Stability of the sum of two solitary waves for (gDNLS) in the energy space
Xingdong Tang Guixiang Xu
























 



 











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gDNLS]
Stability of the sum of two solitary waves for (gDNLS) in the energy space

]Xingdong Tang
-1.15em Xingdong Tang Beijing Computational Science Research Center, No. 10 West Dongbeiwang Road, Haidian
District, Beijing, China, 100193,
txd@csrc.ac.cn

]Guixiang Xu
-1.15em Guixiang Xu Institute of
Applied Physics and Computational Mathematics, P. O.
Box 8009, Beijing, China, 100088.
xu_guixiang@iapcm.ac.cn






[2010]Primary: 35L70, Secondary: 35Q55




In this paper, we continue the study in <cit.>. We use the perturbation argument, modulational analysis and the energy argument in <cit.> to show the stability of the sum of two
solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case σ>1 we considered corresponds to the L^2-supercritical case.


[
    [
    December 30, 2023
=====================








§ INTRODUCTION

In this paper, we consider the stability of the solitary waves for the generalized derivative Schrödinger equation (gDNLS for short) in H^1()

    {   ı u_t+u_xx+ı |u|^2σu_x=0,   t,x∈×,
    
        u0,x=u_0x∈ H^1,   .

where u is a complex valued function of t,x∈×, σ>0.
With σ=1, (<ref>) has appeared as a model for Alfvén waves in plasma physics <cit.>.

The equation (<ref>) is Ḣ^σ-1/2σ critical since the scaling
transformation

    u(t,x)↦ u_λ(t,x)=λ^1/2σu(λ^2t, λ x)

leaves both (<ref>) and Ḣ^σ-1/2σ-norm invariant. The mass, momentum
and energy of the solution u(t,x) of (<ref>) are defined as following

    M(u)(t)=    1/2∫ |u(t,x)|^2 , 
    
    P(u)(t)=    -1/2∫u̅u_xt,x ,
    
    E(u)(t)=   1/2∫|u_xt,x|^2 
      +1/2(σ+1)∫|u|^2σu̅u_xt,x.

They are conserved under the flow (<ref>) according to the
phase rotation invariance, spatial translation invariance and time
translation invariance respectively. Compared with
nonlinear Schrödinger equation, the equation (<ref>) doesn't enjoy the Galilean
invariance and pseudo-conformal invariance any more.

Local well-posedness result for (<ref>) with σ≥ 1 in H^1() has
been worked out by Hayashi and Ozawa <cit.>. They
combined the compactness method with L^4_IW^1,∞()
estimate to construct the local-in-time solution with arbitrary initial data
in the energy space. Since (<ref>)
is Ḣ^1-subcritical, the maximal lifespan interval only depends on
the H^1 norm of initial data. More precisely, we have


 Let σ≥ 1. For any u_0 ∈ H^1() and t_0 ∈, there
exists a unique maximal-lifespan solution u:I×→ to (<ref>) with u(t_0)=u_0, the map u_0→ u is continuous from H^1() to C(I, H^1())∩ L^4_loc(I, W^1,∞()).  Moreover, the solution also
has the following properties:


  * I is an open neighborhood of t_0.


  * The mass, momentum and energy are conserved, that is, for all
t∈ I,

    M(u)(t)=M(u)(t_0),    P(u)(t)=P(u)(t_0),   E(u)(t)=E(u)(t_0).



  * If  sup(I)<+∞,  (or inf(I)>-∞) , then

    lim_t→sup(I)∂_x
    u(t)_L^2=+∞,  (
    lim_t→inf(I)∂_x u(t)_L^2=+∞,
    respectively.)



  * If u_0 _H^1 is sufficiently small,
then u is a global solution.



The local well-posedness result of (<ref>) with σ≥ 1 in H^1/2 is due to Takaoka <cit.> by Fourier restriction norm method and gauge transformation and
Santos <cit.> by local smoothing effect of the Schrödinger operator. The different features between the case σ=1 and the case σ>1 are that the former is the integrable system and has the gauge transformation invariance.  In addtion, there are some numerical stability analysis and blowup results of (<ref>) in the energy space, please refer to <cit.>.


