to understand the interaction among kinks as actually localized, soliton-like interactions.
Let us be more precise. Using the energy (see (1.5)) of the solution u(t), one can control with no additional difficulties the kink solution. This is a consequence of the non negative character of the linearized operator around the kink solution, see [47,37] for more details. However, this quantity is far from being enough to control the behavior of the Gardner solitons. We overcome this difficulty by using the transformation (1.22), which introduces a new functionũ(t), almost solution of a Gardner-like equation (cf. (1.23)). It turns out that the perturbative terms on the right hand side of (1.23) can be controlled provided the solitons are far from the center of the main kink solution, which holds true if we assume that the initial configuration is well prepared (see Proposition 4.2). Additionally, we introduce a new, almost conserved mass (see (4.25)) for the portion on the left of the solutionũ, which allows to control each Gardner soliton by separated. Using this, we avoid the problem of using the natural mass (1.6), which is very bad behaved for H 1 (R) perturbations. This approach is completely general and can be adapted to prove Theorem 1.12. No additional hypotheses are needed, only the single stability of each generalized soliton component of the multi-kink solution. The proof of the asymptotic stability property generalizes the argument used [35], this time to the functionũ.
Finally, concerning Theorem 1.5 -proved in Section 5-, we extend the result of Martel [29]. Most of the proof is similar to the proof of Theorem 1.6, but estimates are easier to carry out since we do not need to control the scaling parameters of each Gardner solitons.

Proof of Theorems 1.2 and 1.3
Proof of Theorem 1.2. LetŨ be another solution of (1.1) satisfying (1.14). Then, from (1.12),Ṽ (t, y) :  [36,35]). Let N ≥ 2, β > 0 and 0 < c 0 1 < c 0 2 < . . . < c 0 N < 2 9β be such that (1.21) holds for all j = 1, . . . , N. There existsα 0 ,Ã 0 ,L 0 ,γ > 0 such that the following is true. Let v 0 ∈ H 1 (R), and assume that there existsL >L 0 ,α ∈ (0,α 0 ) and Moreover, there exist c ∞ j > 0 such that lim +∞x j (t) = c ∞ j and It is important to stress that the well-preparedness restriction on the initial data (2.1) is by now necessary since there is no satisfactory collision theory for the non-integrable cases. 5 However, as explained in [36] for the KdV case, the above argument can be extended to a global-in-time stability result, thanks to the continuity of the Gardner flow in H 1 (R) [24], and the fact that the Gardner equation (1.10) is an integrable model, with explicit N -soliton solutions (see (1.18)-(1.19)), given by the family V (N ) above described. Therefore, a direct consequence of this property and the invariance of the equation under the transformation u(t, x) → u(−t, −x) is the following Corollary 2.2 (H 1 -stability of Gardner multi-solitons, [36,35]).
Let δ > 0, N ≥ 2, 0 < c 0 1 < . . . < c 0 N and x 0 1 , . . . , x 0 N ∈ R. There exists α 0 > 0 such that if 0 < α < α 0 , then the following holds. Let v(t) be a solution of (1.10) Remark. Let us emphasize that the proof of this result requires the existence and the explicit behavior of the multi-soliton solution V (N ) of the Gardner equation, and therefore the integrable character of the equation. In particular, we do not believe that a similar result is valid for a completely general, non-integrable gKdV equation, unless one considers some perturbative regimes (cf. [32,34] for some global H 1 -stability results in the non-integrable setting.) Therefore, using (1.12) and the previous result one has the following more precise version of Theorem 1.3. that if 0 < α < α 0 , then the following holds. Let u(t) be a solution of (1.1) such that

