THE APPROXIMATE SOLUTION FOR BENJAMIN-BONA-MAHONY EQUATION UNDER SLOWLY VARYING MEDIUM

. In this paper, we investigate the soliton dynamics under slowly varying medium for the BBM equation, that is how the solution of this equation evolve when the time goes. We construct the approximate solution of this equation and prove that the error term due to the approximate solution can be controlled. By using the method of Lyapunov and Weinstein functions, we prove that the approximate solution is stable.

Many relevant works have been done. Zabusky and Kruskal in 1965 [29], Fermi, Pasta and Ulam in 1974 [5] had research on numerical work of the KdV equation. After this work in 1987, Le Veque [12] concluded interaction of nearly equal solitons in the KdV equation. Martel, Merle [16,17] discussed inelastic interaction of nearly equal solitons and description of two soliton collision for the quartic gKdV equation. At the same time, there exist another papers of the BBM equation. Benjamin, Bona and Mahony [2] made the study of the regularized long wave range equation. Bona in 1980 [3] got the solitary-wave interaction for this equation. In 1987, Bona, Strauss [4] studied stability and instability of solitary waves of Korteweg-de Vries type. Martel, Merle also made contributions to the BBM equation. Inelastic interaction of nearly equal solitons for the BBM equation was discovered by Martel, Merle [14,15].
Weinstein [26,27,18] considered the Lyapunov stability of ground states of nonlinear dispersive evolution equations and existence and dynamic stability of solitary wave solutions of equations arising in long wave propagations. Of course, Mizumachi [19] got the conclusion of asymptotic stability of solitary wave solutions to the regularized long wave equation.
We can easily see the similar character between nonlinear Schrödinger equation and gKdV equation: complete integrability. So we have the idea of applying this method into BBM equation. We can do the researches of the dynamics, stabilities, the long behaviors and the existence of pure soliton solution.
Note that, we take the limit of the (1.1), x )u t + (u xx − u + 2u 2 ) x = 0, where x → +∞. We take the T ε = ε − 1 100 and discuss the different behaviors of the region of the time.
Next, we recall the numerical character of soliton Q and operator L.
then the operator L satisfies the following two results: (i) the kernel of L is spawned by Q c ; (ii) for all h = h(y) ∈ L 2 (R) such that R hQ c = 0, there exists a uniqueĥ ∈ H 2 (R) such that Rĥ Q c = 0 and Lĥ = h. Moreover, if h is even (resp.odd), thenĥ is even(resp.odd).
Finally, we introduce the definition of (IP) property Y. We say that A c (εt, x) satisfies the (Y) property if and only if: Due to the definition of (IP), we get 3. Construction of a soliton-like solution. In this section, we construct the approximate solution on the interval [−T ε , T ε ] for the generalized BBM equation under slowly varying medium and discuss the estimate due to the error term of the constructed approximate solution. Via binomial theorem, integrable estimate and cut-off function, we can prove that the error term can be controlled by O(ε 3 2 e −γε|t| ) in H 2 (R). Moreover, the integration of the error term is also controlled.

Decomposition of the approximate solution. Let
we look for theũ(t, x), the approximate solution for (1.1) on the interval of time .
The form ofũ(t, x) will be the sum of the soliton plus error term: where A c (εt, x) satisfies (IP) property, we want to measure the size of error term produced by definingũ(t, x) in (3.4) about the equation (1.1). Let we can get the following results. where Proof. This proposition is proved explicitly in the following four Lemmas. where Proof. Recallũ = R+w and this lemma is just proved by the binomial theorem.
Proof. Note that (A 2 c ) ∈ Y , and A c satisfies (IP) property, so . Now we collect the estimate from Lemmas 3.2, 3.3 and 3.4. We finally get Due to Lemma 3.2, 3.3, 3.4, the Proposition 3.1 is proved.
Note that if we want to improve the approximationũ, the unknown function A c must be chosen such that (Ω) Then the error term will be reduced to the second order quantity .
We prove such a solvable result in the next part.

