\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 147, pp. 1--30.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/147\hfil Inelastic collision of two solitons] {Inelastic collision of two solitons for generalized BBM equation with cubic nonlinearity} \author[J. Wei, L. Tian, Z. Zhen, W. Gao \hfil EJDE-2015/147\hfilneg] {Jingdong Wei, Lixin Tian, Zaili Zhen, Weiwei Gao} \address{Jingdong Wei \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China} \email{wjd19871022@126.com} \address{Lixin Tian (corresponding author)\newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China. \newline School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China} \email{tianlx@ujs.edu.cn, tianlixin@njnu.edu.cn} \address{Zaili Zhen \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China} \email{lddcb@126.com} \address{Weiwei Gao \newline Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China} \email{yeahgaoweiwei@163.com} \thanks{Submitted September 26, 2014. Published June 6, 2015.} \subjclass[2010]{35Q53, 35C10, 35B35, 35B40} \keywords{Generalized Benjamin-Bona-Mahony equation; cubic nonlinearity; \hfill\break\indent solitary waves; pure 2-soliton; collision} \begin{abstract} We study the inelastic collision of two solitary waves of different velocities for the generalized Benjamin-Bona-Mahony (BBM) equation with cubic nonlinearity. It shows that one solitary wave is smaller than the other one in the $H^1(\mathbb{R})$ energy space. We explore the sharp estimates of the nonzero residue due to the collision, and prove the inelastic collision of two solitary waves and nonexistence of a pure 2-soliton solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we study the generalized Benjamin-Bona-Mahony (BBM) equation with cubic nonlinearity \begin{equation} (1 - \partial _x^2 )\partial _t u + \partial _x (u + u^3) = 0,\quad (t,x) \in \mathbb{R}\times \mathbb{R}, \label{e1.1} \end{equation} where $u(t,x)$ is a function of time $t$ and a single spatial variable $x$. If the nonlinearity term $u^3$ changes to $u^2$, then the above equation becomes the BBM equation. Equation \eqref{e1.1} was introduced by Peregrine \cite{30} and Benjamin, Bona and Mahony \cite{2}. In particular, it is not completely integrable. No inverse-scattering theory can be developed for this equation \cite{23, 29}. This situation is in contrast with the generalized Korteweg-de Vries equation (gKdV) equations \begin{equation} \partial _t u + \partial _x (\partial _x^2 u + f(u)) = 0,\label{e1.2} \end{equation} which is completely integrable for $ f(u)=u^2 $ (KdV equation), $ f(u)=u^3 $ (mKdV equation) and $ f(u)=u^2-\mu u^3 $ (Gardner equation). Let us review some classical works related to collision problems of solitons for the generalized KdV and BBM equations. The two equations have been studied since the 1960s from both experimental and numerical points of view; see examples \cite{1,3,4,5,8, 10,31,33,34,35}. Many elegant results have been found on the existence of explicit solution and stability, local and global well-posedness, long time dynamical behavior, etc. It is well-known that the KdV equation has explicit pure $N$-soliton solutions by using the inverse scattering transform \cite{9,24,32}. The stability and asymptotic stability of $N$-solitons of KdV equation were studied by Maddocks and Sachs \cite{12} in $H^{N}(R)$ using variational techniques and in $H^1(\mathbb{R})$ by Martel, Merle and Tsai \cite{21}. LeVeque \cite{11} further investigated the behavior of the explicit 2-soliton solution of KdV equation for nearly equal size. Mizumachi \cite{26} considered the large time behavior of two decoupled solitary wave of the generalized KdV equation. Martel and Merle \cite{16} investigated the inelastic collision of two solitons with nearly equal size for gKdV equation \eqref{e1.2} with $f(u)=u^4$. It shows that the 2-soliton structure is globally stable in $H^1(\mathbb{R})$ and the nonexistence of a pure 2-soliton in the regime. They also considered the so-called BBM equation \cite{15}: \[ (1 - \lambda \partial _x^2 )\partial _t u + \partial _x (\partial _x^2 u - u + u^2) = 0,\quad (t,x) \in \mathbb{R}\times \mathbb{R},\;\lambda \in (0,1). \] For the pure 2-soliton, we mean that the solution $u(t,x)$ of KdV equation satisfies \begin{gather*} \| {u(t,x) - \sum_{j = 1}^2 {Q_{c_j } (x - c_j t - x_j )} } \|_{H^1(\mathbb{R})} \to 0\quad \text{as } t \to - \infty , \\ \| {u(t,x) - \sum_{j = 1}^2 {Q_{c_j } (x - c_j t - {x}'_j )} } \|_{H^1(\mathbb{R})} \to 0 \quad \text{as } t \to + \infty, \end{gather*} for some $x_j$ such that the shifts $\Delta_j=x'_{j}-x_j$ depend on $c_1,c_2$. This solution which is called the 2-soliton represents the pure collision of two solitons, with no residue terms before and after the collision. In other words, the collision is elastic. Except for the collision of two solitons with nearly equal size of two equations, the collision of two solitons with different velocities has also been studied in \cite{14, 18,20,28}. In \cite{14}, it was shown that the collision of two stable solitary waves is inelastic but almost elastic. As a consequence, the monotonicity properties are strict: the size of the large soliton increases and the size of the small soliton decreases through the collision with explicit lower and upper bounds. In \cite{18}, they considered the generalized KdV equation with a general nonlinearity $f(u)$: \[ u_t + (u_{xx} + f(u))_x = 0,\quad (t,x) \in \mathbb{R}^ + \times \mathbb{R}, \quad u(0) = u_0 \in H^1(\mathbb{R}), \] assuming that for $p=2,3,4$, $f(u) = u^p + f_1 (u)$ where $\lim_{u \to 0} | {\frac{f_1 (u)}{u^p}} | = 0$, and $f_1$ belongs $C^{p+4}$. In \cite{28}, it is classified on the nonlinearities for which collision are elastic and inelastic. For integrable case, the collision of two solitons is elastic, such as KdV, mKdV and Gardner nonlinearities. For non-integrable case, the collision of two solitons is inelastic, such as gKdV, a general case $f$ and BBM etc. There are many unsolved questions related to the collision problems for partial differential equations. In this study we focus on the existence of a pure 2-soliton for the gBBM equations in the collision regime. To prove the nonexistence of a pure 2-soliton for non-integrable partial differential equations, we should consider four conditions \cite{4,18,20}. The first one is that the related Cauchy problem should be globally well-posed. The second one is that the solutions of the equations should satisfy the mass and energy conservation laws. The third one is that the equations should have solitary wave solutions with certain properties. The last one is that the equations should have asymptotic stability of multi-solitons. In this article, we consider the collision of two solitary waves with different velocities for the gBBM equation with cubic nonlinearity. Our purposes is to study the dynamical behavior of two solitons during the collision and to prove the nonexistence of a pure 2-soliton after the collision. As we know, the Cauchy problem related to \eqref{e1.1} is globally well-posed in $H^1(\mathbb{R})$ \cite{2} and the solutions of \eqref{e1.1} satisfy the following conservation laws: \begin{gather} E(u(t)) = \frac{1}{2}\int_\mathbb{R} {u^2(t,x)dx} + \frac{1}{4}\int_\mathbb{R} {u^4(t,x)dx} = E(u_0 ),\label{e1.3}\\ m(u(t)) = \frac{1}{2}\int_\mathbb{R} {(u^2(t,x) + u_x^2 (t,x))dx} = m(u_0 ). \label{e1.4} \end{gather} The quantity $\int_\mathbb{R} {u(t,x)dx}$ is also formally conserved. However, \eqref{e1.1} admits no more conserved quantities. Recall that \eqref{e1.1} has a two-parameter family of solitary wave solutions $\{ \phi _c (x - ct - x_0 ) :c > 1,\;x_0 \in \mathbb{R}\} $, where $\phi_c$ satisfies \begin{equation} c{\phi }''_c - (c - 1)\phi _c + \phi _c^3 = 0,\quad\text{in }\mathbb{R}. \label{e1.5} \end{equation} The unique even solution of \eqref{e1.5} is given by $$ \phi_c (x) = (c - 1)^{1/2}Q\Big(\sqrt {\frac{c - 1}{c}} x\Big), $$ where $Q(x) = \sqrt 2 \cosh ^{ - 1}(x)$ solves \begin{equation} Q'' + Q^3 = Q. \label{e1.6} \end{equation} Let $N\le1$, $1c_2>1$ and $u$ be the unique solution of \eqref{e1.1} such that \begin{equation} \lim_{t \to - \infty } \| {u(t) - \sum_{j = 1,2} {\phi _{c_j } (x - c_j t)} } \|_{H^1(\mathbb{R})} = 0 .\label{e1.9} \end{equation} There exists $ \varepsilon_0 = \varepsilon_0 (c_1 ) > 0 $ such that if $0 c_2^{+} > 1,\rho_1(t),\rho_2(t)$ and $T_0 ,K > 0$ such that $$ w^{+}(t,x) = u(t,x) - \sum_{j = 1,2} {\phi _{c_j^ {+}}} (x - \rho_j(t)) $$ satisfies \begin{gather} \lim_{t \to + \infty } \| {w^ + (t)} \|_{H^{1}(x > \frac{1}{2}(1 + c_2 )t)} = 0, \label{e1.10}, \\ \frac{1}{K}(c_2 - 1)^{{\frac{9}{2}}} \le c_1^ + - c_1 \le K(c_2 - 1)^{{\frac{7}{2}}},\quad \frac{1}{K}(c_2 - 1)^5 \le c_2 - c_2^ + \le K(c_2 - 1)^4,\label{e1.11}, \\ \frac{1}{K}(c_2 - 1)^{9/4} \le \| {\partial _x w^ + (t)} \|_{L^2(\mathbb{R})} + \sqrt {c_2 - 1} \| {w^{+} (t)} \|_{L^2(\mathbb{R})} \le K(c_2 - 1)^{7/4}, \label{e1.12} \end{gather} for $t \ge T_0$. \end{theorem} \begin{remark} \label{rmk1} \rm Because the mass and energy conservation, the Sobolev space $H^1(\mathbb{R})$ appears to be an ideal space to study long time dynamical properties of \eqref{e1.1}. \end{remark} \begin{remark} \label{rmk2} \rm The results of the Theorem \ref{thm1} mean nonexistence of a pure 2-soliton in the regime. By \eqref{e1.9} and \eqref{e1.10}, we see that an asymptotic 2-soliton at $ - \infty$ cannot be an asymptotic 2-soliton at $+ \infty$. We also see from \eqref{e1.11} that the size of the large soliton increases and the size of the small soliton decreases through the collision, with explicit lower and upper bounds. The bound in \eqref{e1.12} is thus a qualitative version of nonexistence of a pure 2-soliton. \end{remark} This article is organized as follows. In Section 2, we construct an approximate solution to the problem in the collision regime. Section 3 is devoted to preliminary stability results. Section 4 is concerned with the proof of Theorem \ref{thm1}. \section{Construction of an approximate 2-soliton solution} The objective of this section is to construct an approximate solution for the gBBM equation with cubic nonlinearity, which describe the inelastic collision of two solitons $\phi _{c_1 },\phi _{c_2 }$ in the case where $0< c_2 - 1 < \varepsilon _0 $ is small enough. And the approximate solution $z(t,x)$ does only exist in the collision region. Also, the structure of $z(t,x)$ and $S(t)$ are more complicated than that of the BBM equation. \subsection{Reduction of the problem} Let \begin{equation} c_{1} > 1,\quad \lambda = \frac{c_{1} - 1}{c_{1} } \in (0,1). \label{e2.1} \end{equation} We make the change of variables: \begin{equation} \hat {x} = \lambda ^{1/2}(x - \frac{t}{1 - \lambda }),\quad \hat {t} = \frac{\lambda ^{3/2}} {1 - \lambda } t, \quad z(\hat {t},\hat {x}) = \sqrt {\frac{1 - \lambda }{\lambda }}\, u(t,x).\label{e2.2} \end{equation} If $u(t,x)$ is a solution to \eqref{e1.1}, then $z(\hat {t},\hat {x})$ satisfies \begin{equation} (1-\lambda\partial_{\hat {x}}^2 )\partial_{\hat{t}}z +\partial_{\hat{x}}(\partial_{\hat{x}}^2 z - z + z^3) = 0.\label{e2.3} \end{equation} \begin{lemma} \label{lem2.1} (i) Let $c > 1$. By \eqref{e2.2}, a solitary wave solution $\phi _c (x - ct)$ to \eqref{e1.1} is transformed into $\tilde {Q}_\sigma (y_\sigma )$ which is a solution of \eqref{e2.3} where \begin{gather} \tilde {Q}_\sigma (x) = \sqrt {\sigma \theta _\sigma } Q(\sqrt \sigma x),\quad Q(x) = \sqrt 2 \cosh ^{ - 1}(x),\label{e2.4} \\ \sigma = \frac{c - 1}{c\lambda },\quad \theta _\sigma = \frac{1 - \lambda}{1 - \lambda \sigma } = (1 - \lambda )\sum_{j = 0}^\infty {(\lambda \sigma )^j} ,\quad \frac{1}{\theta _\sigma } = \frac{1}{1 - \lambda } - \frac{\lambda }{1 - \lambda }\sigma , \nonumber\\ \mu _\sigma = \frac{1 - \sigma }{1 - \lambda \sigma } = 1 + (\lambda - 1)\sigma \sum_{j = 0}^\infty {(\lambda \sigma )^j} ,\quad y_\sigma = \hat {x} + \mu _\sigma \hat {t}. \nonumber \end{gather} Especially, if $c = c_1$, then $\mu _\sigma = 0$, $y_\sigma = \hat {x}$, $\tilde {Q}_\sigma (y_\sigma ) = Q(\hat {x})$ and \begin{equation} {Q}'' + Q^3 = Q,\quad ({Q}')^2 + \frac{1}{2}Q^4 = Q^2 \quad\text{in }\mathbb{R}. \label{e2.5} \end{equation} (ii) Moreover, $\tilde {Q}_\sigma $ satisfies the equations \begin{equation} \tilde {{Q}}''_\sigma + \frac{1}{\theta _\sigma }\tilde {Q}_\sigma ^3 = \sigma \tilde {Q}_\sigma ,\quad ({\tilde {Q}}'_\sigma )^2 + \frac{1}{2\theta _\sigma }\tilde {Q}_\sigma ^4 = \sigma \tilde {Q}_\sigma ^2.