Inelastic interaction of nearly equal solitons for the BBM equation

This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of the results obtained in arXiv:0910.3204 by the same authors, addressing the same question for the quartic (gKdV) equation. First, we prove that the two solitons are preserved by the interaction and that for all time they are separated by a large distance, as in the case of the integrable (KdV) equation in this regime. Second, we prove that the collision is not perfectly elastic, except in the integrable case (i.e. in the limiting case of the (KdV) equation).


Review on the collision problem for (KdV) type equations
We briefly review some results concerning the problem of collision of solitons for (KdV) type models and we refer to the introduction of [31] for more details.
First, it is very well-known that the (KdV) equation has explicit pure N -soliton solutions ( [17], [42], [34]): for any given c 1 > . . . > c N > 0, y − 1 , . . . , y − N ∈ R, there exists an explicit multi-soliton solution u(t, x) of (KdV) which satisfies for some y + j (such solutions were found using the inverse scattering transform). Stability and asymptotic stability of N -solitons were studied by Maddocks and Sachs [24] in H N by variational techniques and in the energy space H 1 by Martel, Merle and Tsai [33].
However, except for some integrable equations for which special explicit solutions are known, the problem of describing rigorously the collision of two solitons is mainly open. Now, we review some recent rigorous works related to the interaction of two solitons in the nonintegrable situation for the generalized KdV equations Recall that solitons of (gKdV) write R c,y (t, x) = c 1 p−1 Q( √ c(x − ct − y)), for c > 0, y ∈ R where Q satisfies Q ′′ + Q p = Q.
Mizumachi [36] studied rigorously the interaction of two solitons of nearly equal speeds for (gKdV) for p = 3 and p = 4. For initial data u 0 close to Q(x) + c  (1.5) for large time, for some c + 1 < c + 2 close to 1 and ε small in some space. The analysis part in [36] relies on scattering results due to Hayashi and Naumkin [15,16] and on the use of spaces of exponentially decaying functions (introduced in this context by Pego and Weinstein [39]).
From [36], the situation is roughly speaking similar to the one described in the integrable case by LeVeque [23]. However, two main questions were left open in this work in this regime Is the 2-soliton structure stable globally in time in the energy space H 1 ? Does there exist a pure 2-soliton in this regime? As in the integrable case, we call pure 2-solitons, solutions of (gKdV) satisfying Note that if (1.6) holds both at −∞ and +∞, then necessarily c − j = c + j for j = 1, 2 (see [30], pp. 68, 69).
These two questions have been answered in a recent work by the authors [31]. Indeed, in the context of two solitons of almost equal speeds for the quartic (gKdV) equation, by constructing an approximate solution to the problem, we were able to prove first the global stability of the two soliton structure in H 1 and second, the inelastic character of the interaction. See Theorems 1 and 2 in [31].
We also point out some other recent works of the authors ( [29], [30]) concerning the problem of collision of two solitons of (gKdV) for a general nonlinearity g(u) in the case where one soliton, is supposed to be large with respect to the other soliton, i.e. assuming 0 < c 1 ≪ c 2 . See also [32], with T. Mizumachi, extending these results to the (BBM) equation.