At the same time, it is well-known in <cit.> that the equation (<ref>) has a two-parameter family of solitary wave solutions of the form

    ut,x=Q_ω,cx-ct^ıω t,

where 4ω > c^2,

    Q_ω,cx=Φ_ω,c(x)exp{ıc/2x-ı/2σ +
          2∫^x_-∞Φ_ω,c^2σ(y)ỵ},

and

    Φ_ω,c(x)
      =
      (σ+1)(4ω-c^2)/
            2√(ω)(cosh(σ√(4ω-c^2)x)-c/2√(ω))
            ^1/2σ

is the unique positive solution of

    - ∂_x^2Φ_ω,c  + (ω -
      c^2/4)Φ_ω,c +c/2|Φ_ω,c|^2σΦ_ω,c - 2σ + 1/(2σ +
        2)^2|Φ_ω,c|^4σΦ_ω,c  = 0,

up to phase rotation and spatial translation invariance. By the stability criteria in <cit.> <cit.>, it was shown that they are orbitally stable when σ∈ (0, 1), and orbitally unstable when σ≥ 2 in <cit.>.

For the case σ=1 and 4ω > c^2. On one hand, by  the convex analysis in <cit.>, the structure analysis[Since the nonlinearity has derivative in (<ref>), the structure analysis of the solitary waves before using the variational argument in <cit.> is used to transform the quasilinear problem into the semilinear problem in principle in <cit.>.] and the variational characterization of the solitary waves, Miao, Tang and Xu obtained the global wellposedness result in some invariant subset K^+ of the energy space in <cit.>, where the construction of K^+ is related to the variational characterization of the solitary wave. On the other hand,
 Colin and Ohta <cit.> made use of the concentration compactness argument and  proved that the above solitary waves are orbitally stable in the energy space. Because (<ref>) is an integrable system,  Nakamura and Chen obtained the explicit
formula of the multi-soliton solutions of (<ref>) in <cit.> by Hirota's
bilinear transform method. Recently, Miao, Tang and Xu in <cit.> and Le Coz and Wu in <cit.> independently showed the stability of the sum of the multi-soliton waves with weak interactions in the energy space, where the arguments are both based on the perturbation argument, the modulation stability and the energy argument in <cit.>.

For the case σ∈(1,2) and 4ω > c^2.
Fukaya, Hayashi and Inui  showed the variational characterization of the solitary waves of (<ref>) in <cit.>, i.e.
Q_ω,c is a minimizer of the following problem:

    dω,c
        =
        infS_ω,cφφ∈ H^1∖0,
           K_ω,cφ=0

where the action functional is defined by

    S_ω,c(φ)=E(φ)+ ω M(φ) + c P(φ),

and the scaling derivative functional is defined by

    K_ω,c(φ)
      =
      ./λ̣S_ω,c(λφ)|_λ=1.

In addition, they also obtained  the global well-posedness
of the solution to (<ref>) in the similar invariant subset K^+ as that in <cit.>.

Next we consider its stability in the energy space. Let z_0=z_0(σ) be the unique solution in -1,1 of F(z_0;σ) =0,
    where F(z;σ) is defined by
    
    F(z; σ) =     (σ-1)^2 ∫_0^∞ (cosh y -z)^-1/σỵ^2
       - ∫_0^∞ (cosh y -z)^-1/σ-1(z
              cosh y -1)ỵ^2.

By the stability criteria in <cit.>, Liu, Simpson and Sulem numerically showed that the solitary wave is stable for c∈ (-2√(ω), 2z_0√(ω))   and unstable for c∈ (2z_0√(ω), 2√(ω)) in <cit.>. That is,

Let σ∈1,2 and z_0=z_0(σ)∈-1,1 satisfy F(z_0;σ) =0,
    where F(z;σ) is defined by (<ref>). Let (ω^0, c^0)∈^2, satisfy c^0∈-2√(ω^0) ,2 z_0 √(ω^0), the solitary wave  Q_ω^0,c^0(x-c^0t)e^iω^0 t to (<ref>) is
orbitally stable in the energy space. That is, for any ϵ>0,
there exists δ>0 such that if u_0∈ H^1() satisfies

    u_0(·)-
    Q_ω^0,c^0(·-x^0)e^iγ^0
    _H^1()<δ
 for some (x^0, γ^0)∈^2, then
the solution u(t) of (<ref>) exists globally in time and
satisfies

    sup_t≥0inf_(y, γ)∈^2u(t,·)-
    Q_ω^0,c^0(·-y)e^iγ_H^1()<.