\label{e2.6} \end{equation} For $\sigma>0$ small, we have \begin{gather} \| {\tilde {Q}_\sigma } \|_{L^\infty (\mathbb{R})} \sim \sqrt {(1 - \lambda )\sigma } \| Q \|_{L^\infty (\mathbb{R})} ,\quad \| {\tilde {Q}_\sigma } \|_{L^2(\mathbb{R})} \sim \sqrt {(1 - \lambda )} \sigma^{1/4} \| Q \|_{L^2(\mathbb{R})}, \label{e2.7} \\ (\tilde {Q}_\sigma ^3 )'(y_\sigma + \delta ) = \frac{(1 - \lambda )^{3/2}}{\lambda^2}(\phi _c^3 )' (x - ct + \frac{\delta }{\sqrt \lambda }).\label{e2.8} \end{gather} \end{lemma} The proof of the above lemma is similar to the proof of \cite[Claim 2.1]{20}, so we omit it. \subsection{Decomposition of approximate solution} Firstly, we construct an approximate solution $z(t,x)$ of \begin{equation} (1 - \lambda \partial _x^2 )\partial _t z + \partial _x (\partial _x^2 z - z+ z^3) = 0,\label{e2.9} \end{equation} which is the sum of the function $Q(y)$, a small soliton $\tilde {Q}_\sigma(y_\sigma )$ and an error term $w(t,x)$. As in \cite{20}, we introduce the new coordinates and the approximate solution of the form \begin{gather} y_\sigma = x + \mu _\sigma t,\quad y = x - \alpha (y_\sigma), \nonumber \\ \alpha (s) = \int_0^s {\beta (r)} dr, \quad \beta (y_\sigma ) = \sum_{(k,l) \in\Sigma _0 } {a_{k,l} } \sigma ^l\tilde {Q}_\sigma ^k (y_\sigma ), \label{e2.10}\\ z(t,x) = Q(y) + \tilde {Q}_\sigma (y_\sigma ) + w(t,x), \label{e2.11} \end{gather} where \begin{gather} w(t,x) = \sum_{(k,l) \in \Sigma _0 } {\sigma ^l(A_{k,l} (y)\tilde {Q}_\sigma ^k (y_\sigma )} + B_{k,l} (y)(\tilde {Q}_\sigma ^k )'(y_\sigma )),\label{e2.12} \\ \Sigma_0 = \{(k,l) =(1,0),(1,1),(2,0),(2,1),(3,0),(4,0)\}, \label{e2.13} \end{gather} Define the operator $L$ by \begin{equation} Lf = - {f}'' + f - 3Q^2f. \label{e2.14} \end{equation} \begin{definition} \label{def1} \rm Let $\mathcal{M}$ be the set of $C^\infty $ functions $f$ such that \begin{equation} \forall j \in \mathbb{N},\;\exists K_j,\,r_j > 0,\; \forall x \in \mathbb{R},\;| {f^{(j)}(x)} | \le K_j (1 + | x |)^{r_j }e^{ - | x |}. \label{e2.15} \end{equation} Following formulas \eqref{e2.10}-\eqref{e2.13}, we set \begin{gather*} S(z) = (1 - \lambda \partial _x^2 )\partial _t z + \partial _x (\partial _x^2 z - z + z^3) = S_{\rm mKdV} (z) + S_{\rm gBBM} (z), \\ S_{\rm mKdV} (z) = \partial _t z + \partial _x (\partial _x^2 z - z +z^3),\quad S_{\rm gBBM} (z)\; = - \lambda \partial _t \partial _x^2 z. \end{gather*} Then, it gives \begin{equation} S(z(t,x)) = S(Q(y)) + S(\tilde {Q}_\sigma (y_\sigma )) + \delta S(w(t,x)) + S_{int} (t,x), \label{e2.16} \end{equation} where \begin{equation} \delta S(w) = \delta S_{\rm mKdV} (w) + S_{\rm gBBM} (w),~~~~\delta S_{\rm mKdV} (w) = \partial _t w - \partial _x Lw, \label{e2.17} \end{equation} \begin{align*} S_{int} (t,x) &= \partial _x (w^3(t,x) + 3Q^2(y)\tilde {Q}_\sigma (y_\sigma ) + 3Q(y)\tilde {Q}_\sigma ^2 (y_\sigma ) + 3\tilde {Q}_\sigma ^2 (y_\sigma )w(t,x)\\ &\quad + 6Q(y)\tilde {Q}_\sigma (y_\sigma )w(t,x) + 3Q(y)w^2(t,x) + 3\tilde {Q}_\sigma (y_\sigma )w^2(t,x)). \end{align*} Since $\tilde {Q}_\sigma (y_\sigma )_{ }$ is a solution to \eqref{e2.9}, we get $S(\tilde {Q}_\sigma (y_\sigma )) = 0$. \end{definition} \begin{proposition} \label{prop2.1} There holds \begin{align*} S(z) &= \sum_{(k,l) \in \Sigma _0 } {\sigma ^l\tilde {Q}_\sigma ^k (y_\sigma )\left( {a_{k,l} ((\lambda - 3){Q}'' - 3Q^3)' - (LA_{k,l} )' + F_{k,l} } \right)(y)} \\ &\quad + \sum_{(k,l) \in \Sigma _0 } \sigma ^l(\tilde {Q}_\sigma ^k )'(y_\sigma )\Big( (3 - \lambda ){A}''_{k,l} + 3Q^2A_{k,l} + a_{k,l} (2\lambda - 3){Q}'' \\ &\quad - (LB_{k,l} )' + G_{k,l} \Big)(y) + \varepsilon (t,x), \end{align*} where \begin{gather*} F_{1,0} = (3Q^2)',\quad G_{1,0} = 3Q^2,\\ F_{1,1} = (3 - 2\lambda){A}'_{1,0} + (3 - \lambda ){B}''_{1,0} + 3Q^2B_{1,0} + \lambda (\lambda -1)a_{1,0} {Q}''',\\ G_{1,1} = 2a_{1,0} \lambda (\lambda - 1){Q}'',\\ F_{2,0} = a_{1,0} \big\{ {(\lambda - 3){A}''_{1,0} - 3A_{1,0} Q^2 - 3Q^2} \big\}' + (3Q + 9A_{1,0} Q)' + (3 - 2\lambda )a_{2,0}^2 {Q}''', \\ \begin{aligned} G_{2,0} &= \frac{a_{1,0} }{2}\left\{ {(6\lambda - 9){A}'_{1,0} + (\lambda - 3){B}''_{1,0} - 3Q^2B_{1,0} } \right\}' \\ &\quad + (3B_{1,0} Q + 3A_{1,0} B_{1,0} Q)'+ 3Q + 9A_{1,0} Q + \frac{3}{2}(1 - \lambda )a_{1,0}^2 {Q}''. \end{aligned} \end{gather*} In addition, the following statements hold: (i) For all $(k,l) \in \Sigma _0 $ such that $3 \le k + l \le 4$, then $F_{k,l}$ and $G_{k,l}$ depend on $A_{{k}',{l}'}$ and $B_{{k}',{l}'}$ for $1 \le k + l \le 2$. Moreover, if $A_{{k}',{l}'} $ is even and $B_{{k}',{l}'} $ is odd, then $F_{k,l} $ is odd and $G_{k,l} $ is even. (ii) If the functions $A_{{k}',{l}'}$ and $B_{{k}',{l}'} $ are bounded, then $\varepsilon (t,x)$ satisfies \begin{equation} | {\varepsilon (t,x)} | \le K\sigma ^{5/2}O(\tilde {Q}_\sigma (y_\sigma )). \label{e2.18} \end{equation} \end{proposition} Before proving Proposition \ref{prop2.1}, we give some preliminary lemmas on $\phi _c$. \begin{lemma}[Identities of $\phi _c$] \label{lem2.2} For all $c > 1$, \begin{equation} \begin{gathered} \int_\mathbb{R} {\phi _c^2 } = c^{1/2}(c - 1)^{1/2} \int_\mathbb{R} {Q^2} ,\\ \int_\mathbb{R} {\phi _c^4 } = \frac{4}{3}(c - 1)\int_\mathbb{R} {\phi _c^2 } ,\quad \int_\mathbb{R} {({\phi }'_c )^2} \; = \frac{1}{3}\big( {\frac{c - 1}{c}} \big)\int_\mathbb{R} {\phi _c^2 } , \\ E(\phi _c ) = \frac{1}{2}\int_\mathbb{R} {\phi _c^2 } + \frac{1}{4}\int_\mathbb{R} {\phi _c^4 } = \big( {\frac{2c + 1}{6}}\big)c^{1/2}(c - 1)^{1/2}\int_\mathbb{R} {Q^2} , \\ m(\phi _c ) = \frac{1}{2}\Big({\frac{1}{3}\big( {\frac{c - 1}{c}} \big) + 1} \Big)\int_\mathbb{R} {\phi _c^2 } = c^{ - 1/2}(c - 1)^{1/2} \big( {\frac{4c - 1}{6}} \big)\int_\mathbb{R} {Q^2}, \\ E(\phi _c ) - cm(\phi _c ) = - \frac{1}{3}(c - 1)^{3/2}c^{1/2}\int_\mathbb{R} {Q^2} , \\ \frac{d}{dc}E(\phi _c ) = c\frac{d}{dc}m(\phi _c ) > 0. \end{gathered} \label{e2.19} \end{equation} \end{lemma} The proof of the above lemma can be completed by a straightforward calculations. \begin{lemma}[{Properties of $L$, \cite[Lemma 2.2]{14}}] \label{lem2.3} The operator $L$ defined in $L^2(\mathbb{R})$ by \eqref{e2.14} is self-adjoint and satisfies the following properties: \begin{itemize} \item[(i)] by the first eigenfunction, then $LQ^2 = - 3Q^2$; \item[(ii)] by the second eigenfunction, then $L{Q}' = 0$; the kernel of $L$ is $\{ c_1{Q}', c_1 \in \mathbb{R}\}$; \item[(iii)] if a function $h \in L^2(\mathbb{R})$ is orthogonal to ${Q}'$ by the $L^2$ scalar product, there exists a unique function $f \in H^2(\mathbb{R})$ orthogonal to ${Q}'$ such that $Lf = h$. Moreover, if $h$ is even (odd), then $f$ is even (odd). \end{itemize} \end{lemma} \begin{lemma}[{\cite[Claim B.1]{20}}] \label{lem2.4} Let $h(t,x) = g(y) = g(x - \alpha (y_\sigma ))$, where $g$ is a $C^{3}$ function. Then we have \begin{gather*} \partial _t h = - \mu _\sigma \beta (y_\sigma ){g}'(y), \quad \partial _x h = (1 - \beta (y_\sigma )){g}'(y), \\ \partial _x^2 h = (1 - \beta (y_\sigma ))^2{g}''(y) - \beta (y_\sigma ){g}'(y), \\ \partial _x \partial _t h = - \mu _\sigma (1 - \beta (y_\sigma ))\beta (y_\sigma ){g}''(y) - \mu _\sigma {\beta }'(y_\sigma ){g}'(y), \\ \partial _x^3 h = (1 - \beta (y_\sigma ))^3{g}'''(y) - 3(1 - \beta (y_\sigma )){\beta }'(y_\sigma ){g}''(y) - {\beta }''(y_\sigma ){g}'(y), \\ \begin{aligned} \partial _x^2 \partial _t h &= \mu _\sigma \Big\{ { - (1 - \beta (y_\sigma ))^2\beta (y_\sigma ){g}'''(y) + 3\beta (y_\sigma ){\beta }'(y_\sigma ){g}''(y)} \\ &\quad - 2{\beta }'(y_\sigma ){g}''(y) - {\beta}''(y_\sigma ){g}'(y) \Big\} . \end{aligned} \end{gather*} \end{lemma} \begin{lemma} \label{lem2.5} Let $A$ and $q$ be $C^{3}$-functions. Then we have \begin{align*} &\delta S_{\rm mKdV} (A(y)q(y_\sigma ))\\ &= q(y_\sigma )\Big\{ { - (LA)'(y) + \beta (y_\sigma )} ( - 3{A}'' - 3AQ^2 + (1 - \mu _\sigma )A)'(y) - {\beta }'(y_\sigma )(3{A}'')(y)\\ &\quad + \beta ^2(y_\sigma )(3{A}''')(y)+(\beta ^2)'(3{A}'' / 2)(y) { -{\beta }''(y_\sigma ){A}'(y) - \beta ^3(y_\sigma ){A}'''(y)} \Big\}\\ &\quad + {q}'(y_\sigma )\Big\{ {3{A}''} (y) + 3A(y)Q^2(y) + (\mu _\sigma -1)A(y) -\beta (y_\sigma )(6{A}'')(y)\\ &\quad - { {\beta }'(y_\sigma )(3{A}')(y)+\beta ^2(y_\sigma )(3{A}'')(y)}\Big\} +{q}''(y_\sigma )\left\{{3(1 - \beta (y_\sigma )){A}'(y)}\right\} \\ &\quad + {q}'''(y_\sigma )A(y). \end{align*} \end{lemma} \begin{proof} Using Lemma \ref{lem2.4}, we have \[ \partial _t (A(y)q(y_\sigma )) = - \mu _\sigma \beta (y_\sigma ){A}'q(y_\sigma ) + \mu _\sigma A{q}'(y_\sigma ), \] and \begin{align*} &- \partial _x L(A(y)q(y_\sigma )) \\ &= \partial _x \big\{ {(\partial _x^2 A - A + 3AQ^2)q(y_\sigma ) + 2(\partial _x A){q}'(y_\sigma ) + A{q}''(y_\sigma )} \big\}\\ &= \{ {\partial _x (\partial _x^2 A - A + 3AQ^2)} \}q(y_\sigma ) + (\partial _x^2 A - A + 3AQ^2){q}'(y_\sigma )\\ &\quad + 2(\partial _x^2 A){q}'(y_\sigma ) + 3(\partial _x A){q}''(y_\sigma ) + A{q}'''(y_\sigma )\\ &= q(y_\sigma )\Big\{ {(1 - \beta (y_\sigma ))}^3{A}''' - 3(1-\beta (y_\sigma )){\beta }'(y_\sigma ){A}'' - {\beta }''(y_\sigma ){A}' - (1 - \beta (y_\sigma )){A}'\\ &\quad +3(1-\beta (y_\sigma ))(AQ^2)' \Big\} + {q}'(y_\sigma )\big\{ {3(1 - \beta (y_\sigma ))^2{A}'' - 3{\beta}'(y_\sigma ){A}' - A + 3AQ^2} \big\}\\ &\quad +{q}''(y_\sigma )\left\{ {3(1 - \beta (y_\sigma )){A}'} \right\} +{q}'''(y_\sigma )A. \end{align*} Combining the above equalities, we complete the proof. \end{proof} \begin{lemma}[{\cite[Claim B.3]{20}}] \label{lem2.6} Let $A$ and $q$ be $C^{3}$-functions. Then we have \begin{align*} &S_{\rm gBBM} (A(y)q(y_\sigma ))\\ &= \lambda \mu _\sigma q(y_\sigma)\big\{ {\beta (y_\sigma ){A}'''(y) + {\beta }'(y_\sigma )(2{A}''(y))}\big\} + \lambda \mu _\sigma q(y_\sigma )\Big\{ \beta ^2(y_\sigma )( - 2{A}''')(y) \\ &\quad + (\beta ^2)'(y_\sigma )( - 3{A}'' / 2)(y) + {\beta }''(y_\sigma ){A}'(y) + { \beta^3(y_\sigma ){A}'''(y)} \Big\} \\ &\quad + \lambda \mu _\sigma {q}'(y_\sigma) \Big\{ { - {A}''(y)} + \beta (y_\sigma )(4{A}'')(y) { + {\beta}'(y_\sigma )(3{A}')(y) + \beta ^2(y_\sigma )( - 3{A}'')(y)} \Big\}\\ &\quad + \lambda \mu _\sigma{q}''(y_\sigma ) \big\{ { - 2{A}'(y) + \beta (y_\sigma )(3{A}')(y)} \big\} + \lambda \mu _\sigma {q}'''(y_\sigma )( - A)(y). \end{align*} \end{lemma} \begin{lemma} \label{lem2.7} Let \[ \beta = a_{1,0} \tilde {Q}_\sigma + a_{1,1} \sigma \tilde {Q}_\sigma + a_{2,0} \tilde {Q}_\sigma ^2 + a_{2,1} \sigma \tilde {Q}_\sigma ^2 + a_{3,0} \tilde {Q}_\sigma ^3 + a_{4,0} \tilde {Q}_\sigma ^4 . \] Then \begin{gather*} {\beta }' = a_{1,0} (\tilde {Q}_\sigma )' + a_{1,1} \sigma (\tilde {Q}_\sigma )' + a_{2,0} (\tilde {Q}_\sigma ^2 )' + a_{2,1} \sigma (\tilde {Q}_\sigma ^2 )' + a_{3,0} (\tilde {Q}_\sigma ^3 )' + a_{4,0} (\tilde {Q}_\sigma ^4 )', \\ \begin{aligned} {\beta }'' &= \sigma \tilde {Q}_\sigma a_{1,0} + \tilde {Q}_\sigma ^3 ( - \frac{a_{1,0} }{1 - \lambda }) + \sigma \tilde {Q}_\sigma ^2 (4a_{2,0} ) + \tilde {Q}_\sigma ^4 ( - \frac{3a_{2,0} }{1 - \lambda }) \\ &\quad + \sigma \tilde {Q}_\sigma ^3 (\frac{\lambda a_{1,0} }{1 - \lambda } - \frac{a_{1,1} }{1 - \lambda } + 9a_{3,0} ) + \sigma ^2\tilde {Q}_\sigma a_{1,1} + \tilde {Q}_\sigma ^5 ( - \frac{6a_{3,0} }{1 - \lambda }) + \sigma ^{5/2}O(\tilde {Q}_\sigma ), \end{aligned}\\ \begin{aligned} \beta ^2 &= a_{1,0}^2 \tilde {Q}_\sigma ^2 + 2a_{1,0} a_{2,0} \tilde{Q}_\sigma ^3 + 2a_{1,0} a_{1,1} \sigma \tilde {Q}_\sigma ^2 + (2a_{1,0}a_{3,0} + a_{2,0}^2 )\tilde {Q}_\sigma ^4 \\ &\quad + (2a_{2,0} a_{3,0} + 2a_{1,0} a_{4,0} )\tilde {Q}_\sigma ^5 +(2a_{1,0} a_{2,1} + 2a_{1,1} a_{2,0} )\sigma \tilde {Q}_\sigma^3 + \sigma ^{5/2}O(\tilde{Q}_\sigma ), \end{aligned} \\ \begin{aligned} (\beta ^2)' &= a_{1,0}^2 (\tilde {Q}_\sigma ^2 )' + 2a_{1,0} a_{2,0} (\tilde {Q}_\sigma ^3 )' + 2a_{1,0} a_{1,1} \sigma (\tilde {Q}_\sigma ^2 )' + (2a_{1,0} a_{3,0} + a_{2,0}^2 )(\tilde {Q}_\sigma ^4 )' \\ &\quad + \sigma ^{5/2}O(\tilde {Q}_\sigma ). \end{aligned} \end{gather*} \end{lemma} In the next four lemmas, we expand the various terms in \eqref{e2.16}. \begin{lemma} \label{lem2.8} There holds \begin{equation} \begin{aligned} S(Q) &= \sum_{(k,l) \in \Sigma _0 } {\sigma ^l} \Big( \tilde {Q}_\sigma ^k (y_\sigma )a_{k,l} \big\{ {(\lambda - 3){Q}'' - 3Q^3} \big\}' (y) \\ &\quad + (\tilde {Q}_\sigma ^k )'(y_\sigma )a_{k,l} (2\lambda - 3){Q}''(y) \Big)\\ &\quad +\sum_{(k,l) \in \Sigma _0 } {\sigma ^l} \Big( {\tilde {Q}_\sigma ^k (y_\sigma )F_{k,l}^I (y) + (\tilde {Q}_\sigma ^k )'(y_\sigma )G_{k,l}^I (y)} \Big) \\ &\quad + \sigma ^{5/2}O(\tilde {Q}_\sigma (y_\sigma )), \end{aligned} \label{e2.20} \end{equation} where \begin{gather*} F_{1,0}^I = 0,\quad G_{1,0}^I = 0,\quad F_{1,1}^I = \lambda (\lambda - 1)a_{1,0} {Q}''',\quad G_{1,1}^I = 2a_{1,0} \lambda (\lambda - 1){Q}'' \\ F_{2,0}^I = (3 - 2\lambda )a_{2,0}^2 {Q}''',\quad G_{2,0}^I = \frac{3}{2}(1 - \lambda )a_{1,0}^2 {Q}'', \end{gather*} and for all $(k,l) \in \Sigma _0 $ such that $3 \le k + l \le 4$, $F_{k,l}^I \in \mathcal{M}$ is odd, $G_{k,l}^I \in \mathcal{M}$ is even and both depend only on $a_{{k}',{l}'} $ for $1 \le {k}' + {l}' \le 2$. \end{lemma} \begin{proof} By Lemma \ref{lem2.5}, let $A(y) = Q(y)$ and $q = 1$, then \begin{align*} S_{\rm mKdV} (Q(y)) &= \delta S_{\rm mKdV} (Q) - \partial _x (2Q^3)\\&= ({Q}'' - Q + Q^3)' + \beta (y_\sigma)( - 3{Q}'' - 3Q^3 \\ &\quad + (1 - \mu _\sigma )Q)' - {\beta }'(y_\sigma )(3{Q}'') + \beta ^2(y_\sigma )(3{Q}''') + (\beta ^2)'(3{Q}'' / 2) \\ &\quad - {\beta }''(y_\sigma ){Q}' - \beta ^3(y_\sigma ){Q}'''. \end{align*} Using ${Q}'' = Q - Q^3$ and \eqref{e2.4}, we find \begin{align*} &S_{\rm mKdV} (Q(y)) \\ &= \beta (y_\sigma )( - 3{Q}'' - 3Q^3)' - {\beta }'(y_\sigma )(3{Q}'')+ \beta ^2(y_\sigma )(3{Q}''') + (\beta ^2)'(y_\sigma )(3{Q}'' / 2) \\ &\quad - \sigma \beta (y_\sigma )(\lambda - 1){Q}' - {\beta }''(y_\sigma ){Q}' - \beta^3(y_\sigma ){Q}''' - (\lambda - 1)\lambda \sigma ^2\beta (y_\sigma ){Q}' \\ &\quad +\sigma ^{5/2}O(\tilde{Q}_\sigma (y_\sigma )). \end{align*} By Lemma \ref{lem2.6} and \eqref{e2.4}, we have \begin{align*} &S_{gBBM} (Q(y)) \\ &= \lambda \mu _\sigma \Big\{ \beta (y_\sigma ){Q}''' + {\beta }'(y_\sigma )(2{Q}'') + \beta ^2(y_\sigma )( - 2{Q}''') + (\beta ^2)'(y_\sigma )( - 3{Q}'' / 2) \\ &\quad + {\beta }''(y_\sigma ){Q}' + \beta ^3(y_\sigma )({Q}''') \Big\}\\ &= \beta (y_\sigma )(\lambda {Q}''') + {\beta }'(y_\sigma )(2\lambda {Q}'') + \beta ^2(y_\sigma )( - 2\lambda {Q}''') + (\beta ^2)'(y_\sigma )( - 3\lambda {Q}'' / 2) \\ &\quad + \sigma \beta (y_\sigma ) \lambda (\lambda - 1){Q}''' + \sigma {\beta }'(y_\sigma )\left\{2\lambda (\lambda - 1){Q}'' \right\} \\ &\quad + {\beta }''(y_\sigma )(\lambda {Q}') + \beta ^3(y_\sigma )(\lambda {Q}''') + \sigma \beta ^2(y_\sigma )\lambda (\lambda - 1)( - 2{Q}''')\\ &\quad + \sigma (\beta ^2)'(y_\sigma )\lambda (\lambda - 1)( - 3{Q}'' / 2) + \sigma {\beta }''(y_\sigma )\lambda(\lambda - 1){Q}'\\ &\quad + \sigma ^2\beta (y_\sigma )\lambda ^2(\lambda - 1){Q}''' + \sigma ^2{\beta }'(y_\sigma )\lambda^2(\lambda - 1)(2{Q}'') + \sigma ^{5/2}{Q}_\sigma (y_\sigma )). \end{align*} Combining the above discussions, we deduce that \begin{align*} &S(Q) \\ &= \beta (y_\sigma )\left\{ {(\lambda - 3){Q}'' - 3Q^3} \right\}' + {\beta }'(y_\sigma )(2\lambda - 3){Q}'' + \beta ^2(y_\sigma )(3 - 2\lambda ){Q}''' \\ &\quad + (\beta ^2)'(y_\sigma )(1 - \lambda )(3{Q}'' / 2) + {\beta }''(y_\sigma )(\lambda - 1){Q}' + \sigma \beta (y_\sigma )(\lambda - 1)\left\{ {\lambda {Q}'' - Q} \right\}' \\ &\quad + \sigma {\beta }'(y_\sigma )\left\{ {2\lambda (\lambda - 1){Q}''} \right\} + \beta ^3(y_\sigma )(\lambda - 1){Q}''' + \sigma \beta ^2(y_\sigma )\lambda (\lambda - 1)( - 2{Q}''') \\ &\quad + \sigma (\beta ^2)'(y_\sigma )\lambda (\lambda - 1)( - 3{Q}'' / 2) + \sigma {\beta }''(y_\sigma )\lambda (\lambda - 1){Q}' \\ &\quad + \sigma ^2\beta (y_\sigma )\lambda (\lambda - 1)(\lambda {Q}'' - Q)' + \sigma ^2{\beta }'(y_\sigma )\lambda ^2(\lambda - 1)(2{Q}'') + \sigma ^{5/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} By Lemma \ref{lem2.7}, we derive \begin{align*} &S(Q) \\ &= \tilde {Q}_\sigma (y_\sigma )a_{1,0} \left\{ {(\lambda - 3){Q}'' - 3Q^3} \right\}' + {\tilde {Q}}'_\sigma (y_\sigma )a_{1,0} (2\lambda - 3){Q}'' \\ &\quad + \tilde {Q}_\sigma ^2 (y_\sigma )\left( {a_{2,0} \left\{ {(\lambda - 3){Q}'' - 3Q^3} \right\}' + (3 - 2\lambda )a_{2,0}^2 {Q}'''} \right) \\ &\quad + (\tilde {Q}_\sigma ^2 )'(y_\sigma )\Big( {a_{2,0} (2\lambda - 3){Q}'' + \frac{3}{2}(1 - \lambda )a_{1,0}^2 {Q}''} \Big) \\ &\quad + \sigma \tilde {Q}_\sigma (y_\sigma )\left( {a_{1,1} \left\{ {(\lambda - 3){Q}'' - 3Q^3} \right\}' + \lambda (\lambda - 1)a_{1,0} {Q}'''} \right) \\ &\quad + \sigma {\tilde {Q}}'_\sigma (y_\sigma )\left( {a_{1,1} (2\lambda - 3){Q}'' + 2\lambda (\lambda - 1)a_{1,0} {Q}''} \right) \\ &\quad + \sum_{3 \le k + l \le 4} {\sigma ^l} \Big( {\tilde {Q}_\sigma ^k (y_\sigma )a_{k,l} \left\{ {(\lambda - 3){Q}'' - 3Q^3} \right\}' (y) + (\tilde {Q}_\sigma ^k )'(y_\sigma )a_{k,l} (2\lambda - 3){Q}''(y)} \Big) \\ &\quad + \sum_{3 \le k + l \le 4} {\sigma ^l} \left( {\tilde {Q}_\sigma ^k (y_\sigma )F_{k,l}^I + (\tilde {Q}_\sigma ^k )'(y_\sigma )G_{k,l}^I } \right) + \sigma ^{5/2}O(\tilde {Q}_\sigma (y_\sigma )), \end{align*} for $3 \le k + l \le 4$, $F_{k,l}^I \in \mathcal{M}$ and $G_{k,l}^I \in \mathcal{M}$ are as in the statement of Lemma \ref{lem2.8}. \end{proof} \begin{lemma} \label{lem2.9} There holds \begin{align*} &\delta S_{\rm mKdV} (w)\\ &= \sum_{(k,l) \in \Sigma _0 } {\sigma ^l\left( {\tilde {Q}_\sigma ^k (y_\sigma )( - LA_{k,l} )'(y) + (\tilde {Q}_\sigma ^k )'(y_\sigma )(( - LB_{k,l} )' + 3{A}''_{k,l} + 3Q^2A_{k,l} )(y)} \right)} \\ &\quad +\sum_{(k,l) \in \Sigma _0 } {\sigma ^l\left( {\tilde {Q}_\sigma ^k (y_\sigma )F_{k,l}^{II} (y) + (\tilde {Q}_\sigma ^k )'(y_\sigma )G_{k,l}^{II} (y)} \right)} + \sigma^{5/2}O(\tilde{Q}_\sigma (y_\sigma )), \end{align*} where \begin{gather*} F_{1,0}^{II} = 0,\quad G_{1,0}^{II} = 0,\quad F_{1,1}^{II} = 3{A}'_{1,0} + 3{B}''_{1,0} + 3Q^2B_{1,0} ,\quad G_{1,1}^{II} = 0, \\ F_{2,0}^{II} = a_{1,0} (- 3{A}''_{1,0} - 3A_{1,0} Q^2)',\quad G_{2,0}^{II} = - \frac{3a_{1,0} }{2}\left( {3{A}''_{1,0} + ({B}''_{1,0} + Q^2B_{1,0} )'} \right). \end{gather*} For all $(k,l) \in \Sigma _0 $ such that $1 \le {k}' + {l}' \le 2$, $F_{k,l}^{II}$ and $G_{k,l}^{II}$ depend on $A_{{k}',{l}'}$ and $B_{{k}',{l}'}$ for $1 \le {k}' + {l}' \le 2$. Moreover, if $A_{{k}',{l}'}$ is even and $B_{{k}',{l}'}$ is odd, then $F_{k,l}^{II} $ is odd and $G_{k,l}^{II} $ is even. \end{lemma} \begin{proof} Note that \[ \delta S_{\rm mKdV} (w) = \sum_{(k,l) \in \Sigma _0 } {\sigma ^l} \Big( {\delta S_{\rm mKdV} (A_{k,l} (y)\tilde {Q}_\sigma (y_\sigma )) + \delta S_{\rm mKdV} (B_{k,l} (y)(\tilde {Q}_\sigma )'(y_\sigma ))} \Big). \] By Lemmas \ref{lem2.5} and \ref{lem2.7}, we have \begin{align*} &\delta S_{\rm mKdV} (A_{1,0} (y)\tilde {Q}_\sigma (y_\sigma ))\\ &= \tilde {Q}_\sigma (y_\sigma )\Big\{ - (LA_{1,0} )' + a_{1,0} \tilde {Q}_\sigma (y_\sigma ) ( - 3{A}''_{1,0} - 3A_{1,0} Q^2)' - a_{1,0} \tilde {Q}'_\sigma (y_\sigma )(3{A}''_{1,0} )\\ &\quad + a_{1,0}^2 \tilde{Q}_\sigma ^2 (y_\sigma )(3{A}'''_{1,0} ) \Big\} + {\tilde {Q}}'_\sigma (y_\sigma )\left\{ {3{A}''_{1,0} } + 3A_{1,0} Q^2 - {a_{1,0} \tilde {Q}_\sigma (y_\sigma )(6{A}''_{1,0} )}\right\} \\ &\quad + {\tilde {Q}}''_\sigma (y_\sigma )(3{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde{Q}_\sigma (y_\sigma )). \end{align*} Using $({\tilde {Q}}'_\sigma )^2(y_\sigma ) = \sigma ^{3/2} O(\tilde {Q}_\sigma (y_\sigma ))$, by \eqref{e2.4} and \eqref{e2.6}, we have \[ {\tilde {Q}}''_\sigma (y_\sigma )(3{A}'_{1,0} ) = \Big( {\sigma \tilde {Q}_\sigma (y_\sigma ) - \frac{1}{1 - \lambda }\tilde {Q}_\sigma ^3 (y_\sigma )} \Big)(3{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \] Thus, we get \begin{align*} &\delta S_{\rm mKdV} (A_{1,0} (y)\tilde {Q}_\sigma (y_\sigma ))\\ &= \tilde {Q}_\sigma (y_\sigma )( - LA_{1,0} )' + {\tilde {Q}}'_\sigma (y_\sigma )\left( {3{A}''_{1,0} + 3A_{1,0} Q^2} \right)\\ &\quad + \tilde {Q}_\sigma ^2 (y_\sigma ) \left( {a_{1,0} ( - 3{A}''_{1,0} - 3A_{1,0} Q^2)'} \right) + (\tilde {Q}_\sigma ^2 )'(y_\sigma ) \Big( { -\frac{9}{2}a_{1,0} {A}''_{1,0} } \Big)\\ &\quad + \tilde{Q}_\sigma ^3 (y_\sigma )(3a_{1,0}^2 {A}'''_{1,0} - \frac{3{A}'_{1,0} }{1 - \lambda }) + \sigma \tilde {Q}_\sigma (y_\sigma )(3{A}'_{1,0} ) \\ &\quad + \sigma^{5/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} We study $\delta S_{\rm mKdV} (B_{1,0} (y){\tilde {Q}}'_\sigma (y_\sigma ))$ in a similar way: \begin{align*} &\delta S_{\rm mKdV} (B_{1,0} (y){\tilde {Q}}'_\sigma (y_\sigma )) \\ &= {\tilde {Q}}'_\sigma (y_\sigma )\left\{ { - (LB_{1,0} )' + a_{1,0} \tilde {Q}_\sigma (y_\sigma )( - 3{B}''_{1,0} - 3B_{1,0} Q^2)'} \right\} \\ &\quad + {\tilde{Q}}''_\sigma (y_\sigma )(3{B}''_{1,0} + 3B_{1,0} Q^2) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )) \\ &= {\tilde {Q}}'_\sigma (y_\sigma )( - LB_{1,0} )' + (\tilde {Q}_\sigma ^2 )'(y_\sigma )\Big( {a_{1,0} \frac{( - 3{B}''_{1,0} - 3B_{1,0} Q^2)'}{2}} \Big)\\ &\quad + \sigma \tilde {Q}_\sigma (y_\sigma )(3{B}''_{1,0} + 3B_{1,0} Q^2) + \tilde {Q}_\sigma ^3 (y_\sigma )\Big( { - \frac{3{B}''_{1,0} + 3B_{1,0} Q^2}{1 - \lambda }} \Big)+\sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} The proof is complete. \end{proof} \begin{lemma} \label{lem2.10} There holds \begin{align*} S_{gBBM} (w) &= \sum_{(k,l) \in \Sigma _0 } {\sigma ^l} (\tilde {Q}_\sigma ^k )'(y_\sigma )( - \lambda {A}''_{k,l} )(y)\\ &\quad + \sum_{(k,l) \in \Sigma _0 } {\sigma ^l} \left( {\tilde {Q}_\sigma ^k (y_\sigma )F_{k,l}^{III} (y)) + (\tilde {Q}_\sigma ^k )(y_\sigma )G_{k,l}^{III} (y))} \right)\\&\quad +\sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} where \begin{gather*} F_{1,0}^{III} = 0,\quad G_{1,0}^{III} = 0,\quad F_{1,1}^{III} = - 2\lambda {A}'_{1,0} - \lambda {B}''_{1,0} ,\quad G_{1,1}^{III} = 0 \\ F_{2,0}^{III} = \lambda a_{1,0} {A}'''_{1,0} ,\quad G_{2,0}^{III} = 3\lambda a_{1,0} {A}''_{1,0} + \frac{\lambda a_{1,0} }{2}{B}'''_{1,0}. \end{gather*} For all $(k,l) \in \Sigma _0 $ such that $1 \le {k}' + {l}' \le 2,\;\;F_{k,l}^{III}$ and $G_{k,l}^{III}$ depend on $A_{{k}',{l}'}$ and $B_{{k}',{l}'}$ for $1 \le {k}' + {l}' \le 2$. Moreover, if $A_{{k}',{l}'}$ is even and $B_{{k}',{l}'} $ is odd, then $F_{k,l}^{III} $ is odd and $G_{k,l}^{III} $ is even. \end{lemma} \begin{proof} By definition, we see that \[ S_{gBBM} (w) = \sum_{(k,l) \in \Sigma _0 } {\sigma ^l} \left( {S_{gBBM} (A_{k,l} (y)\tilde {Q}_\sigma ^k (y_{_\sigma } )) + S_{gBBM} (B_{k,l} (y)(\tilde {Q}_\sigma ^k )'(y_{_\sigma } ))} \right). \] It follows from Lemma \ref{lem2.6} and \eqref{e2.4}--\eqref{e2.6} that \begin{align*} &S_{gBBM} (A_{1,0} (y)\tilde {Q}_\sigma (y_\sigma ))\\ &= \lambda \mu _\sigma \tilde {Q}_\sigma (y_\sigma )\left\{ {\beta (y_\sigma ){A}'''_{1,0} + {\beta }'(y_\sigma )(2{A}''_{1,0} ) + \beta ^2(y_\sigma )( - 2{A}'''_{1,0} )} \right\} \\ &\quad + \lambda \mu _\sigma {\tilde {Q}}'_\sigma (y_\sigma )\left\{ { - {A}''_{1,0} + \beta (y_\sigma )(4{A}''_{1,0} )} \right\} + \lambda \mu _\sigma {\tilde {Q}}''_\sigma (y_\sigma )( - 2{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )) \\ &= \lambda \tilde {Q}_\sigma (y_\sigma )\left\{ {a_{1,0} \tilde {Q}_\sigma (y_\sigma ){A}'''_{1,0} + a_{1,0} {\tilde {Q}}'_\sigma (y_\sigma )(2{A}''_{1,0} ) + a_{1,0}^2 \tilde {Q}_\sigma ^2 (y_\sigma )( - 