Main results
In the present paper, we extend the results of [31] to the (BBM) model.
There are two main motivations to consider these questions for the (BBM) model: first, the structure of the (BBM) equation is close to the one of the (KdV) equation but it cannot be considered as a perturbation of the (KdV) equation. Second, the present paper on (BBM) proves that our techniques extend to quadratic nonlinearity, unlike [36], based on scattering techniques critical for p = 3.
Comments on the results: 1. The (KdV) case in Theorems 1 and 2.
The value λ = 0 in the BBM equation corresponds to the integrable KdV equation. In this case, estimates (1.9)-(1.10) still hold. Estimate (1.9) corresponds to (1.4) but from the proofs in the present paper, we improve the main result in [23] in this case by computing explicitely the term of size µ 2 0 , see Remark 3.
Note also that for λ = 0, the existence of pure 2-soliton solutions corresponds to µ + 1 = µ 0 and µ + 2 = −µ 0 in (1.11) and (1.12). Moreover, Theorem 2 holds for λ = 0 and it is also a new global stability result for the (KdV) equation in the energy space. This kind of result cannot be proved by scattering theory.
2. Except for the value of the constant α > 0, Theorems 1 and 2 are exactly the same as for the quartic (gKdV) equation. In particular, the orders of size in µ 0 in the various estimates do not depend on the power of the nonlinearity. Moreover, the function (1.16) appears in both problems and has a universal character in this problem. Note that Theorems 1 and 2 can be extended to any two solitons of (1.1) of almost equal sizes using a simple scaling argument. See Appendix C.
Finally, from the present paper and [31], it is clear that the results can be extended to (gKdV) equations with general nonlinearities.
3. As in [31], the lower bounds in (1.11) and (1.12) measur the inelastic character of the collision. Moreover, the different exponents of µ 0 in (1.11) and (1.12) denotes a gap in the estimates which is an open problem.

Strategy of the proofs
We describe briefly the strategy of the proofs of Theorems 1 and 2, which is the same as in [31]. We point out the analogies and the main technical differences between the (BBM) and the quartic (gKdV) case. The proof of Theorem 2 is a consequence of the proof of Theorem 1 and so we focus on the proof of Theorem 1. Let U (t) be the solution of (BBM) defined in Theorem 1.
(1) The first step is the construction of an approximate solution in terms of a series in e −y(t) where y(t) = y 1 (t)−y 2 (t) is the distance between the two solitons, using the exponential decay of the solitons. From Proposition 2.1, the approximate solution contains a tail of order e −y(t) between the two solitons, which is relevant in the description of the exact solution, see Remark 3. This tail of order e −y(t) is not related to inelasticity since it appears also in the integrable case λ = 0. Moreover, it does not prevent the approximate solution to be in the energy space at this order, since it is localized in space between the two solitons.
In contrast, for λ = 0, one cannot build an approximate solution at order e − 3 2 y(t) in the energy space, whereas it is possible for λ = 0. The presence of a nonzero tail at −∞ in space at this order is related to nonintegability and inelasticity.
The construction of the approximate solution for (BBM) in Section 2 is more involved that in the quartic (gKdV) case mainly because the nonlinearity is quadratic rather that quartic.
(2) After the approximate solution is constructed, we introduce the following decomposition of the solution U (t): where Q c 1 (t) (x − y 1 (t)) + Q c 2 (t) (x − y 2 (t)) + W (t, x) is the modulated approximate solution and ε(t) is a rest term. To prove stability of the two soliton structure, we have to control both the parameters c j (t) and y j (t) and the rest term ε(t).
From the construction of the approximate solution, the parameters c j (t) and y j (t) have to satisfy an approximate dynamical system. Remarkably, it is exactly the same dynamical system as for the quartic (gKdV) equation (except the values of the numerical constants). This dynamical system, and the related solution Y (t) of the ODË seem to be universal in this type of problems. The control of the dynamical system satisfied by the parameters is thus exactly the same as in [31] and we will not repeat the arguments in the present paper (see Section 4).
Concerning the control of the rest term ε(t), as in [31], we use variants of techniques developed for large time stability and asymptotic stability of solitons and multi-solitons for the (gKdV) equations in the energy space, [44], [27], [33] and [25], extended to the (BBM) case in [45], [37], [9], [10], [11] and [26]. At this point, we need some new refined arguments and the proofs are more involved than in the quartic (gKdV) case. Note that since the nonlinearity is quadratic, one cannot use scaterring theory from [15], [16] as in [36].
(3) Finally, in Section 5, we prove that for λ = 0, the defect due to the interaction of two solitons is bounded from below, which implies in particular that the collision is not elastic.
Assuming for the sake of contradiction that the lower bound in (1.11) is not satisfied for any positive value of c, we obtain first some symmetry properties (x → −x, t → −t) on the parameters c j (t), y j (t) at a certain order.
Second, using space decay properties of U (t, x), we obtain a gain in the control of the error term in the dynamical system satisfied by c j (t), y j (t). Using this refined version of the dynamical system which is not symmetric for λ = 0 (as a consequence of the tail of order e − 3 2 y(t) in the approximate solution), we find a contradiction.