For the case σ∈(3/2, 2) and c^0=2z_0√(ω^0), Fukaya shown that the traveling wave is still unstable in <cit.>. It is notice that it are still open problem whether the solitary waves with any σ>0 and the critical case c^0= 2√(ω^0) are stable or not. In fact, the solitary waves with the critical parameter c^0= 2√(ω^0) have polynomial decay, and the difficulty is that there is no the spectral gap about the linearized operator around the solitary wave.


In this paper, we consider the stability of the sum of two solitary waves for (<ref>) with σ∈ (1,2) and c^0_k∈ (-2√(ω^0_k), 2z_0(σ)√(ω^0_k)), k=1, 2. As far as we know, the integrability (non-integrability) of (<ref>) is not clear,
the existence (nonexistence) of the explicit multi-solition solutions is not
obvious. Here we use the argument in <cit.> (see also <cit.>) and the references therein.

The main result is the following result.



Let σ∈1,2 and z_0=z_0(σ)∈0,1 satisfy F(z_0;σ) =0,
    where F(z;σ) is defined by (<ref>). Let (ω^0_k, c^0_k)∈^2, k=1,2 satisfy
    
    
  (a) Nonlinear stability:
        c^0_k∈-2√(ω^0_k),2 z_0 √(ω^0_k) for k=1, 2.

    
  (b) Technical assumption: ω^0_2-ω^0_1/c^0_2-c^0_1>0.

    
  (c) Relative speed: c^0_1 <
    ω^0_2-ω^0_1/c^0_2-c^0_1, and 4ω^0_2-ω^0_1/c^0_2-c^0_1<c^0_2.
    
    Then there exist positive numbers C, δ_0, θ_0 and L_0, such that if
    0<δ<δ_0,  L>L_0 and
    
    u_0(·)-∑^2_k=1Q_ω^0_k,
        c^0_k(·-x^0_k)e^iγ^0_k_H^1()≤δ,

    with x^0_2-x^0_1>L, then the solution u(t) of (<ref>)
    exists globally in time and there exist functions
    x_k(t) and γ_k(t), k=1,2 such that for any t≥ 0,
    
    u(t,·)-∑^2_k=1Q_ω^0_k,
        c^0_k(·-x_k(t))e^iγ_k(t)_H^1()≤
        C(δ+e^-θ_0 L/2).





  
  *  The function F(z;σ) and existence of z_0.
  In order to use the abstract functional analysis argument
  in <cit.>, Liu, Simpson and Sulem introduced  the function F(z;σ)  to obtain the stability
   (instability) of single soliton solutions of (<ref>)
   in <cit.>. The function F(z;σ) is closely related to the determinant of the Hessian d”ω,c. It numerically turns out that for any fixed σ∈1,2, the function F(z;σ) is monotonically decreasing with respect to z and has exactly one root z_0 in the interval -1,1. See Figure <ref>.
   

   
  *  The technical assumption:
    ω^0_2-ω^0_1/c^0_2-c^0_1>0. Because of the fact that the radiation term cannot separate from the solitary wave along the flow (<ref>), this technical assumption allows us to deal with some "bad" term with good sign in (<ref>), see the monotonicity formulas in Section <ref> for more details.

   
  *  The relative speed assumption: In fact, it is sufficient that c^0_1
    <
    2ω^0_2-ω^0_1/c^0_2-c^0_1
    <c^0_2 from our proof. However,
    we suppose
    c^0_1 <
    ω^0_2-ω^0_1/c^0_2-c^0_1, and
    4ω^0_2-ω^0_1/c^0_2-c^0_1<c^0_2 for the convenience.
    In addition, combining 4ω^0_2-ω^0_1/c^0_2-c^0_1<c^0_2
    with ω^0_2-ω^0_1/c^0_2-c^0_1>0, we immediately obtains that
    c_2^0 and z_0(σ) need to be positive.

    
  * The stability of the sum of two solitary waves can be easily extended to that of the k solitary
waves case, k≥ 3.