2{A}'''_{1,0} )} \right\} \\ &\quad + \lambda {\tilde {Q}}'_\sigma (y_\sigma )\left\{ { - {A}''_{1,0} + a_{1,0} \tilde {Q}_\sigma (y_\sigma )(4{A}''_{1,0} )} \right\} + \lambda {\tilde {Q}}''_\sigma (y_\sigma )( - 2{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )) \\ &= {\tilde {Q}}'_\sigma (y_\sigma )( - \lambda {A}''_{1,0} ) + \sigma \tilde {Q}_\sigma (y_\sigma )( - 2\lambda {A}'_{1,0} ) + \tilde {Q}_\sigma ^2 (y_\sigma )(\lambda a_{1,0} {A}'''_{1,0} ) \\ &\quad + (\tilde {Q}_\sigma ^2)'(y_\sigma )(3\lambda a_{1,0} {A}''_{1,0} ) + \tilde {Q}_\sigma ^3 (y_\sigma )( - 2\lambda a_{1,0}^2 {A}'''_{1,0} + \frac{2\lambda }{1 - \lambda }{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} Similarly, we can derive that \begin{align*} S_{gBBM} (B_{1,0} (y){\tilde {Q}}'_\sigma (y_\sigma )) &= \lambda {\tilde {Q}}'_\sigma (y_\sigma )a_{1,0} \tilde {Q}_\sigma (y_\sigma ){B}'''_{1,0} + \lambda {\tilde {Q}}''_\sigma (y_\sigma )( - {B}''_{1,0} ) \\ &= \sigma \tilde {Q}_\sigma (y_\sigma )( - \lambda {B}''_{1,0} ) + (\tilde {Q}_\sigma ^2 )'(y_\sigma )\Big( {\frac{\lambda a_{1,0} }{2}{B}'''_{1,0} } \Big) \\ &\quad + \tilde {Q}_\sigma ^3 (y_\sigma )\Big( {\frac{\lambda }{1 - \lambda }{B}''_{1,0} } \Big) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} In view of $2 \le k + l \le 4$, we have \begin{gather*} S_{gBBM} (\sigma ^l\tilde {Q}_\sigma ^k (y_\sigma )A_{k,l} (y)) = \sigma (\tilde {Q}_\sigma ^k )'(y_\sigma )( - \lambda {A}''_{k,l} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )), \\ S_{gBBM} (\sigma ^l(\tilde {Q}_\sigma ^k )'(y_\sigma )B_{k,l} (y)) = \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{gather*} The proof is complete \end{proof} \begin{lemma} \label{lem2.11} There holds $$ S_{int} (w) = \sum_{(k,l) \in \Sigma _0 } {\sigma ^l} \left( {\tilde {Q}_\sigma ^k (y_\sigma )F_{k,l}^{\rm int} (y)) + (\tilde {Q}_\sigma ^k )(y_\sigma )G_{k,l}^{\rm int} (y))} \right) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )), $$ where \begin{gather*} F_{1,0}^{\rm int} =(3Q^2)',\quad G_{1,0}^{\rm int} = 3Q^2,\quad F_{1,1}^{\rm int} = G_{1,1}^{\rm int}= 0, \\ F_{2,0}^{\rm int} = 3{Q}' - 3a_{1,0} (Q^2)' + (9A_{1,0} Q)',\\ G_{2,0}^{\rm int} = 3Q + 9A_{1,0} Q + (3B_{1,0} Q)' + (3A_{1,0} B_{1,0} Q)'. \end{gather*} For all $(k,l) \in \Sigma _0 $ such that $3 \le {k}' + {l}' \le 4$, $F_{k,l}^{\rm int}$ and $G_{k,l}^{\rm int}$, depend on $A_{{k}',{l}'} ,B_{{k}',{l}'} $ for $1 \le {k}' + {l}' \le 2$. Moreover, if $A_{{k}',{l}'} $, are even and $B_{{k}',{l}'} $ are odd then $F_{k,l}^{\rm int} $ are odd and $G_{k,l}^{\rm int} $ are even. \end{lemma} \begin{proof} By a direct calculation, we have \begin{align*} \partial _x (w^3) &= \partial _x (A_{1,0}^3 (y)\tilde {Q}_\sigma ^3 (y_\sigma )) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma ))\\ &= (1 - \beta (y_\sigma ))\left\{ {(A_{1,0}^3)'\tilde {Q}_\sigma ^3 (y_\sigma )} \right\} + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma ))\\ &= \tilde {Q}_\sigma ^3 (y_\sigma )(A_{1,0}^3 )' + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )), \end{align*} \begin{align*} &\partial _x (3Q^2\tilde {Q}_\sigma (y_\sigma ) + 3Q\tilde {Q}_\sigma ^2 (y_\sigma ))\\ &= 3(1 - \beta (y_\sigma ))\left\{ {\tilde {Q}_\sigma (y_\sigma )(Q^2)' + \tilde {Q}_\sigma ^2 (y_\sigma ){Q}'} \right\} + 3Q^2(\tilde {Q}_\sigma )'(y_\sigma ) + 3Q(\tilde {Q}_\sigma ^2 )'(y_\sigma ) \\ &= \tilde {Q}_\sigma (y_\sigma )(3Q^2)' + (\tilde {Q}_\sigma )'(y_\sigma )(3Q^2) + \tilde {Q}_\sigma ^2 (y_\sigma )(3{Q}' - 3a_{1,0} (Q^2)') \\ &\quad + (\tilde {Q}_\sigma ^2 )'(y_\sigma )(3Q) + \tilde {Q}_\sigma ^3 (y_\sigma )(3a_{1,0} {Q}'), \end{align*} $$ \partial _x (3\tilde {Q}_\sigma ^2 (y_\sigma )w) = \tilde {Q}_\sigma ^3 (y_\sigma )(3{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )), $$ and \begin{align*} &\partial _x (6Q\tilde {Q}_\sigma (y_\sigma )w)\\ &= \partial _x (6A_{1,0} Q\tilde {Q}_\sigma ^2 (y_\sigma ) + 3B_{1,0} Q(\tilde {Q}_\sigma ^2 )'(y_\sigma ))\\ &= \tilde {Q}_\sigma ^2 (y_\sigma )(6A_{1,0} Q)' + (\tilde {Q}_\sigma ^2 )'(y_\sigma )\left( {6A_{1,0} Q + (3B_{1,0} Q)'} \right)- \tilde {Q}_\sigma ^3 (y_\sigma )(6a_{1,0} A_{1,0} Q)' \\ &\quad +\sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). \end{align*} Thus, we further get \begin{align*} \partial _x (3Qw^2) &= \partial _x \left( {3A_{1,0} Q\tilde {Q}_\sigma ^2 (y_\sigma ) + 3A_{1,0} B_{1,0} Q(\tilde {Q}_\sigma ^2 )'(y_\sigma )} \right) \\ &= \tilde {Q}_\sigma ^2 (y_\sigma )(3A_{1,0} Q)' + (\tilde {Q}_\sigma ^2 )'(y_\sigma )\left( {(3QA_{1,0} B_{1,0} )' + 3A_{1,0} Q} \right) \\ &\quad + \tilde {Q}_\sigma ^3 (y_\sigma )( - 3a_{1,0} (A_{1,0} Q)') + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma ), \end{align*} $$ \partial _x (3\tilde {Q}_\sigma (y_\sigma )w^2) = \tilde {Q}_\sigma ^3 (y_\sigma )(3{A}'_{1,0} ) + \sigma ^{3/2}O(\tilde {Q}_\sigma (y_\sigma )). $$ By Lemmas \ref{lem2.8}--\ref{lem2.11} and Proposition \ref{prop2.1}, we obtain explicit expressions of $F_{k,l} $ and $G_{k,l} $ for $1 \le k + l \le 2$. immediately. By Proposition \ref{prop2.1}, if the system, \eqref{e2.21}, \begin{equation} \begin{gathered} (LA_{k,l} )' = a_{k,l} ((\lambda - 3){Q}'' - 3Q^3)' + F_{k,l} \\ (LB_{k,l} )' = (3 - \lambda ){A}''_{k,l} + 3Q^2A_{k,l} + a_{k,l} (2\lambda - 3){Q}'' + G_{k,l} \end{gathered} \label{e2.21} \end{equation} is solved for every $(k,l) \in \Sigma _0 $, then $S(z) = \varepsilon (t,x)$ is small. \end{proof} \subsection{Explicit resolution of system \eqref{e2.21}} In this part, we consider \eqref{e2.21} with $k=1$ and $l=0$. We look for explicit solutions of the form \begin{equation} A_{k,l} = \tilde {A}_{k,l} + \gamma _{k,l} ,\quad B_{k,l} = \tilde {B}_{k,l} + b_{k,l} \phi , \label{e2.22} \end{equation} where $\tilde {A}_{k,l} \in \mathcal{M}$ is even and $\tilde {B}_{k,l} \in \mathcal{M}$ is odd. Let \[ P_\lambda = \frac{3}{2}Q + \frac{3 - \lambda }{2}x{Q}',\quad P = P_1 = \frac{3}{2}Q + x{Q}'. \] So we see that \begin{equation} LP = - 2Q - Q^3,\quad LP_\lambda = ((\lambda - 3){Q}'' - 3Q^3). \label{e2.23} \end{equation} \begin{lemma} \label{lem2.12} Assume that \eqref{e2.22} and $(a_{k,l} ,A_{k,l},B_{k,l} )$ satisfy \eqref{e2.21}. Then \[ a_{k,l} = \frac{12}{(\lambda - 3)(\lambda - 7)} \frac{1}{\int_\mathbb{R} {Q^2}} \Big\{ { - \gamma _{k,l} \int_\mathbb{R} {P_\lambda } + \int_\mathbb{R} {G_{k,l} Q} + \int_\mathbb{R} {F_{k,l} } \int_0^x {P_\lambda } } \Big\}. \] \end{lemma} \begin{proof} Multiplying the equation of $B_{k,l} $ by $Q$ and using $L{Q}' = 0$ gives \begin{align*} a_{k,l} (2\lambda - 3)\int_\mathbb{R} {({Q}')^2} &= \int_\mathbb{R} {((3 - \lambda ){Q}'' + 3Q^3)A_{k,l} } + \int_\mathbb{R} {G_{k,l} Q}\\ &= - \int_\mathbb{R} {(LA_{k,l} )P_\lambda } + \int_\mathbb{R} {G_{k,l} Q} . \end{align*} Then, multiplying the equation of $A_{k,l} $ by $\int_0^x {P_\lambda (y)dy} $ yields \begin{align*} \int_\mathbb{R} {(LA_{k,l} )'} \int_0^x {P_\lambda } &= - \int_\mathbb{R} {(LA_{k,l} )} P_\lambda + \gamma _{k,l} \int_\mathbb{R} {P_\lambda }\\ &= - a_{k,l} \int_\mathbb{R} {(LP_\lambda )P_\lambda } + \int_\mathbb{R} {F_{k,l} } \int_0^x {P_\lambda }. \end{align*} By combining the above two identities, we obtain \[ a_{k,l} \Big\{ {(2\lambda - 3)\int_\mathbb{R} {({Q}')^2} + \int_\mathbb{R} {(LP_\lambda )P_\lambda } } \Big\} = - \gamma _{k,l} \int_\mathbb{R} {P_\lambda } + \int_\mathbb{R} {G_{k,l} Q} + \int_\mathbb{R} {F_{k,l} } \int_0^x {P_\lambda }, \] which leads to the formula of $a_{k,l}$. According to Lemma \ref{lem2.2}, \begin{equation} a_{k,l} \Big\{ {(2\lambda - 3)\int_\mathbb{R} {({Q}')^2} + \int_\mathbb{R} {(LP_\lambda )P_\lambda } } \Big\} = a_{k,l} \frac{(\lambda - 3)(\lambda - 7)}{12}\int_\mathbb{R} {Q^2} . \label{e2.24} \end{equation} \end{proof} \begin{lemma} \label{lem2.13} The following is a solution of\eqref{e2.21} with $k=1$ and $l=0$: \[ a_{1,0} = \frac{ - 6\lambda }{(\lambda - 3)(\lambda - 7)}\frac{\int_\mathbb{R} {Q^3} }{\int_\mathbb{R} {Q^2} },\quad A_{1,0} = a_{1,0} \Big( {\frac{3}{2}Q + \frac{3 - \lambda }{2}x{Q}'} \Big) - Q^2. \] \end{lemma} \begin{proof} Recall that from Proposition \ref{prop2.1}, $F_{1,0} = (3Q^2)'$ and $G_{1,0} = 3Q^2$. Thus from Lemma \ref{lem2.12} we obtain \begin{equation} a_{1,0} = \frac{12}{(\lambda - 3)(\lambda - 7)}\frac{1}{\int_\mathbb{R} {Q^2} }\Big( {3\int_\mathbb{R} {Q^3} - 3\int_\mathbb{R} {Q^3P_\lambda } } \Big) = \frac{ - 6\lambda }{(\lambda - 3)(\lambda - 7)}\frac{\int_\mathbb{R} {Q^3} }{\int_\mathbb{R} {Q^2} }. \label{e2.25} \end{equation} Integrating \eqref{e2.21} with $(k,l)= (1,0)$ gives \begin{gather} LA_{1,0} = a_{1,0} ((\lambda - 3){Q}'' - 3Q^3) + 3Q^2, \label{e2.26} \\ (LB_{1,0} )' = (3 - \lambda ){A}''_{1,0} + 3Q^2A_{1,0} + a_{1,0} (2\lambda - 3){Q}'' + 3Q^2. \label{e2.27} \end{gather} Since $L( - Q^2) = 3Q^2$ and $LP_\lambda = (\lambda - 3){Q}'' - 3Q^3$, we have \begin{equation} A_{1,0} = a_{1,0} P_\lambda - Q^2, \label{e2.28} \end{equation} where $P_\lambda = (\frac{3}{2}Q+\frac{(3-\lambda)}{2})xQ'$. \end{proof} \subsection{Resolution of Systems \eqref{e2.21} for $2 \le k + l\le 4$.} \begin{proposition} \label{prop2.2} Let $F \in \mathcal{M}$ be odd and $G \in \mathcal{M}$ be even. Let $\gamma \in \mathbb{R}$. Then, there exists $a,b \in \mathbb{R},\;\tilde {A} \in \mathcal{M}$ being even, and $\tilde {B} \in \mathcal{M}$ being odd such that \[ A = \tilde {A} + \gamma ,\quad B = \tilde {B} + b\phi \] and satisfy \begin{equation} \label{Omega} \begin{gathered} (LA)' + a((3 - \lambda ){Q}'' + 3Q^3)' = F \\ (LB)' + a(3 - 2\lambda ){Q}'' - (3 - \lambda ){A}'' - 3Q^2A = G. \end{gathered} \end{equation} \end{proposition} \begin{proof} Using $(L1)' = (1 - 3Q^2)' = - 3(Q^2)'$, we obtain $\tilde {A}$ and $\tilde {B}$, respectively. Since $F \in \mathcal{M}$ is odd, and $H(x) = \int_{ - \infty }^x {F(z)dz} + 3\gamma Q^2$ belongs to M and is even, We have \begin{gather*} L\tilde {A} + a((3 - \lambda ){Q}'' + 3Q^3) = H, \\ (L\tilde {B})' + a(3 - 2\lambda ){Q}'' - (3 - \lambda )\tilde {{A}''} - 3Q^2\tilde {A} = G + 3\gamma Q^2 - b(L\phi )'. \end{gather*} Since $\int_\mathbb{R} {H{Q}'} = 0$ and $H \in \mathcal{M}$, by Lemma \ref{lem2.3}, there exists $\bar {H} \in \mathcal{M}$ such that $L\bar {H} = H$. It follows that $\tilde{A} = - aP_\lambda + \bar {H}$ is even, and belongs to $M$. \end{proof} We need to find $\tilde {B} \in \mathcal{M}$ be odd, such that $(L\tilde{B})' = - aZ_0 + D - b(L\phi )'$, where \begin{gather*} D = (3 - \lambda ){\bar {H}}'' + 3Q^2\bar {H} + G + 3\gamma Q^2 \in \mathcal{M},\\ Z_0 = (3 - 2\lambda ){Q}'' + (3 - \lambda ){P}''_\lambda + 3Q^2P_\lambda \in \mathcal{M}. \end{gather*} Let $$ E = \int_0^x {(D - aZ_0 )} (z)dz - bL\phi . $$ Since \[ \int_\mathbb{R} {Z_0 Q} = (2\lambda - 3)\int_\mathbb{R} {({Q}')^2} - \int_\mathbb{R} {(LP_\lambda )P_\lambda } = \frac{(\lambda - 3)(\lambda - 7)}{12}\int_\mathbb{R} {Q^2} \ne 0, \] if we choose $a = \int_\mathbb{R} {DQ} / {\int_\mathbb{R} {Z_0 Q} }$, and $b = \int_0^{ + \infty} {(D - aZ_0 )} (z)dz$, then we have the following lemma. \begin{lemma} \label{lem2.14} There exist $a$ and $b $ such that $E \in \mathcal{M}$ and $\int_\mathbb{R} {E{Q}'} = 0$. \end{lemma} \subsection{Recomposition of the approximate solution after the collision} Let $1 < c_2 < c_1 $, where $0 < c_2 - 1 < \varepsilon _0 $ is small and set \[ \lambda = \frac{c_1 - 1}{c_1 },\quad \sigma = \frac{c_2 - 1}{c_2 \lambda}. \] Consider the function $z(t,x)$ defined by \eqref{e2.10}-\eqref{e2.13}, where for all $(k,l) \in \Sigma _0$, $a_{k,l}$, $A_{k,l}$, $B_{k,l}$ are chosen as in Lemmas \ref{lem2.13} and \ref{lem2.14}. Set \begin{equation} \tau _\sigma = \sigma ^{ - \frac{1}{2} - \frac{1}{100}} = \big( {\frac{c -1}{c\lambda }} \big)^{ - \frac{1}{2} - \frac{1}{100}},\quad d(\lambda ) = b_{3,0} (\lambda ) + \frac{1}{6(1 - \lambda )}b_{1,0}^3 (\lambda ). \label{e2.29} \end{equation} \begin{lemma} \label{lem2.15} There holds: for all $t,,$, $z(t,x) = z( - t, - x)$, \begin{gather} \| {(1 - \lambda \partial _x^2 )\partial _t z + \partial _x (\partial _x^2 z - z + z^3)} \|_{H^1(\mathbb{R})} \le C\sigma ^{\frac{11}{4}}\quad \forall t, \label{e2.30} \\ \begin{aligned} &\| {z(\tau _\sigma )} - \Big\{ {Q(x - {\frac{1}{2}}\delta )} + \tilde {Q}_\sigma (x + \mu _\sigma \tau _\sigma - {\frac{1}{2}}\delta _\sigma )\\ &\quad - d(\lambda )(\tilde {Q}_\sigma ^3 )'(x + \mu _\sigma \tau _\sigma - {\frac{1}{2}}\delta _\sigma ) \Big\} \|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}, \end{aligned} \label{e2.31} \\ \begin{aligned} &\| {z( - \tau _\sigma )} - \Big\{ Q(x + {\frac{1}{2}}\delta ) + \tilde {Q}_\sigma (x - \mu _\sigma \tau _\sigma + {\frac{1}{2}}\delta _\sigma )\\ &\quad + d(\lambda )(\tilde {Q}_\sigma ^3 )'(x - \mu _\sigma \tau _\sigma + {\frac{1}{2}}\delta _\sigma ) \Big\} \|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}, \end{aligned}\label{e2.32} \end{gather} where \begin{equation} \delta = \sum_{(k,l) \in \Sigma_{0} } {a_{k,l} } \sigma ^l\int_\mathbb{R} {\tilde {Q}_\sigma ^k } , \quad \tilde {b}_{1,1} = b_{1,1} - (1/6)b_{1,0}^3 ,\quad \delta _\sigma = 2(b_{1,0} + \sigma \tilde {b}_{1,1}).