Construction of an approximate solution
We denote by Y the set of functions f ∈ C ∞ (R, R) such that , σ ≥ 3 and 0 < µ * < 1/10 such that for any 0 < µ 0 < µ * , the following hold.

9)
and for some C = C(K) > 0, Note that the function V 0 is not in L 2 since B j have non zero limits at −∞. We now introduce an L 2 approximation of V 0 , using a suitable cut-off function. Let ψ : R → [0, 1] be a C ∞ function such that As a consequence of Proposition 2.1, we obtain the following result. Then, (i) Closeness to the sum of two solitons.

16)
and for some C = C(K) > 0, (2.17) The proof of Proposition 2.2 being very similar to the one of Proposition 2.2 in [31], it is omitted.

Preliminary expansion
We set and similarly for Λ 2 R j , where ΛQ µ , Λ 2 Q µ are defined in Claim A.2. We introduce the notation 20) and M j , N j as in (2.9), for α, β, δ and a, b j , d j to be determined. We look for an approximate solution of S(v) = 0 under the form v(t, where w(t, x) = w(x; Γ(t)) so that using the equation of Q µ (see (A.5)) and ∂ ∂µ j R j = Λ R j , where In the rest of this section, we give preliminary expansions of F and F .
Note that the term F Q does not exist in the quartic case (see Lemma 2.1 in [31]). The proof of Lemma 2.1 is given in Appendix A.
The proof of this result is the same as the one of Lemma 2.2 in [31], thus it is omitted.

Determination of
(2.23) Then, Proof. Proof of (i). First, we determine α. Multiplying the equation of A 1 by Q, integrating and using L(Q ′ ) = 0, we obtain by (A.9) and (A.8) (2.26) Second, we find the value of θ A . For θ A to be chosen, set To findÂ 1 in Y, we need First, Using the estimate and (A.24), we have Thus, using the expressions of F A and F A in Lemmas 2.1 and 2.2 and the equations of A 1 and A 2 , we find Finally, we compute µ 1 Thus, using (2.6), For this term, we use Claim A.3 (see Appendix A.2), i.e.
We obtain Combining these computations, we obtain and similarly for the other scalar products in (2.25).

Nonlocalized term of order O 3/2
Lemma 2.4 (Approximate solution at order O 3/2 with localized error tem). Let Then Proof. The proof is based on Claim A.5 in Appendix A. First, arguing as in the proof of Lemma 2.3, we have Moreover, since x = 1 2 (x − y 1 + x − y 2 ) + 1 2 (y 1 + y 2 ), using (2.6), we have Therefore, using Claim A.5 and the asymptotics of Q from (A.14), we get where S 1 and S 2 satisfy the desired conditions.

Determination of
Then, Proof. We follow the strategy of the proof of Lemma 2.3. The only difference is that we now look for solutions B 1 , B 2 both with limit 0 at +∞.
Proof of (i). We find the value of β from the equation of B 1 multiplied by Q, using (A.9) and (A.18), Next, from (A.12), (A.9), (A.8), we have and we find θ B by integrating the equation of We now obtain the existence ofB 1 ∈ Y as in the proof of Lemma 2.3, with b 1 uniquely chosen so that B 1 (1 − λ∂ 2 x )Q = 0 and B 1 (1 − λ∂ 2 x )Q ′ =0. Proof of (ii). We solve the equation of B 2 exactly in the same way. We check that the values of β and θ B are suitable to solve the problem, and we obtain uniqueB 2 ∈ Y and b 2 so Then, multiplying the equation of B by ΛQ, integrating and using and so, by QΛQ = 1 4 (λ + 3) Q, and thus, in view of the expression of θ B , we obtain Proof of (iii). We finish the proof of Lemma 2.5 as the one of Lemma 2.3. In particular, using the limits of B 1 and B 2 at ±∞ This, combined with the equations of B 1 and B 2 and Lemmas 2.1, 2.2, 2.3 and A.5 proves (2.30). Note that w B is not in L 2 since it has a nonzero limit at −∞. However, it has exponential decay as x → +∞. This allows us to prove that all rest terms are indeed of the form O 2 (see notation O 2 in (2.19)).
The control of the various scalar products is easily obtained as in Lemma 2.3 from the properties of B 1 , B 2 .
Finally, we claim without proof the following result. Then, We do not need to compute d 1 − d 2 , this is the reason why the exact expression of S 1 and S 1 are not needed.