At last, the paper is organized as following.  In Section
<ref>,  we introduce the linearized operator around
the solitary wave, and show the coercivity property of the
linearized operator under the geometric constraints; In Section <ref>, we
give the modulation analysis of the solution around the sum of two
solitary waves with weak interactions. In Section <ref>, we introduce some extra monotonicity formulas and their
variance along the flow (<ref>). In Section <ref>,  we firstly introduce
a localized action functional, which is almost conserved by the
monotonicity formula and the conservation laws of mass, momentum and
energy, to refine the energy estimate about the radiation term in the
modulation analysis of the solution; secondly, we use some
monotonicity formulas to refine the estimates of the parameter variance
|ω_k(t)-ω_k(0)|+|c_k(t)-c_k(0)|, k=1, 2 besides of the
conservation laws of mass and momentum. These refined estimates improve
the energy estimate of the radiation term in the modulation analysis and imply Theorem <ref> together with the bootstrap argument in <cit.> (see also <cit.>). In Appendix A, for all solitary waves Q_k^0(x)=Q_ω^0_k,
    c^0_k(x), k=1, 2, which satisfy the conditions in Theorem <ref>, we verify the fact that

    2MQ_k^0∂_x Q_k^0_L^2^2 - 4[PQ_k^0 ]^2≠ 0,

which is used to show the non-degenerate condition (<ref>). In Appendix B, we give the expansion of the action functional 𝒮(t) (i.e., Lemma <ref>) in details.



§ PRELIMINARY RESULTS

In this section, we give some basic facts about the solitary waves for (<ref>). Let (ω, c)∈^2 with 4ω>c^2, and ut,x=φ_ω,cx-ct^ıω t be a solution of (<ref>), it is easy to check that φ_ω,c satisfies

    ωφ_ω,c-∂^2_xφ_ω,c +ı c ∂_xφ_ω,c - ıφ_ω,c^2σ∂_x φ_ω,c=0.

Now define the set 𝒢_ω,c of the solitary waves to (<ref>)

    𝒢_ω,c
        =
        {φ_ω,c∈ H^1∖0 : φ_ω,c satisfies (<ref>)}.

and let

    Q_ω,cx=Φ_ω,c(x)exp{ıc/2x-ı/2σ +
          2∫^x_-∞Φ_ω,c^2σ(y)ỵ},

with

    Φ_ω,c(x)
      =
      (σ+1)(4ω-c^2)/
            2√(ω)(cosh(σ√(4ω-c^2)x)-c/2√(ω))
            ^1/2σ.



The first result is the variational characterization lemma of the solitary waves.


Suppose ω,c∈^2 satisfies 4ω>c^2. Let dω,c, S_ω,c and K_ω,c be defined by (<ref>), (<ref>) and (<ref>) respectively. Then we have

    𝒢_ω,c=     {φ∈ H^1()
    ∖0:S_ω,c(φ)=  dω,c, K_ω,c(φ)=0 }
    
    =     {Q_ω,c(·-y)^ıθ : θ∈ [0,2π), y∈}.



  
    
  * By the Lagrange multiplier argument in <cit.>, we have
    S_ω,c'(Q_ω,c)=0, which implies that
    d'ω,c = MQ_ω,c,PQ_ω,c
    and
    
    d”ω,c
          =
          [ ∂_ωMQ_ω,c ∂_cMQ_ω,c; ∂_ωPQ_ω,c ∂_cPQ_ω,c;           ].


    
  * By the explicit formula of the solitary waves, the following nondegenerate condition
        
    [ d”ω,c]<0

       holds with σ∈1,2 and
        c∈ -2√(ω) ,2z_0√(ω),  see Theorem 4.3 in <cit.>. This nondegenerate condition is important to show the stability result of the solitary waves by the perturbation argument, the modulation stability and the energy method.
  