Resolution of Ω.
Lemma 3.5 (Existence theory for Ω). Suppose F ∈ Y even and satisfy the orthogonal condition Without loss of generality, we can suppose the constant term γ = 1+c 1+λc β.

The problem of Ω is solvable if and only if
Namely recall that LQ c = 0, thus there exists a solution A 1 (y) satisfying 1+c R F . This finishes the proof. According to the Lemma 3.5, it suffices to verify the orthogonal condition.
Lemma 3.6. There exists a solution A c of the problem (Ω) satisfying (IP) and such that Proof. We prove this Lemma in next three Lemmas.
Lemma 3.7 (The imposed condition). To get orthogonal condition R F Q c = 0, the parameters of c, a satisfy the following condition Proof. Note that We just compute these three terms So put these three parts together and from (2.5)-(2.10) to get orthogonal condition, we impose Lemma 3.8. Proof.

3.3.
Correction to the solution of problem of (Ω). Consider the cut-off function η ∈ C ∞ (R) satisfying the following properties, (3.10) (3.11) The following Proposition, which deals with the error associated to the cut-off function and the new approximate solutionũ, is the main result.
The error associated to the new functionũ satisfies 14) and the integral estimate holds Proof. The proof of first part about this proposition is similar to the proof of Proposition 4.6 in [17], so we omit this part. We will prove the second part of this proposition in the next Lemma. where Proof.
Note that From the (IP) property to estimate as follows Therefore So, we get .
(3.15) is just from integration of the formula (3.14).
This finishes the second proof of Proposition 3.2.

4.
First stability results. In this section,our aim is to prove that the approximate solutionũ describes the actual dynamics of interaction on the interval [−T ε , T ε ]. We analyze the energy conservation with the method of Lyapunov energy function.
Proposition 4.1. There exist constants K, k, ε > 0, assume u(t) is a H 1 (R) solution of (1.1) in a vicinity of t = T ε satisfying The aim is to prove T * = T ε that for K large enough. To achieve this, we argue by assuming T * < T ε and reaching on a contradiction with the definition of T * by proving some independent estimates for Proof. (4.1) is a consequence of the Implicit Function Theorem. Note that where z t = (1 + ρ 1 )u t −ũ t is the solution of the equation (1.1) andũ is the approximate solution. Substituting z t into the former formula, we can get (4.2). We take time derivative of orthogonal formula in (4.1) Integrating by parts this finishes Lemma 4.1.

Lemma 4.2 (Almost conversation of energy).
Consider the constructed approximate solution, In particular, there exists K > 0 such that From Cauchy-Schwarz inequality, we have We can get (4.5) from (3.15).
Proof. For the equation of (1.1), the conserved energy

WENXIA CHEN, PING YANG, WEIWEI GAO AND LIXIN TIAN
proof is finished. We consider the Lyapunov energy function, let This completes the proof of Lemma 4.3. Proof.
We use the Taylor expansion of a ε for the term of Note that |ε a a R yQ c z 2 | ≤ Kεe −r |t| z 2 L 2 (R) , So we can get (4.7).
we have Replace z t with (4.2) (4.10) Note the term of (4.9), we find that So, we can simplify (4.9) into and integrate by parts,

Moreover
where Rũ Integrating by parts for the term of (4.11), We can get the conclusion form (3.15) and (4.2), Finally, we simplify the formula of (4.12), From the estimates of (4.9)-(4.12) and the integration of F (t), we can gain This finishes the proof.
From the lemmas 4.3, 4.4 and due to F (−T ε ) < Kε 2k , According to Using the Gronwall inequality, we conclude that for some large constant K > 0, Obviously, when K * large enough, we obtain that for all t ∈ [−T ε , T ε ], . This estimate contradicts the definition of T * and concludes the proof of Proposition 4.1.
In this section, our aim is to prove that the approximate solutionũ describe the actual dynamics. We analyze the energy conservation and the first stability results with the method of Lyapunov energy function.