End of the proof of Proposition 2.1
Set From the preliminary expansion (2.22), we have Thus, Proposition 2.1 is proved.
Γ(t) satisfies the following estimates The next proposition presents almost monotonicity laws which are essential in proving long time stability results in the interaction region. They will allow us to compare the approximate solution V (t, x) with exact solutions. The functional is different depending on whether µ 1 (t) > µ 2 (t) or µ 1 (t) < µ 2 (t).
The constant 0 < ρ < 1/32 to be fixed later, set , (3.7) Proposition 3.1 (Almost monotonicty laws). For ρ > 0 small enough, and under the assumptions of Lemma 3.1, let There exists C > 0 such that Remark 1. The introduction of almost monotone variants of the energy and mass is related to Weinstein's approach for stability of one soliton [45] and to Kato identity for the (gKdV) equation (see [19]). These techniques have been developed in [27], [33] and then extended in [28], [9], [36] and [11].

Stability of the two soliton structure for large time
In this section, we present a stability result for the two soliton structure for large time, i.e. far away from the interaction time. The argument, similarly to the one of Propositions 3.1, is based on almost monotone variant of energy and mass. As a corollary, we obtain a sharp estimate for large negative time on the pure two solution solution considered in Theorem 1.
Proposition 3.2 (Stability for large time). For 0 < ρ < 1/32 small enough, there exist C > 0 and such that for µ 0 > 0 and ω > 0 small enough, if u(t) is an H 1 solution of (BBM) satisfying See the proof of this result in Appendix B.
Remark 2. Using the invariance of the BBM equation by the transformation it follows that a statement similar to Proposition 3.2 holds for t 0 > (ρµ 0 ) −1 | log µ 0 |.

Stability of the 2-soliton structure
In this section, using the approximate solution constructed in Propositions 2.1 and 2.2 and the asymptotic arguments of Section 3, we prove the stability part of Theorem 1 and Theorem 2.

Conclusion of the proof of the stability of the 2-soliton structure
In this section, we finish the proof of the stability part of Theorem 1.
where for t close to 0, the term e −y (A 1 (x − y 1 ) + A 2 (x − y 2 )) is indeed relevant as a correction term in the computation of U (t). In view of the behavior at ±∞ of the functions A 1 and A 2 (see Lemma 2.3), this term decays exponentially for x > y 1 (t) and x < y 2 (t) but contains a tail for y 2 (t) < x < y 1 (t). Note that this tail also appears in the integrable case i.e. for λ = 0, and thus it is not related to the lack of integrability.
For future reference, we observe that the following hold for t ∈ [−T, T ]

Nonexistence of a pure 2-soliton and interaction defect
In this section, we complete the proof of Theorem 1 by proving the lower bounds in (1.11) and (1.12).

Refined control of the translation parameters
Now, we introduce specific functionals J j (t) related to the translation parameters y j (t) to obtain a refined version of the dynamical system.
Then J j (t) is well-defined and the following hold (i) Estimates on J j .
Remark 4. The constant ((1 − λ∂ 2 x )ΛQ)Q is not zero (see (A.9)). Note also that Λ = 0 (see (A.8)), and so the functions J j (x) are bounded but have no decay at +∞ in space. Therefore, J j (t) is not well-defined for a general ε ∈ H 1 . Part of the proof of Lemma 5.1 consists on obtaining decay in space for ε(t) in order to give a rigorous sense to J j .
Remark 5. Estimate (5.3) says formally that µ j −ẏ j − N j is of order O 7/4 , which is an decisive improvement with respect to (4.16) (gain of a factor e − 1 2 Y 0 ).
Proof. Preliminary estimates. We work under the assumptions of Proposition 4.1, and on the interval [−T, T ]. First, we claim exponential decay properties of U (t) on the right (x > y 1 (t)).
Estimate of J j . Note that J j does not belong to L 2 (see Remark 4) but satisfies sup x∈R 1 + e − 1 2 (x−y j (t)) |J j (t, x)| ≤ C.
Moreover, using y 1 (t) − y 2 (t) = y(t) ≤ Y (T ) ≤ CY 0 , one gets by similar arguments To prove (5.3), we make use of the equation of ε (see (3.4)), and of the special algebraic structure of the approximate solution V (t, x) introduced in Propositions 2.1 and 2.2. We have First observe that