  
  Let σ, z_0 be as that in Theorem <ref>, and (ω, c)∈^2 with
  c∈ -2√(ω) ,2z_0√(ω).
  If ∈ H^1() satisfies the orthogonality conditions

    ı Q_ω,c
      =
      ∂_xQ_ω,c
      =
      Q_ω,c
      =
      ı∂_xQ_ω,c
      =
      0,

  then we have
  
    S_ω,c”Q_ω,c≥ C_abs_H^1^2,

where

    S_ω,c”(Q_ω,c) := T_ω,c + N_ω,c

with T_ω,c
  :=
  ∫_x^2+ω ^2
   -c_x , and 
    N_ω,c
        :=
           ∫[ Q_ω,c^2σ_x
          +
          σQ_ω,c^2σ-2(
                Q̅_ω,c∂_xQ_ω,c^2
                +
                Q_ω,c∂_xQ_ω,c^2
             )
            ].


 We follow the argument in <cit.> (see also <cit.>) and the references therein, and divide the proof into several steps.


  * Spectral distribution of S_ω,c”(Q_ω,c).
On one hand, by Hölder's inequality, we have

    T_ω,c
      =
         ∫∂_x^2+ı c ∂_x+ω^2
        
    
      =
         ∫∂_x^2+ı c ∂_x+c^2/4 ^2
        
       +
       ω-c^2/4∫^2
         
    ≥   ω-c^2/4∫^2,

which means that σ_T_ω,c⊂[ω-c^2/4 , ∞).
On the other hand, by the exponential decay of Q_ω,c and the similar argument of Proposition 2.9 in <cit.>, we know that the operator N_ω,c is relatively compact with respect to T_ω,c. By Weyl's theorem in <cit.>, we have

    σ_ S_ω,c”(Q_ω,c)  =  σ_T_ω,c⊂[ω-c^2/4 , ∞).



  * 

    We claim that for any φ∈ H^1∖0 with
    K'_ω,cQ_ω,cφ=0,
      
    S”_ω,cQ_ω,cφφ≥0.

    
In deed,  notice that K'_ω,cQ_ω,c≠ 0, we can choose ψ
   such that K'_ω,cQ_ω,cψ≠ 0. We now define
   for any φ∈ H^1∖0 with K'_ω,cQ_ω,cφ=0, 

    κm,s:=K_ω,cQ_ω,c +m ψ+sφ.

   Applying the Implicit Function Theorem to κm,s with
   κ0,0= K_ω,cQ_ω,c
   and
   .
        ∂_mκm,s|_m,s=0,0≠ 0
    yields that there exists δ>0 such that
    m:-δ,δ↦ is of class 𝒞^1 with m0=0,
    and
   
    κms, s=K_ω,cQ_ω,c +msψ+sφ≡0  for  s∈-δ,δ.

Differentiating on s, we have

    ṁ0∂_mκ0,0 + ∂_sκ0,0=0,

   which implies that
K'_ω,cQ_ω,cφ
        +
    ṁ0 K'_ω,cQ_ω,cψ=0.
Consequently we have
ṁ0=0.

Based on the above argument, we can define the function ι: -δ,δ↦
as following:

    ιs:=S_ω,c Q_ω,c+m sψ+sφ.

It means from Lemma <ref> and (<ref>) that
0 is a local minimum point of ι,
and implies that the function ι is convex around 0,
i.e.
  ι”0= S”_ω,c Q_ω,cφφ≥0.


  * 

S_ω,c”(Q_ω,c) has at least one negative eigenfunction.
For this purpose, we only need to show that there exists a function U in H^1 with
S_ω,c”(Q_ω,c) UU<0. Indeed, it follows from
K_ω,c(Q_ω,c)=0 and 4ω>c^2 that

    S”_ω,c(Q_ω,c)  Q_ω,c Q_ω,c
        =
          
        -
        2σ∫∂_xQ_ω,c^2 + ωQ_ω,c
        + ı c ∂_xQ_ω,cQ̅_ω,c
        <0.



  * 
S_ω,c”(Q_ω,c) has at most one-dimensional negative eigenspace.
We argue by contradiction. Suppose that there exist two linearly independent eigenfunctions
χ_1 and χ_2 of S_ω,c”(Q_ω,c). Since S_ω,c”(Q_ω,c) is a self-adjoint operator, without of generality, one may assume that
χ_1χ_2=0. It is easy to check that

    S_ω,c”(Q_ω,c) χ_1 χ_2 =0.