5.
Stability. In this section, we give the stability results of the error term between the exact solution and the constructed approximate solution with the methods of Virial estimates and the new mass and energy conservation when t goes to +∞.
Here u(t) is a H 1 (R) solution of the equation (1.1). Then, for all t > t 1 , there exist a constant K > 0 and a function ρ 2 (t) defined on [t 1 , +∞), ∀t > t 1 , we have The proof of this proposition is similar to the proof of Proposition 4.1. Let us recall that for large time t > T ε , the soliton-like solution is far away from the region where a ε varies. In particular, the stability and asymptotic stability properties will follow from the fact that in this region (1.1) behaves like the gBBM equation, Obviously, the argument is so sketchy, now we give the rigorous statement. Let us assume that for some fixed K > 0, In order to simplify the calculations, note that the function v = 2u solves, At the same time, (5.2) is equal to the inequality . Now we make the research of the stability of (5.3) and rename v = u. Let The aim is to prove T * = +∞. Therefore, for the sake of contradiction, in what follows we shall suppose T * < +∞. The first step to reach a contradiction is now to decompose the solution on [t 1 , T * ) using modulation theory around the soliton. In particular, we find a special ρ 2 (t) satisfying its definition but with a contradiction with the definition of T * .
Lemma 5.1. There exist C 1 -functions ρ 2 (t), c 2 (t) defined on [t 1 , T * ), with c 2 (t) > 0, the function z(t) given by, Proof. The proof of (5.5) is just like the formula (4.1) in the Lemma 4.1 with implicit function theorem. According to Where R t = λQ c2 c 2 −c 2 Q c2 and R is the solution of (1−λ∂ 2 x )R t +(R xx −R+R 2 ) x = 0. So, we take time derivative of (5.5) Replace z t with (5.6) (5.6) can be simplified into the following form According to the Lemma 3.7, scaling So, we take time derivative of We can get we replace z t , R t and integrate by parts, We can obtain Combining with the estimate of This finishes the proof of Lemma 5.1.

Lemma 5.2.
For all t ∈ [t 1 , T * ], we gain the boundary estimate, Obviously, we get Note that Proof of (5.8) is over.
≤ 0, thus the mass is not conserved. In order to avoid this problem, we shall introduce a virial-type identity.
Lemma 5.3. Virial estimate Let φ(x) ∈ C(R) be an even function satisfying the following properties: Note that, set ψ = x 0 φ, it is an odd function. Moreover, for all |x| ≥ 2, Finally, for all A > 0, denote We omit the proof of Lemma 5.3, because it is the same as the lemma in the paper [20].
Proof. According to the (5.6)

WENXIA CHEN, PING YANG, WEIWEI GAO AND LIXIN TIAN
(5.9) can be proved by the plus of these estimates. Corollary 1. For the formula (5.8) in the Lemma 5.2, we have another conclusion, Proof. Taking the A 0 large enough in formula (5.7) and Lemma 5.4, we take the integration of (5.9), . Via Taylor expansion of the former formula at c 2 (t 1 ), is a limited quality. So, (5.10) can be obtained from the formula (5.8).
Lemma 5.5. For all t ∈ [t 1 , T * ], we have So, this finishes the proof.
We can also get φ R0 (t, x) > e R0 , on the region of |x − ρ 2 (t)| > R 0 . Moreover, for some constants K, γ > 0 satisfying |(1 − φ R0 )Q c2 | < Ke −γR0 , Taking R 0 large enough, there exists λ 0 > 0 from the Anderson localization argument, we have Finally, taking ε smaller if necessary, we have (5.12). Now we prove that our assumption T * < +∞ leads to a contradiction. Indeed, from Lemma 5.5, 5.6, for all t ∈ [t 1 , T * ] and for K > 0, We know that the mass function is non-increasing Taking ε small and D 0 = D 0 (K) large enough, we have z H 1 (R) ≤ 1 4 D 2 0 ε, which contradicts the definition of T * . So, we obtain T * = +∞ and the conclusion is that, Proposition 5.1 is proved.
6. Main theorems. In this section, we summarize the dynamics, stability results and give the estimate of the soliton solution on the boundary of time.