6)
with U e the 2N -kink solution defined in (1.20). Then there exist x j (t) ∈ R, j = 1, . . . , N , such that Moreover, there exist c ∞ j > 0 such that and x j (t) are C 1 for all |t| large enough, with x j (t) → c ∞ j ∼c j as t → +∞. A similar result holds as t → −∞, with the obvious modifications.
Remark. Let us recall, for the sake of completeness, that estimate (2.8) is deduced from (2.5) by using the transformation (1.12).
3. Proof of Theorems 1.7 and 1.9 Proof of Theorem 1.7. Thanks to (a)-(b), there exists a generalized transformation of the form (1.12), such that (3.1) with b 1 given by (1.29), and such thatũ(t, y) satisfies (1.30)-(1.31). Moreover, note that a Taylor expansion gives us thatf is a subcritical perturbation of the pure power nonlinearity: for some ξ in between b 0 and b 0 + b 1 s. Note in addition that lim s→0f b0 (s) |s| k0 = 0.
According to Berestycki and Lions [5],f is an admissible nonlinearity for the existence of small solitons, in the sense that there exists c * > 0 (depending on f and b 0 fixed), such that for all 0 < c < c * , there exists a solutionũ =ũ(t, y) of (1.30)-(1.31), of the formũ (t, y) = Q c (y − ct), and such that Q c = Q c (s) satisfies Moreover, Q c can be chosen even and exponentially decreasing as s → ±∞.
Therefore, from the classification theorem for the regime 0 < c 1 c 2 c * showed in [32,33,39], one can conclude that the 2-soliton structure is globally H 1stable, but the solution U e constructed in Theorem 1.7 is never pure as t → −∞. The final conclusion follows after applying (3.1). The proof is complete.
Note that a simple continuity argument, using the local Cauchy theory developed in [37] shows that there exists t 0 > 0 such that (4.6) for somex j (t) ∈ R, j = 1, . . . , N . Therefore, given K * > 2, we can define the following quantity T * := sup T > 0, for all t ∈ [0, T ], (4.6) is satisfied with 2 replaced by K * , and for somex j (t) ∈ R. . Our objective is to show that for some K * > 0 large enough, one has T * = +∞. Following a contradiction argument, we will assume T * < +∞. This allows to prove the following modulation property.
In what follows, we introduce some useful notation. Let us consider Q cj ,β (x +c j t + x j (t)), (= the Gardner solitons) (4.13) andũ(t, y) defined by the relation where, for the sake of clarity, we have defined c := c 0 N . In particular, =:R(t, y) +z(t, y). A simple computation shows thatũ =ũ(t, y) satisfies the modified Gardner equation (compare with (1.23)) In this last equation ϕ c is a function of the variable y in the sense that ϕ c (x + ct + x N (t)) = ϕ c (y − 2ct + x N (t)). The following result gives an explicit expansion of the energy of u(t).

Lemma 4.4 (Expansion of the energy).
Consider the energy E[u](t) defined in (1.5), for c = c 0 N = 1 9β . Then, for any t ∈ [0, T * ], one has the following decomposition with F(t) the following second order functional Proof. From (4.8) and (4.13), one has: First of all, note that the term E[ϕ c ](t) actually does not depend on t. Additionally, from (1.2) and (1.4) one has In order to obtain some estimates of the above quantities, we need the following Lemma 4.5 (Identities for R(t)). Let R be the sum of N decoupled Gardner solitons defined in (4.13). Then one has the following identities: . (4.20) and Proof. Let us prove (4.20). From (4.13) and the fact that c = 1 9β , one has l.h.s. of (4.20) = β Using the equation for Q c,β (cf. (B.1)), one has l.h.s. of (4.20 as desired. Now we consider (4.21). From (4.13) and (B.4), one has l.h.s. of (4.21) = β The proof is complete.
Let us come back to the proof of Lemma 4.4. From the above results, the orthogonality conditions (4.10), (4.18) and (4.19) we have Finally, the last two lines in the above identity, namely (4.22)-(4.23), are exponentially small. Indeed, one has e.g.
The other terms can be bounded in a similar fashion. From these estimates, (4.17) follows directly.
Finally, let, for j = 1, . . . , N − 1, the modified mass (4.14). Note that this quantity considers the mass on the left of each soliton, which represents the main difference, compared with the standard arguments included in [36,31].
There exist K > 0 and L 0 > 0 such that, for all L > L 0 , the following is true. (4.26) Proof. The proof is similar to [29,36], so we sketch the main steps. Let j ∈ {1, . . . , N − 1}. Using equation (4.16) and integrating by parts several times, we have j (4.27) Let us consider the term (4.27). By definition of ψ, |ψ (3) | ≤ σ0 4 ψ , so that In order to bound the term R ( 4 3ũ 3 − 3 2 βũ 4 )ψ j , one follows the argument of [36], splitting the real line in two different regions according to the position of each soliton, and the rest. Following that argument, one finds Now we consider the term (4.28). Note that one has , and a similar estimate is valid for the term |(1 − ψ j )( √ c + ϕ c )|. Therefore Let us consider the term (4.29). In this case, it is enough to recall that and |x N (t)| ≤ Kα. Finally, we obtain for some K > 0, Thus, by integrating between 0 and t, we get the conclusion. Note that K and L are chosen independently of t.
Lemma 4.7 (Quadratic control of the variation of c j (t)).
There exists K > 0 independent of K * , such that for all t ∈ [0, T * ], Proof. We proceed in several steps, following the proof given in [36].
1. Note that from (4.17), and using a Taylor expansion of the function around the point s 0 := M [Q cj (0),β ], 6 one has for some K > 0, Note in addition that for α 0 small and L 0 > 0 large, from (B.5), (4.32) In other words, the left hand side above is of quadratic variation in z. 6 In particular, a simple computation using (B.5) and c = 1 9β shows that (2c − c j (0)) = β(2c − c j (0)). We claim that there exists K > 0 such that, for all j = 1, . . . , N − 1, is nonnegative, there is nothing to prove. Let us assume that d j (t) − d j (0) < 0, therefore we have to show that