Moreover by the nonnegative property in <ref> and S_ω,c”(Q_ω,c) χ_1 χ_1 <0
and
S_ω,c”(Q_ω,c) χ_2 χ_2 <0
, we have

    K'_ω,cQ_ω,cχ_1≠0
    ,  and  K'_ω,cQ_ω,cχ_2≠0,

which implies that there exists ξ_0∈∖0 with χ_0=χ_1+ξ_0χ_2 such that

    K'_ω,cQ_ω,cχ_0 =0.

By the nonnegative property in <ref>, we have
S_ω,c”(Q_ω,c) χ_0 χ_0 ≥0,
which is in contradiction with

    S_ω,c”(Q_ω,c) χ_0 χ_0 
      =
           S_ω,c”(Q_ω,c) χ_1 χ_1 
        +
        ξ_0^2  S_ω,c”(Q_ω,c) χ_2 χ_2 
        <0.


  * 
    S_ω,c”(Q_ω,c) ={ı Q_ω,c, ∂_x Q_ω,c}.
It follows from Proposition 3.6 in
<cit.>.


  * 
Positivity of the quadratic form S_ω,c”(Q_ω,c).
In fact, we have


  For any ∈ H^1∖0 with (<ref>)
we have
S_ω,c”Q_ω,c>0.

 This is a consequence of
<ref>–<ref> and the standard spectral decomposition arguments for the quadratic form
S_ω,c”Q_ω,c.
In this proof, we will ignore the subscript ω and c for convenience and write S_ω,c”Q_ω,c and Q_ω,c
as S”Q and Q respectively.

First, we infer, from <ref>–<ref> together with (<ref>), that
the space H^1 can be decomposed as a direct sum of three subspaces:

    H^1= N⊕ K⊕ P ,

with
K:={ı Q , ∂_x Q },
P:=S”Q>0
and
N:=χ,
where χ is the L^2-normalized negative eigenfunction
corresponding to the negative eigenvalue -λ^2.
According to (<ref>), we can decompose any function ∈ H^1
satisfying (<ref>) into

    = κχ +

with ∈ P and κ = χ.

Now, we turn to the decomposition of some special functions related to the non-degenerate condition
d”ω,c <0.
On one hand, by (<ref>), we have

    S”Q∂_ωQ = - Q,  and 
      S”Q∂_cQ = -ı Q,

which implies that

    S”Q∂_ωQ∂_ωQ
      =
      -∂_ω MQ,
        S”Q∂_ωQ∂_cQ
      =
      -∂_c MQ,
      
        S”Q∂_cQ∂_ωQ
      =
      -∂_ω PQ,
        S”Q∂_cQ∂_cQ
      =
      -∂_c PQ,

and

    S”Q∂_ωQ =  S”Q∂_cQ=0.

On the other hand, the non-degenerate condition d”ω,c<0 implies that
there exists ξ=(ξ_1, ξ_2)∈^2 such that d”ω,cξξ<0,
which together with (<ref>), (<ref>)-(<ref>) and setting
U = ξ_1∂_ωQ + ξ_2∂_cQ yields that

    S”Q U  U 
      <0.

Using the decomposition (<ref>), we decompose the function U as following:

    U :=αχ + ζ + 𝚢,

with α = Uχ, ζ∈ S”(Q) and 𝚢∈ P.
From (<ref>), (<ref>)-(<ref>), we have

    0> S”Q U  U    
      =
      -α^2λ^2 +  S”Q𝚢𝚢,
    
    
      0= S”Q U    
      =
      -ακλ^2 +  S”Q𝚢.

Now inserting (<ref>) into
S”Q, and taking into account (<ref>)-(<ref>),
we obtain by the Cauchy–Schwarz inequality that

    S”Q
      =
        -κ^2λ^2
        +
        S”Q≥
      -κ^2λ^2
      +
      S”Q𝚢^2/S”Q𝚢𝚢
      >
      -κ^2λ^2
      +
      α^2κ^2λ^4 /α^2λ^2 
      =
      0.

This completes the proof.



By <ref> to <ref>,  the coercivity property of S_ω,c”(Q_ω,c) can be obtained by the argument in
  <cit.> and <cit.> (see also <cit.>). This concludes the proof of the proposition.
  