Preliminary symmetry arguments
First, we claim the following additional information obtained on the parameters of the solution U (t), under the assumptions of Proposition 4.1.

Claim 5.2. For all t ∈ [−T, T ],
Proof of (5.9). From (4.6) andẎ (0) = 0, we have |µ Thus, by (4.16) and the expression of M j in (2.9), we obtain The next lemma claims that if the asymptotic 2-soliton solution U (t) considered in Proposition 4.1 has an approximate symmetry property (i.e. U (t, x) − U (−t + t 0 , −x + x 0 ) is small for some t 0 , x 0 ) then the corresponding decomposition parameters (i.e. Γ(t) in Proposition 4.1) also have some symmetry properties, despite the fact that the decomposition itself is not symmetric (see the definition of V (t, x) in Propositions 2.1-2.2).
On the other hand, from (5.9) and (5.12), we have |µ Using Section 2, the proof is similar to the one of Lemma 5.2 in [31] and it is omitted.

Lower bound on the defect
In this section, we prove the following result.
Proof. The proof is the same as the one of Proposition 5.2 in [31] but we repeat it here since the argument is the key of the nonexistence of a pure 2-soliton solution. It suffices to prove the estimate on w(t). The estimates on the final parameters then follow from Lemma 4.1.
Let ǫ > 0 arbitrary, and suppose for the sake of contradiction that Step 1. We claim that for someT (t),X(t), for all t ∈ R, Proof of (5.15). By Lemma 4.1, it follows that In particular, for all t From (5.14) and the behavior of U , it follows that there exist T 1 , T 2 > T and X such that for Y 0 large enough. From Proposition 4.2, it follows that there existT (t) andX(t) such that (5.15) holds.

A Appendix to the construction of an approximate solution
A.1 Linearized operator, identities and asymptotics for solitons Recall that we set We recall the following well-known spectral properties of L (see [45] and Lemma 2.

from [29])
Claim A.1 (Properties of the operator L). The operator L defined in L 2 (R) by is self-adjoint and satisfies the following properties: (ii) Second eigenfunction : LQ ′ = 0; the kernel of L is {c 1 Q ′ , c 1 ∈ R}; (iii) For any function h ∈ L 2 (R) orthogonal to Q ′ for the L 2 scalar product, there exists a unique function f ∈ H 2 (R) orthogonal to (1 − λ∂ 2 x )Q ′ such that Lf = h; moreover, if h is even (respectively, odd), then f is even (respectively, odd).
(iv) Suppose that f ∈ H 2 (R) is such that Lf ∈ Y. Then, f ∈ Y.
(v) There exists c 1 > 0 such that for all f ∈ H 1 (R), Claim A.2 (Preliminary computations on solitons). (i) Scaling. Then, Then, Proof. (i) First, we check that Q µ (. + x 0 ) solves the following equation Indeed, we have We have by direct computations The expressions of ΛQ and Λ 2 Q then follow.
(ii) Differentiating (A.15) with respect to µ and then with respect to x 0 , we obtain (iii) These identities are readily obtained from (A.2). Note for example: The identities on Q 2 µ , (Q ′ µ ) 2 and Q 3 µ follows directly from (A.1). Now, we prove (A.11). We first observe that is a consequence of (A.15), multiplied by ΛQ µ and integrated over R. Then, we check d dµ M (Q µ ) > 0. For µ small, the result is true by (A.9) and a perturbation argument. In fact, it is true for all µ > −1 (see Weinstein [45]). Indeed, by the expressions of (∂ x Q µ ) 2 and Q 2 µ , we have Differentiating with respect to µ, we find (iv)-(v) These identities and asymptotic properties are easily obtained from the explicit expression of Q: In particular, we observe that We obtain in particular Moreover, Proof. We distinguish the two regions x − y 2 > y 2 and x − y 2 < y 2 .