Lemma 4.8 (Bootstrap).
There exists K > 0, independent of K * , such that for all t ∈ [0, T * ], Proof. By (4.17), On the other hand, note that from (4.32) and Lemma 4.6, Thereforẽ withF(t) given by the formulã We prove in Appendix A that this quadratic form is coercive, in the sense that there exists λ 0 > 0 independent of t and K * such that, thanks to (4.9) and (4.10), Therefore, from (4.40), (4.31), and taking α 0 smaller if necessary, we obtain , for some constant K > 0, independent of K * . Thus, the proof of Lemma 4.8 is complete.
We conclude the proof of Proposition 4.2. From (4.11), Lemmas 4.7 and 4.8, we have , where K > 0 is a constant independent of K * . Finally, choosing K * = 4K, we get the desired contradiction. The proof is complete.

Sketch of proof of Theorem 1.5
The proof follows the lines of [29, Theorem 1] and the proof of Proposition 4.2 from the previous section. Let us assume the hypotheses of Theorem 1.5. Let T n → +∞ be an increasing sequence, and It is clear that R(t) − ϕ c 0 N (· + c 0 N t) is uniformly bounded in any H s (R), s ≥ 0. Now we consider the following Cauchy problem From [37], one has global existence of a unique solution u n (t) for (5.1), satisfying u n − ϕ c 0 N ∈ C(R, H 1 (R)), with conserved energy (1.5). The main part of the proof is to establish the following uniform estimates: There exist K, n 0 > 0 such that for all n ≥ n 0 , and for all t ∈ [T n0 , T n ], one has with σ 0 > 0 defined in (4.5).
The proof of this result is similar to the proof of Proposition 4.2, but it is easier since we do not need to modulate the scaling parameters c j (t) in Lemma 4.3. In particular, the term ∼ j c 3/2 j (t) in (4.17) is constant. Lemma 4.6 holds with no modifications. In order to control the directions Q cj ,β in (4.43), we only use Lemma 4.6, so Lemma 4.7 is not needed. The reader may consult [29] for a detailed proof.
As a consequence of the above estimate, one has, up to a subsequence, and for all t ≥ T n0 , satisfies the equation (1.23). Arguing as in [29, eqn. (14)], one has the following Lemma 5.2 (Egorov estimate).
Uniqueness. Using once again the transformation (1.22), and the equation (1.23), we claim that from [29], one has the following Lemma 5.3 (Exponential decay).
Let v ∈ ϕ c 0 N (·+c 0 N t)+C(R, H 1 (R)) be a solution of (1.1) satisfying (1.26). Then there exists K, T 0 > 0 such that Using this property, the uniqueness result is just a consequence of the analysis carried out in [29]. We skip the details.
Appendix A. Proof of (4. 43) In this section we sketch the proof of (4.43). See e.g. [36] for a detailed, similar proof. First of all, note that from (4.41), (4.13) and (4.15) one has with c(t, x) given in (4.42).
1. We recall the following well-known result.
Lemma A.1 (Positivity of the Zhidkov functional, see [47,37]). There exists λ 0 > 0 such that for all z ∈ H 1 (R), with R zϕ c = 0, one has Lemma A.2 (Localized coercivity, see e.g. [36,46]). There exists B 0 , λ 0 > 0 such that, for all B > B 0 , if z ∈ H 1 (R) satisfies Note that a similar argument can be followed in order to prove a localization property for the Zhidkov functional considered in Lemma A.1. We skip the details.
3. Now we perform a localization argument, as in [36]. One has from (A.1), Each term above can be treated following the lines of the proof of Lemma 4 in [36], and it is proved that for all B large, the above terms can be estimated by The final conclusion is that for B large enough, but independent of z, for some λ 0 > 0 independent of z(t) and B. Thus the proof of (4.43) is complete.
Appendix B. Proof of some identities Proof. The first identity in (B.1) is just the elliptic equation for Q c,β , obtained by replacing in (1.10). The second one follows from the first identity in (B.1), after multiplication by Q c,β and integration in space.
On the other hand, the first identity in (B.2) follows after integration of (B.1). In the same form, the second identity in (B.2) is a consequence of the first identity in (B.1) and the integration of the second one in (B.1) against Q c,β .
Let us prove (B.3). Note that from (B.1), Using the definition of Q c,β from (1.11), and integrating, one gets Acknowdlegments. I would like to thank Yvan Martel, Frank Merle, Miguel Angel Alejo, Luis Vega and Manuel del Pino for several remarks and comments on a first version of this paper. This work was in part written at the University of the Basque Country, and the University of Versailles. The author has been partially funded by grants Anillo ACT 125 CAPDE and Fondo Basal CMM.