\label{e2.33} \end{equation} \end{lemma} \begin{proof} The symmetry property $z(t,x) = z( - t, - x)$ follows from \eqref{e2.10}--\eqref{e2.13}, since the transformation $x \to - x$, $t \to - t$ gives $y_\sigma \to - y_\sigma $ and $y \to - y$. Note that $a_{k,l}$, $A_{k,l}$, $B_{k,l} \in\Sigma_0 $ solve $\Omega _{k,l}$, and $S(z) =\varepsilon (t,x)$. It follows that \[ \| {S(t)} \|_{H^1(\mathbb{R})} \le C\sigma ^{5/2}\| {\tilde {Q}_\sigma } \|_{H^1(\mathbb{R})} \le C\sigma ^{\frac{11}{4}}. \] To prove \eqref{e2.31}, we begin with some preliminary estimates \begin{equation} \| {\alpha (s)} \|_{L^\infty (\mathbb{R})} \le C,\quad \| {{\alpha }'(s)} \|_{L^\infty (\mathbb{R})} \le C\sqrt \sigma . \label{e2.34} \end{equation} For $t = \tau _\sigma $, $f \in \mathcal{M}$ and the small $\sigma > 0$, we have \begin{gather} \| {f(y)\tilde {Q}_\sigma (y_\sigma )} \|_{H^1(\mathbb{R})} \le C\sigma ^{10}, \label{e2.35} \\ \| {Q(y) - Q(x - \frac{1}{2}\delta )} \|_{H^1(\mathbb{R})} \le C\sigma ^{10}. \label{e2.36} \end{gather} To prove \eqref{e2.34}, by the definition of $\tilde {Q}_\sigma $ (see Lemma \ref{lem2.1}), we have \begin{equation} 0 \le \tilde {Q}_\sigma (x) \le C\sqrt \sigma e^{ - \sqrt \sigma | x |}, \quad \forall x \in \mathbb{R}. \label{e2.37} \end{equation} Let $f \in \mathcal{M}$, so that $| {f(y)} | \le C| y |e^{ - | y |}$ on $\mathbb{R}$. Note that $t = \tau _\sigma $, since $\mu _\sigma > 1/2$, we have \[ \sqrt \sigma | {y_\sigma } | \ge \sqrt \sigma (\mu _\sigma \tau _\sigma - | y | - | {\alpha (y_\sigma )} |) \ge \frac{1}{2}\sigma ^{ - \frac{1}{100}} - \sqrt \sigma | y | - 1. \] Thus, by \eqref{e2.37}, \[ | {\tilde {Q}_\sigma (y_\sigma )f(y)} |^2 \le C\sigma e^{ - \sigma ^{ - \frac{1}{100}}}| y |^{2r}e^{ - 2(1 - \sqrt \sigma )| y |} \le Ce^{ - \sigma ^{ - \frac{1}{100}}}e^{ - | y|}. \] Using $\int_\mathbb{R} {e^{ - | y |}dx} = \int_\mathbb{R} {e^{ - | y |}} \frac{dy}{1 - {\alpha }'(y_\sigma )} \le C$, we obtain \[ \| {\tilde {Q}_\sigma (y_\sigma )f(y)} \|_{L^2(\mathbb{R})} \le Ce^{-\frac{1}{2}\sigma ^{ - 1/100}} \le C\sigma ^{10}. \] For $t = \tau _\sigma $ and $x > -\tau_\sigma/2$, we have $| {\alpha (y_\sigma ) - \frac{1}{2}\delta } | \le K\sigma ^{10}$. Indeed, $| {\alpha (y_\sigma ) - \frac{1}{2}\delta} | \le C\int_{y_\sigma }^{ + \infty } {\tilde {Q}_\sigma } \le Ce^{ - \sqrt \sigma y_\sigma }$ holds for $t = \tau _\sigma $ and $x > - \frac{1}{2}\tau _\sigma $, so we have $y_\sigma \ge \frac{1}{4}\tau _\sigma $ and so $e^{ - \sqrt \sigma y_\sigma } \le e^{ - \frac{1}{4}\sigma ^{ - \frac{1}{100}}} \le C\sigma ^{ - 10}$. For $t = \tau _\sigma $, we get \[ \| {Q(y) - Q(x - \frac{1}{2}\delta )} \|_{H^1(x > - \frac{1}{2}\tau _\sigma )} \le C\sigma ^{10}. \] To prove \eqref{e2.36}, it suffices to use the decay of $Q$. Note that if $x < - \frac{1}{2}\tau _\sigma $, since $| {\alpha (y_\sigma )} | \le 1$, we have $y < - \frac{1}{2}\tau _\sigma + 1$ and \begin{align*} &\| {Q(y) - Q(x - \frac{1}{2}\delta )} \|_{H^1(x < - \frac{1}{2}\tau _\sigma )} \\ &\le \| {Q(y)} \|_{H^1(y < - \frac{1}{2}\tau _\sigma + 1)} + \| {Q(x - \frac{1}{2}\delta )} \|_{H^1(x < - \frac{1}{2}\tau _\sigma )} \le C\sigma ^{10}. \end{align*} From the expression of $z(\tau _\sigma )$, the structure of the functions $A_{k,l} ,\;B_{k,l} ,_{}$ as well as $\lim_{y \to - \infty } \phi (y) = - 1$, we have \begin{equation} \begin{aligned} &\big\| z(\tau _\sigma ) - \Big\{ Q(y) + \tilde {Q}_\sigma - b_{1,0} {\tilde {Q}}'_\sigma + \gamma _{2,0} \tilde {Q}_\sigma ^2 - b_{2,0} (\tilde {Q}_\sigma ^2 )' + \gamma _{1,1} \sigma \tilde {Q}_\sigma \\ &\quad - b_{1,1} \sigma {\tilde {Q}}'_\sigma - b_{3,0} (\tilde {Q}_\sigma^3 )' + \gamma _{3,0} \tilde {Q}_\sigma ^3 + \gamma _{2,1} \sigma \tilde {Q}_\sigma ^2 + \gamma _{4,0} \tilde {Q}_\sigma ^4 \Big\} \big\|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}. \end{aligned} \label{e2.38} \end{equation} Note that $\sigma ^{9/4}$ corresponds to the sizes of $b_{2,1} \sigma (\tilde {Q}_\sigma ^2 )'$ and $b_{4,0} (\tilde {Q}_\sigma ^4 )'$ in $H^1(\mathbb{R})$, where $b_{2,1}$ and $b_{4,0}$ are bounded. Let us expand $\tilde {Q}_\sigma (y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} )$ and $(\tilde {Q}_\sigma ^3 )'(y_\sigma - b_{1,0} -\sigma \tilde {b}_{1,1} )$ up to the order $\sigma ^\frac{13}{4}$ in $H^1(\mathbb{R})$ as: \begin{gather} \begin{aligned} &\big\| \tilde {Q}_\sigma (y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} ) - \Big\{ \tilde {Q}_\sigma - b_{1,0} \tilde {Q}'_\sigma - \tilde {b}_{1,1} \sigma {\tilde {Q}}'_\sigma \\ &+ \frac{1}{2}b_{1,0}^2 {\tilde {Q}}''_\sigma - \frac{1}{6}b_{1,0}^3 {\tilde {Q}}'''_\sigma \Big\} \big\|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}, \end{aligned} \label{e2.39} \\ \| {(\tilde {Q}_\sigma ^3 )'(y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} ) - (\tilde {Q}_\sigma ^3 )'} \|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}. \label{e2.40} \end{gather} By \eqref{e2.39} and \eqref{e2.40}, we have \[ {\tilde {Q}}''_\sigma = \sigma \tilde {Q}_\sigma - \frac{1}{1 - \lambda }\tilde {Q}_\sigma ^3 + \frac{\lambda }{1 - \lambda }\sigma \tilde {Q}_\sigma ^3 ,\quad {\tilde {Q}}'''_\sigma = \sigma {\tilde {Q}}'_\sigma - \frac{1}{1 - \lambda }(\tilde {Q}_\sigma ^3 )' + \sigma ^\frac{5}{2}O(\tilde {Q}_\sigma ), \] and \begin{equation} \begin{aligned} &\big\| \Big\{ {\tilde {Q}_\sigma (y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} ) - d(\lambda )(\tilde {Q}_\sigma ^3 )'(y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} )} \Big\} \\ & - \Big\{ {\tilde {Q}_\sigma } - b_{1,0} {\tilde {Q}}'_\sigma + (1/2) b_{1,0}^2 \sigma \tilde {Q}_\sigma - (\tilde {b}_{1,1} + (1/6) b_{1,0}^3 )\sigma {\tilde {Q}}'_\sigma \\ &- ({b_{1,0}^2 }/2(1 - \lambda )) \tilde {Q}_\sigma ^3 - b_{3,0} (\tilde {Q}_\sigma ^3 )' \Big\} \big\|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}. \end{aligned} \label{e2.41} \end{equation} Combing \eqref{e2.37}, \eqref{e2.39} with \eqref{e2.40} yields \begin{align*} &\big\| z(\tau _\sigma ) - \Big\{ Q(y) + \tilde {Q}_\sigma - b_{1,0} {\tilde {Q}}'_\sigma + \gamma _{2,0} \tilde {Q}_\sigma ^2 - b_{2,0} (\tilde {Q}_\sigma ^2 )' + \gamma _{1,1} \sigma \tilde {Q}_\sigma - b_{1,1} \sigma {\tilde {Q}}'_\sigma \\ &- b_{3,0} (\tilde {Q}_\sigma ^3 )' + \gamma _{3,0} \tilde {Q}_\sigma ^3 + \gamma _{2,1} \sigma \tilde {Q}_\sigma ^2 +\gamma _{4,0} \tilde {Q}_\sigma ^4 \Big\} \big\|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}, \\ &\big\| z(\tau _\sigma ) - \Big\{Q(y) + \tilde {Q}_\sigma (y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} ) - d(\lambda )(\tilde {Q}_\sigma ^3 )'(y_\sigma - b_{1,0} - \sigma \tilde {b}_{1,1} ) \\ & + (\gamma _{1,1} - (1/2) b_{1,0}^2 )\sigma \tilde {Q}_\sigma + ( - b_{1,1} + \tilde {b}_{1,1} + (1 /6)b_{1,0}^3 ) \sigma {\tilde {Q}}'_\sigma + \gamma _{2,0} \tilde {Q}_\sigma ^2 - b_{2,0} (\tilde {Q}_\sigma ^2 )' \\ & + \gamma _{2,1} \sigma \tilde {Q}_\sigma ^2 + (\gamma _{3,0} + (b_{1,0}^2 /2(1 - \lambda ))\tilde {Q}_\sigma ^3 +\gamma _{4,0} \tilde {Q}_\sigma ^4 \Big\} \big\|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}. \end{align*} By choosing \begin{gather*} \gamma _{1,1} = \frac{1}{2}b_{1,0}^2,\quad \tilde {b}_{1,1} = b_{1,1} - \frac{1}{6}b_{1,0}^3,\quad \gamma _{2,0} = 0,\\ b_{2,0} = 0,\quad \gamma _{2,1} = 0,\quad \gamma _{3,0} = - \frac{b_{1,0}^2}{2(1-\lambda)},\quad \gamma _{4,0} = 0, \end{gather*} together with \eqref{e2.34}, we arrive at \eqref{e2.31}. \end{proof} \subsection{Existence of approximate 2-soliton solutions} \begin{proposition} \label{prop2.3} There exists a function $z_\# $ of the form \eqref{e2.10}-\eqref{e2.11} such that for all $t \in [ - \tau _\sigma ,\tau _\sigma ]$, \begin{gather} \| {(1 - \lambda \partial _x^2 )\partial _t z_\# + \partial _x (\partial _x^2 z_\# - z_\# + z_\# ^3 )} \|_{H^1(\mathbb{R})} \le C\sigma ^{7/4}, \label{e2.42} \\ \begin{aligned} &\big\| z_\# (\tau _\sigma ) - \Big\{ Q(x - (1/2) \delta ) + \tilde {Q}_\sigma (x + \mu _\sigma \tau _\sigma - (1/2)\delta _\sigma ) \\ &- 2d(\lambda )(\tilde {Q}_\sigma ^3 )' (x + \mu _\sigma \tau _\sigma - (1/2)\delta _\sigma ) \Big\} \big\|_{H^1} \\ &+ \| z_\# ( - \tau _\sigma ) - \big\{ {Q(x + (1/2)\delta ) + \tilde {Q}_\sigma (x - \mu _\sigma \tau _\sigma + (1/2)\delta _\sigma )} \big\} \|_{H^1(\mathbb{R})} \le C\sigma ^{7/4}, \end{aligned} \label{e2.43} \end{gather} where for all $\lambda \in (0,1)$ and $d(\lambda ) \ne 0$, \begin{equation} \big| {\delta - \frac{ - 6\lambda (1 - \lambda )^{1/2}}{(\lambda - 3)(\lambda - 7)}\frac{(\int {Q)^2} }{\int {Q^2} }} \big| \le C\sigma ^{1/2},\quad | {\delta _\sigma - 2b_{1,0} } | \le C\sigma . \label{e2.44} \end{equation} Making the change of variable \eqref{e2.2}, we define \begin{gather} v(t,x) = \sqrt {\frac{\lambda }{1 - \lambda }} z_\# (\hat {t},\hat {x}),\quad D = \frac{(1 - \lambda )^{3/2}}{\lambda ^{5/2}}d(\lambda ),, \label{e2.45}\\ T = \frac{1 - \lambda }{\lambda ^{3/2}}\tau _\sigma = (\frac{c_2 - 1}{c_2 })^{ - \frac{1}{2} - \frac{1}{100}}(\frac{1 - \lambda }{\lambda })\lambda ^{\frac{1}{100}}. \label{e2.46} \end{gather} \end{proposition} From the above proposition, we have the following result. \begin{proposition} \label{prop2.4} For a positive constant $C$, the function $v$ defined by \eqref{e2.45} satisfies \begin{gather*} \| {(1- \partial _x^2 )\partial _t v + \partial _x (v + v^3)} \|_{H^1(\mathbb{R})} \le C(c_2 - 1)^{5/2}, \quad \forall t \in [ - T,T],\\ \begin{aligned} &\big\| v(T) - \big\{ \phi _{c_1 } (x - c_1 T - \frac{1}{2}\Delta _1 ) + \phi _{c_2 } (x - c_2 T - \frac{1}{2}\Delta _2 ) \\ &- 2D(\phi _{c_2 }^3 )'(x - c_2 T - \frac{1}{2}\Delta _2 ) \big\} \big\|_{H^1(\mathbb{R})} + \big\| v( - T) - \big\{ \phi _{c_1 } (x + c_1 T + \frac{1}{2}\Delta _1 )\\ &\quad + \phi _{c_2 } (x + c_2 T + \frac{1}{2}\Delta _2 ) \big\} \big\|_{H^1(\mathbb{R})} \le K(c_2 - 1)^{9/4}, \end{aligned} \end{gather*} where for all $c_1 > 1$, $D = D(c_1 ) \ne 0$, and \begin{equation} \big| {\Delta _1 - \frac{ - 6\lambda (1 - \lambda )^{1/2}}{(\lambda - 3)(\lambda - 7)}\frac{(\int_\mathbb{R} Q )^2}{\int_\mathbb{R} {Q^2} }} \big| \le C(c_2 - 1)^{1/2},\quad | {\Delta _2 - 2b_{1,0} } | \le C(c_2 - 1). \label{e2.47} \end{equation} \end{proposition} To prove Propositions \ref{prop2.3} and \ref{prop2.4}, we see that the first estimate is a consequence of $$ (1 - \partial _x^2 )\partial _t v + \partial _x (v + v^3) = \frac{\lambda^2} {(1 - \lambda )^{3/2}}\{(1-\lambda\partial_{\hat {x}}^2 ) \partial_{\hat{t}}z +\partial_{\hat{x}}(\partial_{\hat{x}}^2 z - z + z^3)\}. $$ Since $\sqrt {\frac{\lambda }{1 - \lambda }} \tilde {Q}_\sigma (y_\sigma + \delta ) = \phi _c (x - ct + \frac{1}{\sqrt \lambda }\delta )$ for $c = c_1 $ or $c_2$, we have \begin{gather*} \begin{aligned} &Q(\hat {x} - \frac{\delta }{2}) + \tilde {Q}_\sigma (\hat {x} + \mu _\sigma \tau _\sigma - \frac{\delta _\sigma }{2}) - 2d(\lambda )(\tilde {Q}_\sigma ^3 )'(\hat {x} + \mu _\sigma \tau _\sigma - \frac{\delta _\sigma }{2}) \\ &= \sqrt {\frac{1 - \lambda }{\lambda }} \Big\{ \phi _{c_1 } (x - c_1 T - \frac{\delta }{2\sqrt \lambda }) + \phi _{c_2 } (x - c_2 T - \frac{\delta _\sigma }{2\sqrt \lambda }) \\ &\quad - 2\frac{(1 - \lambda)^{3/2}} {\lambda ^{^{5/2}}}(\phi _{c_2 }^3 )'(x - c_2 T - \frac{\delta _\sigma }{2\sqrt \lambda }) \Big\} , \end{aligned}\\ \begin{aligned} &Q(\hat {x} + \frac{\delta}{2} ) + \tilde {Q}_\sigma (\hat {x} - \mu _\sigma \tau _\sigma + \frac{1}{2}\delta _\sigma )\\ &= \sqrt {\frac{1 - \lambda }{\lambda }} \Big\{ {\phi _{c_1 } (x + c_1 T + \frac{\delta }{2\sqrt \lambda }) + \phi _{c_2 } (x + c_2 T + \frac{\delta _\sigma }{2\sqrt \lambda })} \Big\}. \end{aligned} \end{gather*} Using these identities and the estimates for $z_\# $, we complete the proof of Proposition \ref{prop2.4}. Let \[ z_\# (t,x) = z(t,x) + w_\# (t,x),\quad w_\# (t,x) = - d(\lambda )(\tilde {Q}_\sigma ^3 )'(y_\sigma )(1 - P(y)) \] where $P$ is defined in \eqref{e2.23}. To prove \eqref{e2.43}, we replace $z = z_\# - w_\# $ in \eqref{e2.31} and obtain \begin{align*} &\| {z_\# (\tau _\sigma )} - \big\{ {Q(x - \frac{1}{2}\delta )} + \tilde {Q}_\sigma (x + \mu _\sigma \tau _\sigma - \frac{1}{2}\delta _\sigma ) \\ & - d(\lambda )(\tilde {Q}_\sigma ^3 )'(x + \mu _\sigma \tau _\sigma - \frac{1}{2}\delta _\sigma ) \big\} - w_\# (\tau _\sigma ) \|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}. \end{align*} Using \eqref{e2.35} ($P \in \mathcal{M})$ gives \begin{align*} &\big\| {z_\# (\tau _\sigma )} - \big\{Q(x -\frac{1}{2}\delta ) + \tilde {Q}_\sigma (x + \mu _\sigma \tau _\sigma - \frac{1}{2}\delta _\sigma ) \\ &- 2d(\lambda )(\tilde {Q}_\sigma ^3 )'(x + \mu _\sigma \tau _\sigma - \frac{1}{2}\delta _\sigma ) \big\} \big\|_{H^1(\mathbb{R})} \\ &\le C\sigma ^{9/4} + \| {d(\lambda )(\tilde {Q}_\sigma ^3 ) '(x + \mu _\sigma \tau _\sigma - \frac{1}{2}\delta _\sigma ) - w_\# (\tau _\sigma )} \|_{H^1(\mathbb{R})}\\ &\le C\sigma ^{9/4} + C\| {(\tilde {Q}_\sigma ^3 )' (x - \frac{1}{2}\delta _\sigma ) - (\tilde {Q}_\sigma ^3 )'} \|_{H^1(\mathbb{R})} \le C\sigma ^{7/4}. \end{align*} Similarly, we have \begin{align*} & \| {z_\# ( - \tau _\sigma )} - \{ Q(x + \frac{1}{2}\delta ) + \tilde {Q}_\sigma (x - \mu _\sigma \tau _\sigma + \frac{1}{2}\delta _\sigma ) + d(\lambda )(\tilde {Q}_\sigma ^3 )'(x - \mu _\sigma \tau _\sigma +\frac{1}{2}\delta _\sigma ) \} \\ &- w_\# ( - \tau _\sigma ) \|_{H^1(\mathbb{R})} \le C\sigma ^{9/4}, \end{align*} so that \begin{align*} &\| z_\# ( - \tau _\sigma ) - \{ Q(x + \frac{1}{2}\delta ) + \tilde {Q}_\sigma (x - \mu _\sigma \tau _\sigma + \frac{1}{2}\delta _\sigma )\} \|_{H^1(\mathbb{R})} \\ &\le C\sigma^{9/4} + C\| (\tilde {Q}_\sigma ^3 )'(x + \frac{1}{2}\delta _\sigma ) - (\tilde {Q}_\sigma ^3 )' \|_{H^1(\mathbb{R})} \le C\sigma ^{7/4}. \end{align*} Note that \eqref{e2.44} is a consequence of \eqref{e2.33}. To prove \eqref{e2.42}, we let \begin{align*} S_\# (t,x) &= (1 - \lambda \partial _x^2 )\partial _t z_\# + \partial _x (\partial _x^2 z_\# - z_\# + z_\# ^3 ) \\ &= S(z(t,x)) + \delta S(w_\# ) + \partial _x ((z + w_\# )^3 - z^3 - 3Q^2w_\#). \end{align*} We claim that \begin{equation} \| {\delta S(w_\# )} \|_{H^1(\mathbb{R})} \le C\sigma ^{7/4}. \label{e2.48} \end{equation} It follows from Lemmas \ref{lem2.9} and \ref{lem2.10} that, the lower order term in $\delta S(w_\# ) = \delta S_{\rm mKdV} (w_\# ) + S_{gBBM} (w_\# )$ is $d(\lambda )(\tilde {Q}_\sigma^3 )'(y_\sigma )$ $(L(1 - P))'$. This term is controlled in $H^1(\mathbb{R})$ by $\sigma ^{7/4}$. Note that \begin{equation} \| {\partial _x ((z + w_\# )^3 - z^3 - 3Q^2w_\# )} \|_{H^1(\mathbb{R})} \le C\sigma ^{\frac{25}{12}}. \label{e2.49} \end{equation} This follows from the expressions of $z $ and $w_{\# }$. \section{Stability of 2-soliton structure} \subsection{Dynamic stability in the interaction region} For any $c > 1$ sufficiently close to 1, we consider the function $z_\#(t)_{ }$ of the form \[ z_\# (\hat {t},\hat {x}) = Q(y) + Q(y_\sigma ) + \sum_{(k,l) \in\Sigma _0 } {\sigma ^l(\tilde {Q}_\sigma ^k (y_\sigma )A_{k,l} (y)} + (\tilde {Q}_\sigma ^k )'(y_\sigma )B_{k,l} (y)), \] which is used in Proposition \ref{prop2.3} (recall that $y,y_\sigma $ are defined in \eqref{e2.10}). As in Proposition \ref{prop2.4}, we set \begin{equation} \begin{aligned} v(t,x) &= \sqrt {\frac{\lambda }{1 - \lambda }} z_\# (\hat {t},\hat {x}) \\ &= \phi _{c_1 } (y_1 ) + \phi _{c_2 } (y_2 ) \\ &\quad +\sum_{(k,l) \in\Sigma _0 } \sigma ^l\big\{ {\phi _{c_2 }^k (y_2 )\tilde {A}_{k,l} (y_1 ) + (\phi _{c_2 }^k )'(y_2 )\tilde {B}_{k,l} (y_1 )} \big\} , \end{aligned} \label{e3.1} \end{equation} where \begin{equation} \begin{gathered} y_2 = x - c_2 t,\quad \;y_1 = \frac{y}{\sqrt \lambda } = x - c_1 t - \tilde {\alpha }(y_2 ),\quad \tilde {\alpha }(y_2 ) = \frac{1}{\sqrt \lambda }\alpha (\sqrt \lambda y_2 ),\\ \tilde {A}_{k,l} (y_1 ) = (\frac{1 - \lambda }{\lambda })^{k - \frac{1}{2}}A_{k,l} (\sqrt \lambda y_1 ),\quad \tilde {B}_{k,l} (y_1 ) = (\frac{1 - \lambda }{\lambda })^{k - \frac{1}{2}}\frac{1}{\sqrt \lambda }B_{k,l} (\sqrt \lambda y_1 ). \end{gathered} \label{e3.2} \end{equation} Now, we set \[ S(t) = (1 - \partial _x^2 )\partial _t v + \partial _x (v + v^3). \] \begin{proposition} \label{prop3.1} Let $\theta > 1/2$. Suppose that there exists $\varepsilon _0 >0$ such that the following holds for $0 < c_2 - 1 < \varepsilon _0$ and all $t \in [ - T,T]$, \begin{equation} \| {S(t)} \|_{H^1(\mathbb{R})} \le K\frac{(c_2 - 1)^\theta }{T}, \label{e3.3} \end{equation} and for some $T_0 \in [ - T,T]$, \begin{equation} \| {u(T_0 ) - v(T_0 )} \|_{H^1(\mathbb{R})} \le K(c_2 - 1)^\theta , \label{e3.4} \end{equation} where $u(t)$ is a $H^1(\mathbb{R})$ solution of \eqref{e1.1}. Then there exist $K_0 = K_0 (\theta ,K,\lambda )$ and a function $\rho : [ - T,T] \to \mathbb{R}$ such that, for all $t \in [ - T,T]$, \begin{equation} \| {u(t) - v(t,x - \rho (t))} \|_{H^1(\mathbb{R})} \le K_0 (c_2 - 1)^\theta ,\quad | {{\rho }'(t)} | \le K_0 (c_2 - 1)^\theta . \label{e3.5} \end{equation} \end{proposition} \begin{proof} We prove the result on $[ {T_0 ,T}]$. Using the transformation $x \to - x,\;t \to - t$, then the proof is the same on $[ {T_0 ,T}]$. Let $K^\ast > K$ be a constant to be fixed. Since $\| {u(T_0 ) - v(T_0 )} \|_{H^1(\mathbb{R})} \le K(c_2 - 1)^\theta$, by continuity in time in $H^1(\mathbb{R})$, there exists $T^\ast > T_0$ such that \begin{align*} T^\ast = \sup \Big\{ &{T_1 } \in [T_0 ,T]\;| \exists r \in C^1([T_0 ,T_1 ]) : \\ &\sup_{t \in [T_0 ,T_1 ]} \| {u(t) - v(t,x - r(t))} \|_{H^1(\mathbb{R})} \le K^\ast (c_2 - 1)^\theta \Big\} . \end{align*} \end{proof} Next we give some estimates related to $v$. \begin{lemma} \label{lem3.1} There holds \begin{gather} \begin{aligned} &\| {(1 - \partial _x^2 )(\partial _t v + c_1 \partial _x v)(t)} \|_{L^\infty (\mathbb{R})} + \| {\partial _t^2 \partial _x^2 v(t) + c_1 \partial _t \partial _x^3 v(t)} \|_{L^\infty (\mathbb{R})} \\ &\le K(c_2 - 1)^{1/2}, \end{aligned} \label{e3.6} \\ \| {\partial _t v(t) + c_1 \partial _x v(t) + (c_1 - c_2 ){\tilde {\alpha }}'(y_2 ){\phi }'_{c_1 } (y_1 )} \|_{L^2(\mathbb{R})} \le K(c_2 - 1)^{\frac{3}{4}}, \\ \| {\partial _t v(t) + c_1 \partial _x v(t) + (c_1 - c_2 ){\tilde {\alpha }}'(y_2 ){\phi }'_{c_1 } (y_1 )} \|_{L^\infty (\mathbb{R})} \le K(c_2 - 1), \label{e3.7} \\ \| {\partial _x v - {\phi }'_{c_1 } (y_1 )} \|_{L^2(\mathbb{R})} \le K(c_2 - 1)^{1/2}, \label{e3.8} \\ \| {{\tilde {\alpha }}''(y_2 )} \|_{L^\infty (\mathbb{R})} + \frac{1}{c_2 - 1}\| {\tilde {\alpha }^{(4)}(y_2 )} \|_{L^\infty (\mathbb{R})} \le K(c_2 - 1). \label{e3.9} \end{gather} \end{lemma} Note that these estimates are consequence of the \eqref{e3.1}. \begin{lemma}[Modulation] \label{lem3.2} There exists a $C^1$ function $\rho :[T_0,T^\ast ] \to \mathbb{R}$ such that, for all $t \in [T_0 ,T^\ast ]$, the function $\varepsilon (t,x)$ defined by $\varepsilon (t,x) = u(t,x + \rho(t)) - v(t,x)$ satisfies \[ \int_\mathbb{R} {\varepsilon (t,x)(1 - \partial _x^2 )({\phi }'_{c_1 } (y_1 ))dx} = 0,\quad \forall t \in [T_0 ,T^\ast ], \] and for $K$ independent of $K^\ast$, \begin{equation} \begin{gathered} \| {\varepsilon (t)} \|_{H^1(\mathbb{R})} \le 2K^\ast (c_2 - 1)^\theta ,\\ {\rho (T_0 )} | + \| {\varepsilon (T_0 )} \|_{H^1(\mathbb{R})} \le K(c_2 - 1)^\theta, \\ | {{\rho }'(t)} | \le K\| {\varepsilon (t)} \|_{H^1(\mathbb{R})} + K\| {S(t)} \|_{H^1(\mathbb{R})} . \end{gathered}\label{e3.10} \end{equation} \end{lemma} \begin{proof} Let \[ \zeta (U,r) = \int_\mathbb{R} {(U(x + r) - v(t,x)) (1 - \partial _x^2 )({\phi}'_{c_1 } (y_1 ))dx} . \] Then $$ \frac{\partial \zeta }{\partial r}(U,r) = \int_\mathbb{R} {{U}'(x + r)(1 - \partial _x^2 )({\phi }'_{c_1 } (y_1 ))dx} , $$ so that from Lemma \ref{lem3.1} (see\eqref{e3.8}), for $(c_2 - 1)$ small enough, we get \begin{align*} \frac{\partial \zeta }{\partial r}(v(t),0) &= \int_\mathbb{R} {(\partial _x v)(t,x)(1 - \partial _x^2 )({\phi }'_{c_1 } (y_1 ))dx}\\ &> \int_\mathbb{R} {[({\phi }''_{c_1 } )^2 +({\phi }'_{c_1 } )^2]dx} - K(c_2 - 1)^{1/2} \\ &> \frac{1}{2}\int_\mathbb{R} {[({\phi }''_{c_1 } )^2 + ({\phi }'_{c_1 } )^2]dx} . \end{align*} Since $\zeta (v,0) = 0$, for $U$ is near $v(t)$ in $L^2(\mathbb{R})$ norm, the existence of a unique $\rho (U)$ satisfying $\zeta (U(x - \rho (U))$,$\rho(U))$ $ = 0$ can be seen by the Implicit Function Theorem. From the definition of $T^\ast $, it follows that there exists $\rho (t) =\rho (u(t))$, such that $\zeta (U(x - \rho (t)),\rho (t)) = 0$. We set \begin{equation} \varepsilon (t,x) = u(t,x + \rho (t)) - v(t,x), \label{e3.11} \end{equation} then $\int_\mathbb{R} {\varepsilon (t)(1 - \partial _x^2 )({\phi }'_{c_1 } (y_1))dx} = 0$ follows from the definition of $\rho (t)$. Estimate \eqref{e3.10} follows from the Implicit Function Theorem and the definition of $K^\ast $. Moreover, since \eqref{e3.4}, we have $| {\rho (T_0 )} | + \| {\varepsilon (T_0 )} \|_{H^1(\mathbb{R})} \le K(c_2 - 1)^\theta $, where $K$ is independent of $K^\ast$. To prove \begin{equation} | {{\rho }'(t)} | \le K\| {\varepsilon (t)} \|_{H^1(\mathbb{R})} + K\| {S(t)} \|_{H^1(\mathbb{R})}, \label{e3.12} \end{equation} by the definition of $\varepsilon(t)$, we have \begin{equation} \begin{aligned} &(1 - \partial _x^2 )\partial _t \varepsilon + \partial _x (\varepsilon + (\varepsilon + v)^3 - v^3)\\ &= - [(1 - \partial _x^2 )\partial _t v + \partial _x (v + v^3)] + {\rho }'(t)(1 - \partial _x^2 )\partial _x (v + \varepsilon ) \\ &= - S(t) + {\rho }'(t)(1 - \partial _x^2 )\partial _x (v + \varepsilon ). \end{aligned} \label{e3.13} \end{equation} Since $\int_\mathbb{R} {\varepsilon (t,x)(1 - \partial _x^2 )({\phi }'_{c_1 }(y_1 ))dx} = 0$, we have \begin{align*} 0 &= \frac{d}{dt}\int_\mathbb{R} {\varepsilon (t)(1 - \partial _x^2 )({\phi }'_{c_1 } (y_1 ))dx} \\&= \int_\mathbb{R} {[(1 - \partial _x^2 )\partial _t \varepsilon (t)]({\phi }'_{c_1 } (y_1 ))} + \int_\mathbb{R} {\varepsilon (t)(1 - \partial _x^2 )[\partial _t {\phi }'_{c_1 } (y_1 )]}\\ &= - \int_\mathbb{R} {\partial _x (\varepsilon + (\varepsilon + v)^3 - v^3){\phi }'_{c_1 } (y_1 )} - \int_\mathbb{R} {S(t){\phi }'_{c_1 } (y_1 )} \\ &\quad + {\rho }'(t)\int_\mathbb{R} {(1 - \partial _x^2 )\partial _x (v + \varepsilon ){\phi }'_{c_1 } (y_1 )} \\ &\quad +\int_\mathbb{R} {\varepsilon (1 - \partial _x^2 )[ - c_1 {\phi }''_{c_1 } (y_1 ) + c_2 {\tilde {\alpha }}'(y_2 ){\phi }''_{c_1 } (y_1 )]} . \end{align*} Integrating by parts, we have \begin{equation} \begin{aligned} &\rho '(t)\int_\mathbb{R} (v + \varepsilon ) [(1 - \partial _x^2 )\partial _x({\phi }'_{c_1 } (y_1 ))] \\ &= - \int_\mathbb{R} S(t){\phi }'_{c_1 } (y_1 ) + \int_\mathbb{R} \varepsilon ( (1 + \varepsilon ^2 + 3\varepsilon v + 3v^2)[\partial _x {\phi }'_{c_1 } (y_1 )] \\ &\quad +(1 - \partial _x^2 ) [ - c_1 {\phi }''_{c_1 } (y_1 ) + c_2 {\tilde {\alpha }}'(y_2 ){\phi }''_{c_1 } (y_1)]) , \end{aligned}\label{e3.14} \end{equation} and so \begin{equation} \big| {{\rho }'(t)\int_\mathbb{R} {(v + \varepsilon )[(1 - \partial _x^2 )\partial _x ({\phi }'_{c_1 } (y_1 ))]} } \big| \le C(\| {\varepsilon (t)} \|_{L^2(\mathbb{R})} + \| {S(t)} \|_{L^2(\mathbb{R})} ). \label{e3.15} \end{equation} it is not difficult to check that \begin{align*} &\int_\mathbb{R} {(v + \varepsilon )[(1 - \partial _x^2 )\partial _x ({\phi }'_{c_1 } (y_1 ))]} \\ &= \int_\mathbb{R} {[(1 - \partial _x^2 )(\phi _{c_1 } (y_1 ))][\partial _x ({\phi }'_{c_1 } (y_1 ))]} + \int_\mathbb{R} {(v - \phi _{c_1 } (y_1 ) + \varepsilon )[(1 - \partial _x^2 )\partial _x ({\phi }'_{c_1 } (y_1 ))]} , \end{align*} and for $c_2 - 1 < \varepsilon _0$ small enough, \begin{align*} - \int_\mathbb{R} {[(1 - \partial _x^2 )(\phi _{c_1 } (y_1 ))][\partial _x ({\phi }'_{c_1 } (y_1 ))]} &\ge - \frac{3}{4}\int_\mathbb{R} {(\phi _{c_1 } - {\phi }''_{c_1 } ){\phi }''_{c_1 } }\\ & = \frac{3}{4}\int_\mathbb{R} {({\phi }'_{c_1 } )^2 + ({\phi }''_{c_1 } )^2} > 0, \end{align*} so that \[ | {\int_\mathbb{R} {(v + \varepsilon )(1 - \partial _x^2 )\partial _x ({\phi }'_{c_1 } (y_1 ))} } | \ge \frac{1}{2}\int_\mathbb{R} {({\phi }'_{c_1 } )^2 + ({\phi }''_{c_1 } )^2} \] holds for $c_2 - 1 < \varepsilon_0$ small, and \eqref{e3.12} follows from \eqref{e3.15}. \end{proof} \begin{lemma}[Control of the negative direction] \label{lem3.3} For all $t \in[T_0 ,T^\ast ]$, there holds \begin{equation} \begin{aligned} &\big| {\int_\mathbb{R} {\varepsilon (t)(1 - \partial _x^2 )\phi _{c_1 } (y_1 )dx} } \big|\\ &\le K(c_2 - 1)^\theta + K(c_2 - 1)^{1/4}\| {\varepsilon (t)} \|_{L^2(\mathbb{R})} + K\| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2. \end{aligned} \label{e3.16} \end{equation} \end{lemma} \begin{proof} Since $v(t)$ is an approximate solution of \eqref{e1.1}, $m(v(t))$ has a small variation. Indeed, by multiplying the equation $S(t)$ by $v(t)$ and integrating, we obtain \[ | {\frac{d}{dt}m(v(t))} | = | {\int_\mathbb{R} {S(t,x)v(t,x)dx} } | \le K\|{S(t)} \|_{L^2(\mathbb{R})} . \] Thus for all $t \in [T_0 ,T^\ast ]$, \begin{equation} | {m(v(t)) - m(v(T_0 ))} | \le KT\sup_{t \in [ - T,T]} \| {S(t)} \|_{H^1(\mathbb{R})} \le K(c_2 - 1)^\theta . \label{e3.17} \end{equation} Since $u(t)$ is a solution of \eqref{e1.1}, we have \begin{equation} m(u(t)) = m(v(t) + \varepsilon (t)) = m(u(T_0 )) = m(v(T_0 ) + \varepsilon (T_0 )). \label{e3.18} \end{equation} By expanding \eqref{e3.18}, and using \eqref{e3.17} and \eqref{e3.10}, we derive \begin{align*} &2| {\int_\mathbb{R} {((1 - \partial _x^2 )v(t))\varepsilon (t)} } |\\ &\le K(c_2 - 1)^\theta + 2| {\int_\mathbb{R} {((1 - \partial _x^2 )v(T_0 ))\varepsilon (T_0 )} } | + \| {\varepsilon (T_0 )} \|_{H^1(\mathbb{R})}^2 + \| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2 \\ &\le K(c_2 -1)^\theta + \| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2 . \end{align*} Using this and $\| {(1 - \partial _x^2 )(v(t) - \phi _{c_1 } (y_1 ))}\|_{L^2(\mathbb{R})} \le K(c_2 - 1)^{1/4}$, so we obtain \begin{align*} & | {\int_\mathbb{R} {\varepsilon (t)((1 - \partial _x^2 )\phi _{c_1 } (y_1 ))dx} } | \\ &\le | {\int_\mathbb{R} {\varepsilon (t)[(1 - \partial _x^2 )(v(t) - \phi _{c_1 } (y_1 )]} } | + | {\int_\mathbb{R} {\varepsilon (t)((1 - \partial _x^2 )v(t))} } | \\ &\le K(c_2 - 1)^\theta + K(c_2 - 1)^{1/4}\| {\varepsilon (t)} \|_{L^2(\mathbb{R})} + K\| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2 . \end{align*} Set \begin{align*} F(t) &= \frac{1}{2}\int_\mathbb{R} {((c_1 - 1)\varepsilon ^2 + c_1 (\partial _x \varepsilon )^2 - \frac{1}{2}((v + \varepsilon )^4 - v^4 - 4v^3\varepsilon ))} \\ &\quad + \frac{1}{2}(c_1 - c_2 )\int_\mathbb{R} {{\tilde {\alpha }}'(y_2 )((\partial _x \varepsilon )^2 + \varepsilon ^2)} . \end{align*} \end{proof} \begin{lemma}[Coercivity of $F$] \label{lem3.4} There exists $k_0 > 0$ such that for $c_2 - 1$ small enough, there holds \begin{equation} \| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2 \le k_0 F(t) + k_0 \Big| {\int_\mathbb{R} {\varepsilon (t)(1 - \partial _x^2 )\phi _{c_1 } (y_1 )dx} } \Big|^2. \label{e3.19} \end{equation} \end{lemma} The proof of the above lemma can be found in \cite{6,14}. \begin{lemma}[Control of the variation of the energy functional] \label{lem3.5} There holds \begin{equation} | {{F}'(t)} | \le K(c_2 - 1)\| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2 + K\| {\varepsilon (t)} \|_{H^1(\mathbb{R})} \| {S(t)} \|_{H^1(\mathbb{R})} , \label{e3.20} \end{equation} where $K$ is independent of $c_2 $. \end{lemma} \begin{proof} First, we compute \begin{align*} {F}'(t) &= \int_\mathbb{R} {(\partial _t \varepsilon )\left( {(c_1 - 1)\varepsilon - c_1 \varepsilon _{xx} - ((v + \varepsilon )^3 - v^3)} \right)} \\ &\quad - \int_\mathbb{R} {(\partial _t v)(\varepsilon ^3 + 3v\varepsilon ^2)} + \frac{1}{2}(c_1 - c_2 )\Big\{ { - c_2 \int_\mathbb{R} {{\tilde {\alpha }}''(y_2 )(\varepsilon _x^2 + \varepsilon ^2)} } \\ &\quad {+ \int_\mathbb{R} {{\tilde {\alpha }}'(y_2 )\partial _t (\varepsilon _x^2 + \varepsilon ^2)} } \Big\}\\ &= F_1 + F_2 + F_3. \end{align*} We claim that \begin{equation} \begin{aligned} &\Big| {F_1 + F_2 - \Big( {{\rho }'(t)\int_\mathbb{R} {\varepsilon [(1 - \partial _x^2 )(\partial _t v + c_1 \partial _x v)]} - \int_\mathbb{R} {(\varepsilon ^3 + 3v\varepsilon ^2)(\partial _t v + c_1 \partial _x v)} } \Big)} \Big|\\ &\le K\| {\varepsilon (t)} \|_{L^2(\mathbb{R})} \| {S(t)}\|_{H^1(\mathbb{R})}, \end{aligned} \label{e3.21} \end{equation} and \begin{equation} \label{e3.22} \begin{aligned} &| {F_3 - (c_1 - c_2 )\Big\{ {{\rho }'(t)\int_\mathbb{R} {\varepsilon [(1 - \partial _x^2 )({\tilde {\alpha }}'(y_2 ){\phi }'_{c_1 } )] - \int_\mathbb{R} {(\varepsilon ^3 + 3v\varepsilon ^2){\tilde {\alpha }}'(y_2 ){\phi }'_{c_1 } } } } \Big\}} | \\ &\le K(c_2 - 1)\| \varepsilon \|_{H^1(\mathbb{R})}^2 + K\| \varepsilon \|_{H^1(\mathbb{R})} \| {S(t)} \|_{H^1(\mathbb{R})}. \end{aligned} \end{equation} To prove \eqref{e3.21}, using the equation of $\varepsilon (t)$ (that is \eqref{e3.13}), we find \begin{align*} F_1 &= c_1 \int_\mathbb{R} {\varepsilon ((1 - \partial _x^2 )\partial _t \varepsilon )} - \int_\mathbb{R} {(\partial _t \varepsilon )(\varepsilon + ((v + \varepsilon )^3 - v^3))}\\ &= c_1 \Big\{ {\int_\mathbb{R} {( - \partial _x (\varepsilon + (\varepsilon + v)^3 - v^3))\varepsilon } - \int_\mathbb{R} {S(t)\varepsilon }+ {\rho }'(t)\int_\mathbb{R} {[(1 - \partial _x^2 )\partial _x (v + \varepsilon )]\varepsilon } } \Big\}\\ &\quad + \int_\mathbb{R} {[(1 - \partial _x^2 )^{ - 1}\partial _x (\varepsilon + (\varepsilon + v)^3 - v^3)](\varepsilon + ((v + \varepsilon )^3 - v^3))}\\ &\quad + \int_\mathbb{R} {[(1 - \partial _x^2 )^{ -1}S(t)] (\varepsilon + ((v + \varepsilon )^3 - v^3))}\\ &\quad - {\rho }'(t)\int_\mathbb{R} {[\partial _x (v + \varepsilon )](\varepsilon + ((v + \varepsilon )^3 - v^3))} , \end{align*} \begin{align*} F_1 &= - c_1 \int_\mathbb{R} {\varepsilon ^3(\partial _x v)} - \frac{3}{2}c_1 \int_\mathbb{R} {\varepsilon ^2(\partial _x v^2)} - c_1 \int_\mathbb{R} {S(t)\varepsilon } + c_1 {\rho }'(t)\int_\mathbb{R} {[(1 - \partial _x^2 )\partial _x (v)]\varepsilon } \\ &\quad + \int_\mathbb{R} {[(1 - \partial _x^2 )^{ - 1}S(t)](\varepsilon + ((v + \varepsilon )^3 - v^3))} - {\rho }'(t)\int_\mathbb{R} {(\partial _x v)\varepsilon } + {\rho }'(t)\int_\mathbb{R} {(\partial _x \varepsilon )v^3}. \end{align*} Thus, we have \begin{align*} &| {F_1 - \Big( { - c_1 \int_\mathbb{R} {\varepsilon ^3(\partial _x v)} - \frac{3}{2}c_1 \int_\mathbb{R} {\varepsilon ^2\partial _x (v^2)} } \Big) + {\rho }'(t)\int_\mathbb{R} {\varepsilon \partial _x (c_1 (1 - \partial _x^2 )v - v - v^3)} } |\\ &\le K\| {\varepsilon (t)} \|_{L^2(\mathbb{R})} \| {S(t)} \|_{H^1(\mathbb{R})} . \end{align*} Using $S = (1 - \partial _x^2 )\partial _t v + \partial _x (v + v^3)$, we find that \begin{align*} &\Big| {F_1 - \Big( { - c_1 \int_\mathbb{R} {\varepsilon ^3(\partial _x v)} - \frac{3}{2}c_1 \int_\mathbb{R} {\varepsilon ^2\partial _x (v^2)} } \Big) + {\rho }'(t)\int_\mathbb{R} {\varepsilon [(1 - \partial _x^2 )(\partial _t v + c_1 \partial _x v)]} } \Big|\\ &\le K\| {\varepsilon (t)} \|_{L^2(\mathbb{R})} \| {S(t)} \|_{H^1(\mathbb{R})}. \end{align*} So \eqref{e3.21} follows from the definition of $F_2 $. To prove \eqref{e3.22}, from \eqref{e3.9} we have \begin{gather*} \big| {\int_\mathbb{R} {{\tilde {\alpha }}''(y_2 )(\varepsilon _x^2 + \varepsilon ^2)} } \big| \le \| {\tilde {\alpha }}'' \|_{L^\infty } \| \varepsilon \|_{H^1(\mathbb{R})}^2 \le K(c_2 - 1)\| \varepsilon \|_{H^1(\mathbb{R})}^2, \\ \frac{1}{2}\int_\mathbb{R} {{\tilde {\alpha }}'(y_2 )\partial _t (\varepsilon _x^2 + \varepsilon ^2)} = \int_\mathbb{R} {{\tilde {\alpha }}'(y_2 )(\partial _t (\varepsilon - \partial _x^2 \varepsilon ))\varepsilon } - \int_\mathbb{R} {{\tilde {\alpha }}''(y_2 )(\partial _x \varepsilon )(\partial _t \varepsilon )} . \end{gather*} Using Lemma \ref{lem3.1}, we have \[ \big| {\int_\mathbb{R} {{\tilde {\alpha }}''(y_2 )(\partial _x \varepsilon )(\partial _t \varepsilon )} } \big| \le C(c_2 - 1)\| \varepsilon \|_{H^1(\mathbb{R})} (\| \varepsilon \|_{H^1(\mathbb{R})} + \| S \|_{H^1(\mathbb{R})} ). \] Using the equation of $\varepsilon$ gives \begin{align*} &\int_\mathbb{R} {{\tilde {\alpha }}'(y_2 )(\partial _t (\varepsilon - \partial _x^2 \varepsilon ))\varepsilon }\\ &= \int_\mathbb{R} {\tilde {\alpha }}'(y_2 )( - \varepsilon _x \varepsilon - (\varepsilon ^3)_x \varepsilon - 3(v^2\varepsilon )_x \varepsilon - 3(v\varepsilon ^2)_x \varepsilon \\ &\quad - S(t)\varepsilon + {\rho }'(t)\varepsilon ((1 - \partial _x^2 )\partial _x (v + \varepsilon ))) \\ &= \int_\mathbb{R} {{\tilde {\alpha }}''(y_2 )(\frac{1}{2}\varepsilon ^2 + \frac{3}{4}\varepsilon ^4 + \frac{3}{2}v^2\varepsilon ^2 + 2v\varepsilon ^3 - {\rho }'(t)\frac{1}{2}(\varepsilon ^2 + 3\varepsilon _x^2 )) + {\rho}'(t)\int_\mathbb{R} {\tilde {\alpha }^{(4)}(y_2 ) \frac{1}{2}\varepsilon ^2} }\\ &\quad + \int_\mathbb{R} {{\tilde {\alpha }}'(y_2 )( - \varepsilon ^3v_x - \frac{3}{2}\varepsilon ^2(\partial _x v^2) + {\rho }'(t)\varepsilon [(1 - \partial _x^2 )v_x ] - S(t)\varepsilon )} . \end{align*} The coefficients of ${\tilde {\alpha }}''(y_2 )$ and $\tilde {\alpha }^{(4)}(y_2 )$ are controlled, so we get \eqref{e3.22}. By Lemma \ref{lem3.3} and \eqref{e3.10}, we have \begin{align*} &\big| \int_\mathbb{R} \varepsilon (T^\ast )(1 - \partial _x^2 )\phi _{c_1 } (y_1 )dx \big| \\ &\le K(c_2 - 1)^\theta + K(c_2 - 1)^{1/4}\| \varepsilon (T^\ast ) \|_{L^2(\mathbb{R})} + K\| {\varepsilon (T^\ast )} \|_{H^1(\mathbb{R})}^2\\ &\le (K+ 1)(c_2 - 1)^\theta , \end{align*} for $0 < c_2 - 1 < \varepsilon _0 $ small enough. Thus, by Lemma \ref{lem3.4}, we obtain \[ \| {\varepsilon (T^\ast )} \|_{H^1(\mathbb{R})}^2 \le k_0 F(T^\ast ) + K(c_2 - 1)^{2\theta }. \] Integrating \eqref{e3.20} on $[T_0 ,T^\ast ]$, by \eqref{e3.10} and \eqref{e3.3}, there exists $K_1 > 0$ independent of $K^\ast $ such that \begin{align*} | {F(T^\ast )} | &\le | {F(T_0 )} | + K(c_2 - 1)^{\frac{3}{4}}T\sup_{t \in [T_0 ,T^\ast ]} \| {\varepsilon (t)} \|_{H^1(\mathbb{R})}^2 \\ &\quad + KT\sup_{t \in [T_0 ,T^\ast ]} (\| {\varepsilon (t)} \|_{H^1(\mathbb{R})} \| {S(t)} \|_{H^1(\mathbb{R})} ) \\ &\le K_1 (c_2 - 1)^{2\theta } + K(K^\ast )^2(c_2 - 1)^{2\theta + \frac{3}{4}} + K_1 K^\ast (c_2 - 1)^{2\theta }. \\ \end{align*} Thus, for $0 < c_2 - 1 < \varepsilon _0 $ small enough, we obtain \[ \| {\varepsilon (T^\ast )} \|_{H^1(\mathbb{R})}^2 \le C(c_2 - 1)^{2\theta }(2 + K^\ast ). \] By fixing $K^\ast $ such that $C(2 + K^\ast ) < \frac{1}{2}(K^\ast )^2$, we see $\| {\varepsilon (T^\ast )} \|_{H^1(\mathbb{R})}^2 \le \frac{1}{2}(K^\ast )^2(c_2 - 1)^{2\theta }$. This contradicts the definition of $T^\ast $, thus we have $T^\ast = T$, and arrive at \eqref{e3.5}. \end{proof} \subsection{Stability and asymptotic stability for large time} In this section, we consider the stability of the 2-soliton structure after the collision. For $v \in H^1(\mathbb{R})$, we denote $\| v \|_{H_{c_2 }^1 (\mathbb{R})} = \sqrt {\int_\mathbb{R} {\left( {({v}'(x))^2 + (c_2 - 1)v^2(x)}\right)dx} }$, which corresponds to the natural norm to study the stability of $_{ }\phi _{c_2 } $. \begin{proposition}[Stability of the two decoupled solitons] \label{prop3.2} Let $c_1 > 1$. Let $u(t)$ be an $H^1(\mathbb{R})$ solution of \eqref{e1.1} such that for some $w > 0$, $X_0 \ge \frac{1}{2}(c_1 - c_2 )T$, there holds \begin{equation} \| {u(0) - \phi _{c_1 } - \phi _{c_2 } (x + X_0 )} \|_{H^1(\mathbb{R})} \le (c_2 - 1)^{\frac{3}{4} + w}. \label{e3.23} \end{equation} There exists $\varepsilon _0 > 0$ such that $1 < c_2 - 1< 1 + \varepsilon _0$. Then there exist the $C^1$ functions $\rho _1 (t), \rho _2 (t)$ defined on $[0, + \infty )$ and $K > 0$ such that (i) (stability) \begin{gather} \sup_{t \ge 0} \| {u(t) - (\phi _{c_1 } (x - \rho _1 (t)) + \phi _{c_2 } (x - \rho _2 (t)))} \|_{H_{c_2 }^1 (\mathbb{R})} \le K(c_2 - 1)^{\frac{3}{4} + w}, \label{e3.24} \\ \frac{c_1 }{2} \le {\rho }'_1 (t) - {\rho }'_2 (t) \le \frac{3c_1 }{2} \quad \forall t \ge 0, \nonumber \\ | {\rho _1 (t_0 )} | \le K(c_2 - 1)^{\frac{3}{4} +w},\quad | {\rho _2 (t_0 ) + X_0 } | \le K(c_2 - 1)^w. \label{e3.25} \end{gather} (ii) (asymptotic stability) There exist $c_1^ + ,\;c_2^ + > 1$ such that \begin{gather} \lim_{t \to + \infty } \| {u(t) - (\phi _{c_1^ + } (x - \rho _1 (t)) + \phi _{c_2^ + } (x - \rho _2 (t)))} \|_{H^1(x > \frac{1}{2}(1 + c_2 )t)} = 0, \label{e3.26} \\ | {c_1^ + - c_1 } | \le K(c_2 - 1)^{\frac{3}{4} + w},\quad | {c_2^ + - c_2 } | \le K(c_2 - 1)^{1 + w + \min (\frac{1}{2},w)}. \label{e3.27} \end{gather} \end{proposition} The proof of the above proposition is similar to the one of \cite[Theorem 1.1]{6}, so we omit it. \section{Proof of Theorem \ref{thm1}} \begin{proposition} \label{prop4.1} Let $c_1 > 1$ and $1 < c_2 < 1 + \varepsilon _0 $, for $\varepsilon _0 > 0$ small enough. 1. Existence and exponential decay: Let $x_1 ,x_2 \in \mathbb{R}$. There exists a unique $H^1(\mathbb{R})$ solution $u(t) = u_{c_1 ,c_2 ,x_1 ,x_2 } (t)$ of \eqref{e1.1} such that \begin{equation} \lim_{t \to - \infty } \| {u(t) - \phi _{c_1 } (x - c_1 t - x_1 ) - \phi _{c_2 } (x - c_2 t - x_2 )} \|_{H^1(\mathbb{R})} =0. \label{e4.1} \end{equation} Moreover, for all $t \le - \frac{T}{32}$, \begin{equation} \| {u(t) - \phi _{c_1 } (x - c_1 t - x_1 ) - \phi _{c_2 } (x - c_2 t - x_2 )} \|_{H^1(\mathbb{R})} \le Ke^{\frac{1}{4}\sqrt {c_2 - 1} (c_1 - 1)t}. \label{e4.2} \end{equation} 2. Uniqueness of the asymptotic 2-soliton solution at $ - \infty $: if $w(t)$ is an $H^1(\mathbb{R})$ solution of \eqref{e1.1} satisfying \begin{equation} \lim_{t \to - \infty } \| {w(t) - \phi _{c_1 } (x - \rho _1 (t)) - \phi _{c_2 } (x - \rho _2 (t))} \|_{H^1(\mathbb{R})} = 0, \label{e4.3} \end{equation} for $\rho _1 (t)$ and $\rho _2 (t)$, then there exist $x_1 , x_2 \in \mathbb{R}$ such that $w(t) = u_{c_1 ,c_2 ,x_1 ,x_2 } (t)$. \end{proposition} The above proposition is essentially the same as \cite[Theorem 1.3]{6}. Recall that such a result was first proved for the generalized KdV equations in \cite{19}, and refined techniques were introduced in \cite{13,21}. \begin{lemma} \label{lem4.1} Let $c_1 > 1$. Let $1 < c_2 < 1 + \varepsilon _0 $ and $\varepsilon _0 = \varepsilon _0 (c_1 ) > 0$. We suppose that $u(t)$ is a solution of \eqref{e1.1} satisfying: for some $\rho _1 (t),\;\rho _2 (t)$, \begin{gather} \lim_{t \to - \infty } \| {u(t) - \phi _{c_1 } (x - \rho _1 (t)) - \phi _{c_2 } (x - \rho _2 (t))} \|_{H^1(\mathbb{R})} = 0, \label{e4.4} \\ \lim_{t \to + \infty } \| {u(t) - \phi _{c_1 } (x - \rho _1 (t)) - \phi _{c_2 } (x - \rho _2 (t)) - w^ + (t)} \|_{H^1(\mathbb{R})} = 0, \label{e4.5} \end{gather} where $| {c_j^ + - c_j } | \le \varepsilon _0 | {c_j - 1}|$ and \begin{equation} \lim_{t \to + \infty } \| {w^ + (t)} \|_{H^1(x > \frac{1}{2}(1 + c_2 )t)} = 0,\quad \limsup _{t \to + \infty } \| {w^ + (t)} \|_{H^1(\mathbb{R})} \le \varepsilon _0 | {c_2 - 1} |^{1/2}. \label{e4.6} \end{equation} Then, there exist $C = C(c_1 )$ such that \begin{gather} \frac{1}{C}\limsup _{t \to + \infty } \| {w^ + (t)} \|_{H_{c_2 }^1 (\mathbb{R})}^2 \le c_1^ + -c_1 \le C\liminf _{t \to + \infty } \| {w^ + (t)} \|_{H_{c_2 }^1 (\mathbb{R})}^2, \nonumber \\ \begin{aligned} \frac{1}{C}(c_2 - 1)^{-1/2}\limsup _{t \to + \infty } \| {w^ + (t)} \|_{H^1(\mathbb{R})}^2 &\le c_2 - c_2^ + \\ &\le C(c_2 - 1)^{-1/2}\liminf _{t \to + \infty } \| {w^ + (t)} \|_{H^1(\mathbb{R})}^2 . \end{aligned}\label{e4.7} \end{gather} \end{lemma} \begin{proof} By \eqref{e1.3}, \eqref{e1.4}, \eqref{e4.4}, \eqref{e4.5} and \eqref{e4.6}, we have that for the large $t$, \begin{gather} m(u(0)) = m(\phi _{c_1 } ) + m(\phi _{c_2 } ) = m(\phi _{c_1^ + } ) + m(\phi _{c_2^ + } ) + m(w^ + (t)) + o(1), \label{e4.8}\\ E(u(0)) = E(\phi _{c_1 } ) + E(\phi _{c_2 } ) = E(\phi _{c_1^ + } ) + E(\phi _{c_2^ + } ) + E(w^ + (t)) + o(1). \label{e4.9} \end{gather} Let $(j = 1,2)$, \[ \bar {a}_j = \frac{E(\phi _{c_j^ + } ) - E(\phi _{c_j } )}{m(\phi _{c_j^ + } ) - m(\phi _{c_j } )}, \] so that \begin{equation} | {\bar {a}_j - c_j } | \le C| {c_j^ + - c_j } |. \label{e4.10} \end{equation} Indeed, by \eqref{e2.19}, one has \[ \frac{E(\phi _{c_j^ + } ) - E(\phi _{c_j } )}{m(\phi _{c_j^ + } ) - m(\phi _{c_j } )} = {\frac{\frac{d}{dc}E(\phi _c )}{\frac{d}{dc}N(\phi _c )}} |_{c = c_j } + O(| {c_j^ + - c_j } |) = c_j + O(| {c_j^ + - c_j } |). \] Considering $\bar {a}_2E$ times \eqref{e4.8} - \eqref{e4.9} and then $\bar {a}_1$ times \eqref{e4.8} - \eqref{e4.9}, we find that for the large $t$ \begin{gather} [E(\phi _{c_1^ + } ) - \bar {a}_2 m(\phi _{c_1^ + } )] - [E(\phi _{c_1 } ) - \bar {a}_2 m(\phi _{c_1 } )] = \bar {a}_2 m(w^ + (t)) - E(w^ + (t)) + o(1), \label{e4.11} \\ [\bar {a}_1 m(\phi _{c_2 } ) - E(\phi _{c_2 } )] - [\bar {a}_1 m(\phi _{c_2^ + } ) - E(\phi _{c_2^ + } )] = \bar {a}_1 m(w^ + (t)) - E(w^ + (t)) + o(1). \label{e4.12} \end{gather} Note that \[ \int_\mathbb{R} {| {w^ + } |^4} \le C\| {w^ + } \|_{H^1(\mathbb{R})}^2 \int_\mathbb{R} {(w^ + )^2} \le C\varepsilon _0 | {c_2 - 1} |\int_\mathbb{R} {(w^ + )^2} \] so that \[ \bar {a}_2 m(w^ + (t)) - _{ }E(w^ + (t)) > \frac{1}{4}((c_2- 1)\int_\mathbb{R} {(w^ + )^2} + \int_\mathbb{R} {(w_x^ + )^2} ). \] Now, let $\beta _1 = \frac{d}{dc}m(\phi _c ) |_{c = c_1 } >0$, by \eqref{e2.19}, we have \[ (\frac{d}{dc}E(\phi _c ) - \bar {a}_2 \frac{d}{dc}m(\phi _c )) |_{c = c_1 } = (c_1 - \bar {a}_2 )\frac{d}{dc}m(\phi _c ) |_{c = c_1 }, \] and so \[ \frac{1}{2}(c_1 -1)\beta _1 < {(\frac{d}{dc}E(\phi _c ) - \bar {a}_2 \frac{d}{dc}m(\phi _c ))} |_{c = c_1 } < (c_1 - 1)\beta _1 . \] Thus, from \eqref{e4.11}, we obtain that for the large $t$, \begin{gather*} c_1^ + - c_1 \ge C[(c_2 - 1)\int_\mathbb{R} {(w^ + (t))^2} + \int_\mathbb{R} {(w_x^ + (t))^2} ] + o(1) \ge C\| {w^ + (t)} \|_{H_{c_2 }^1 (\mathbb{R})}^2 + o(1),\\ c_1^ + - c_1 \le {C}'\| {w^ + (t)} \|_{H_{c_2 }^1 (\mathbb{R})}^2 + o(1). \end{gather*} Similarly, we have \begin{gather*} \begin{aligned} c_2 - c_2^ + &\ge \frac{C}{(c_2 - 1)^{1/2}}[(c_1 - 1) \int_\mathbb{R} {(w^ + (t))^2} + \int_\mathbb{R} {(w_x^ + (t))^2} ] + o(1)\\ &\ge \frac{C}{(c_2 - 1)^{1/2}}\| {w^ + (t)} \|_{H^1(\mathbb{R})}^2 + o(1), \end{aligned} \\ c_2 - c_2^ + \le \frac{{C}'}{(c_2 - 1)^{1/2}}\| {w^ + (t)} \|_{H^1(\mathbb{R})}^2 + o(1). \end{gather*} Estimate \eqref{e4.7} follows. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] Let $c_1 > 1$ and $\varepsilon _0 =\varepsilon _0 (c_1 )$ be small enough. Let $1 < c_2 < 1 + \varepsilon _0 $ and $T$ be defined by \eqref{e2.46}. Let $\tilde {u}(t)$ be the unique solution of \eqref{e1.1} such that \[ \lim_{t \to - \infty } \| {\tilde {u}(t) - \phi _{c_1 } (x - c_1 t) - \phi _{c_2 } (x - c_2 t)} \|_{H^1(\mathbb{R})} = 0. \] (1) By Proposition \ref{prop4.1}, for all $t \le - \frac{T}{32}$, \begin{equation} \| {\tilde {u}(t) - \phi _{c_1 } (x - c_1 t) - \phi _{c_2 } (x - c_2 t)} \|_{H^1(\mathbb{R})} \le Ke^{\frac{1}{4}\sqrt {c_2 - 1} (c_1 - 1)t}. \label{e4.13} \end{equation} Let $\Delta _1 ,\Delta _2 $ be defined in Proposition \ref{prop2.4}, and let \[ T^ - = T + \frac{1}{2}\frac{\Delta _1 - \Delta _2 }{c_1 - c_2 }. \] Since $| {\Delta _1 } | + | {\Delta _2 } | \le C = C(c_1 )$, and $c_1 - c_2 > c_1 - 1 - \varepsilon _0 \ge \frac{1}{2}(c_1 - 1)$, we have $ - T^- < -\frac{1}{32}T$, for small $c_2 - 1$ and so \begin{equation} \begin{aligned} \| {\tilde {u}( - T^ - ) - \phi _{c_1 } (x + c_1 T^ - ) - \phi _{c_2 } (x + c_2 T^ - )} \|_{H^1(\mathbb{R})} &\le Ke^{ - \frac{1}{4}\sqrt {c_2 - 1} (c_1 - 1)T^ - } \\ &\le (c_2 - 1)^{10}, \end{aligned} \label{e4.14} \end{equation} for $\varepsilon _0 $ small enough. Let \begin{equation} u(t,x) = \tilde {u}(t + T - T^ - ,\;x + \frac{1}{2}\Delta _1 + c_1 (T - T^ -)). \label{e4.15} \end{equation} Then, $u(t)$ is a solution of \eqref{e1.1} and satisfies \begin{equation} \| {u( - T) - \phi _{c_1 } (x + c_1 T + \frac{1}{2}\Delta _1 ) - \phi _{c_2 } (x + c_2 T + \frac{1}{2}\Delta _2 )} \|_{H^1(\mathbb{R})} \le (c_2 - 1)^{10}. \label{e4.16} \end{equation} It is easily checked that the results obtained for $u(t)$ imply the desired results on $\tilde {u}(t)$. (2) By Proposition \ref{prop2.4} and \eqref{e4.16}, we have \[ \| {u( - T) - v( - T)} \|_{H^1(\mathbb{R})} \le K(c_2 -1)^{9/4}. \] By Proposition \ref{prop2.3} and the above estimate, we can apply Proposition \ref{prop3.1} with $\theta = \frac{5}{2} - \frac{1}{2} - \frac{1}{100} = 2 - \frac{1}{100}$. There exists $\rho (t)$ such that for all $t \in [ - T,T]$, \[ \| {u(t) - v(t,x - \rho (t))} \|_{H^1(\mathbb{R})} + | {{\rho }'(t)} | \le C(c_2 - 1)^{2 -\frac{1}{100}}. \] In particular, for $r = \rho (T),\;| r | \le C(c_2 - 1)^{2 - \frac{1}{50}}$, we have \[ \| {u(T) - v(t,x - r)} \|_{H^1(\mathbb{R})}\le _{ }C(c_2 - 1)^{2 - \frac{1}{100}}, \] and using Proposition \ref{prop2.4}, we obtain \begin{equation} \begin{aligned} &\| {u(T) - \{ {\phi _{c_1 } (x - r_1 ) + \phi _{c_2 } (x - r_2 ) - 2D(\phi _{c_2 }^3 )'(x - r_2 )} \}} \|_{H^1(\mathbb{R})} \\ &\le C(c_2 - 1)^{2 - \frac{1}{100}}, \end{aligned} \label{e4.17} \end{equation} where $r_1 = c_1 T + \frac{1}{2}\Delta _1 + r$ and $r_2 = c_2 T + \frac{1}{2}\Delta _2 + r$, so that \[ \frac{1}{2}(c_1 - c_2 )T \le r_1 - r_2 \le \frac{3}{2}(c_1 - c_2 )T. \] Moreover, since $\| {(\phi _{c_2 }^3 )'} \|_{H^1(\mathbb{R})} \le C(c_2 - 1)^{7/4}$, we also obtain \begin{equation} \| {u(T) - \{ {\phi _{c_1 } (x - r_1 ) + \phi _{c_2 } (x - r_2 )} \}} \|_{H^1(\mathbb{R})} \le C(c_2 - 1)^{7/4}. \label{e4.18} \end{equation} In what follows, \eqref{e4.18} will serve us to prove that $u(t)$ is close to the sum of two solitons for $t > T$, whereas \eqref{e4.17} will allow us to prove that $u(t)$ is not a pure 2-soliton solution at $ + \infty $. (3) By using Proposition \ref{prop3.2} with $w =1$, it follows from \eqref{e3.24}, \eqref{e3.26} and \eqref{e3.27} that there exists $\rho _1(t),\rho _2 (t),c_1^ + ,c_2^ + $ such that \begin{equation} \begin{gathered} c_1^ + = \lim_{t \to + \infty } \bar {c}_1(t),\quad c_2^ + = \lim_{t \to + \infty } \bar {c}_2 (t),\quad | {c_1^ + - c_1 } | \le C(c_2 - 1)^{7/4}, \\ |{c_2^ + - c_2 } | \le C(c_2 - 1)^{5/2}, \quad w^ + (t,x) = u(t,x) - \{ {\phi _{c_1^ + } (x - \rho _1 (t)) + \phi _{c_2^ + } (x - \rho _2 (t))} \}, \\ \sup_{t \ge T} \| {w^ + (t)} \|_{H_{c_2 }^1 (\mathbb{R})} \le C(c_2 - 1)^{7/4},\quad \lim_{t \to + \infty } \| {w^ + (t)} \|_{H^1(x > \frac{1}{2}(1 + c_2 )t)} = 0. \end{gathered} \label{e4.19} \end{equation} From Lemma \ref{lem4.1}, we obtain \[ 0 \le c_1^ + - c_1 \le C(c_2 - 1)^{\frac{7}{2}},\quad 0 \le c_2^ + - c_2 \le C(c_2 - 1)^4. \] (4) There exists $K_0 > 0$ such that \begin{equation} \liminf _{t \to + \infty } \| {w^ + (t)} \|_{H_{c_2 }^1 (\mathbb{R})} \ge K_0 (c_2 - 1)^{9/4}. \label{e4.20} \end{equation} This follows the proof of the BBM case in \cite{20} immediately, so we omit it. 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