A.3 Proof of Lemma 2.1
First, we claim the following estimates.
Claim A.4. The following holds (ω ≥ 0) Proof. We have Note that, from the proof of (A.3) and elementary computations: Proof of (A.25). For y 2 < x < y 1 , we have Arguing similarly for the case x < y 2 < y 1 , we prove (A.24) and (A.25).
The existence, uniqueness and continuity of Γ(t) is a consequence of Claim B.1. The C 1 regularity of Γ(t) is obtained by standard regularization arguments and the equation of ε(t) which is deduced easily from (BBM) and (2.15).
The proof of Proposition 3.1 is inspired by the proof of Proposition 3.1 in [31]. However, it is technically more involved in the BBM case. We refer to [37], [9], [11] and [32] for previous similar arguments for the (BBM) equation.
The proof of (3.11) is standard, see for example Lemma 4 in [33] and [11]. Recall that it is based on coercivity property of the operator L under orthogonality conditions, see Claim A.1 (v).
Using (3.4) and then by direct computations and estimates, we claim the following estimates, which imply immediately (3.13).
First, we claim the following control of the scaling directions of ε(t).
Now, we use a functional F similar to F − . Let where, ϕ being defined in (3.7), We perform similar (and simpler) computations as the ones of Propositions 3.1 and 3.1 (scaling parameters and Φ j are time independent here). We obtain, for some ρ > 0 small enough d dt F(t) ≤ C ε L 2 e −2ρy ε L 2 + e − 3 4 y . (B.30) From this point, the end of the proof is the same as the one of Proposition 3.2 in [31] and it is omitted.
Proof of (3.18). It is completely similar.

B.4 Proof of Proposition 4.2
For X 1 , X 2 ∈ R, let U X 1 ,X 2 be the unique solution of (BBM) such that Then, for any Y 1 , Y 2 ∈ R, one has In particular, the map (X 1 , X 2 ) → U X 1 ,X 2 is smooth and We assume T 1 ∈ (−T, T ), the case |T 1 | > T being similar, and we prove the stability result for t ∈ (−∞, T 1 ], the stability proof for t > T 1 following from similar arguments.
For C 1 > 2 to be chosen, we define By the assumption on u(T 1 ), C 1 > 2 and continuity of u(t) in H 1 , T * < T 1 is well-defined. We prove that T * = −∞ by using a contradiction argument : we assume T * > −∞ and we obtain a contradiction by strictly improving the estimate of inf X 1 ,X 2 u − U X 1 ,X 2 H 1 on t ∈ [T * , T 1 ]. By Proposition 4.1, U X 1 ,X 2 is close for all time to the sum of two distant solitons. Thus, on [T * , T 1 ], for ω small enough, we can use modulation theory (as in Lemma 3.1) to obtain (X 1 (t), X 2 (t)) ∈ R 2 , such that u(t, x) =Ũ (t, x) +ε(t, x),Ũ (t, x) = U X 1 (t),X 2 (t) (t, Note that there existsμ j (t) andỹ j (t) such that for all t: Moreover, as in Proposition 4.1, there exists t 0 such thatμ 1 (t) >μ 2 (t) if t > t 0 andμ 1 (t) < µ 2 (t) if t < t 0 . We assume that t 0 < T * .
To controlε(t) on [t 0 , T 1 ] (i.e. to prove that T * < t 0 ), we use the functional forΦ j defined fromμ j (t) as in (3.10). We follow the same computations as in the proof of Propositions 3.1 and (3.2), except that here there is no error term E(t, x), and no scaling parameter; thus we get