§ MODULATION ANALYSIS

Following the modulation analysis in <cit.> <cit.> (also <cit.>), we will show the geometrical decomposition of the solutions to (<ref>) close to the sum of two solitary waves with weak interactions. Now let (σ, z_0) be as in Theorem <ref>, (ω_j^0,
c_j^0)∈^2 be such that -2 √(ω_j^0) <c_j^0
<2z_0√(ω_j^0), j=1, 2, then by Theorem 4.3 in <cit.>, we have the non-degenerate condition
        
    d”ω^0_j,c^0_j  <0,  for  j=1, 2.

Let α < α_0 be small
enough, and L>L_0 be large enough, where α_0, L_0 will be
determined later. We first consider the tube of size α in the energy space H^1()

    αω^0 c^0L
      :=
      u∈ H^1∖0inf_x_2-x_1>L,
    γ_1,γ_2∈ u-∑_j=1^2 Q_ω_j^0,c_j^0·-x_j^ıγ_j_H^1<α

with
ω^0= ω_1^0 , ω_2^0 and 𝐜^0= c_1^0 , c_2^0 .
We denote
Q_j^0=Q_ω_j^0,c_j^0, Q_j=Q_ω_j,c_j for convenience,
and let
ω, 𝐜, 𝐱 and γ
be the vectors ω_1, ω_2,
c_1, c_2, x_1, x_2
and γ_1, γ_2 respectively.

By the Implicit Function Theorem, we have

There exist L_ large enough, α_ small enough,
such that for any L>L_ , α<α_, if u∈αω^0 c^0L,
then there exist unique 𝒞^1 functions ω, 𝐜, 𝐱, γ such that the following decomposition holds:

    ux=∑_j=1^2 Q_jx-x_j^ıγ_j + x ,

with -2 √(ω_j) <c_j
<2z_0√(ω_j), j=1,2 and

    R_j =ı∂_xR_j=ı R_j=∂_xR_j=0,
         j=1,2,

where
R_jx=Q_jx-x_j^ıγ_j.
Moreover, we have

    _H^1+∑^2_j=1(ω_j-ω_j^0  +  c_j-c_j^0 )
      <C_α,
         j=1,2,
    
       x_2-x_1 >L/2.

and

    1/2<√( 4ω_j-c_j^2 )/√( 4ω_j^0-c_j^0^2 )<2,   j=1,2,



First of all, by the definition of  αω^0 c^0L,
there exist 𝐱^0:=x_1^0, x_2^0 ∈^2 with x_2^0 -x_1^0≥ L
and γ^0:=γ_1^0,γ_2^0 ∈^2 such that

    u-∑_j=1^2 Q_ω_j^0,c_j^0·-x_j^0^ıγ_j^0_H^1<α.

Let  = ω, 𝐜, 𝐱, γ and

    ^0 =    ω^0, 𝐜^0, 𝐱^0, γ^0 ,  𝐐^0 (x)=    ∑_j=1^2 Q_ω_j^0,c_j^0x-x_j^0^ıγ_j^0.

For any u with (<ref>) and , we define

    x; , u :=ux-∑_j=1^2 Q_jx-x_j^ıγ_j.

It is easy to see that

    εx; ^0, 𝐐^0
        ≡ 0.



Defining
P, u
    :=
    ϱ_1^1, ϱ_1^2, ϱ_1^3, ϱ_1^4, ϱ_2^1, ϱ_2^2, ϱ_2^3, ϱ_2^4
    , u
by
  
    ϱ_j^1,u 
            :=· ;,u  Q_j·-x_j^ıγ_j,
       ϱ_j^2,u
            :=· ;,u ı∂_xQ_j·-x_j^ıγ_j,
        
       ϱ_j^3,u
            :=· ;,u ıQ_j·-x_j^ıγ_j,
       ϱ_j^4,u 
            :=· ;,u  ∂_x Q_j·-x_j^ıγ_j,

 where k=1, 2.  By simple calculations, we have
  
    ∂/∂ω_j= -∂_ω_jQ_jx-x_j^ıγ_j,
             ∂/∂ c_j=-∂_c_jQ_jx-x_j^ıγ_j,
    ∂/∂ x_j=  ∂_xQ_jx-x_j^ıγ_j,
             ∂/∂γ_j=  -ıQ_jx-x_j^ıγ_j,

and
  
    ∫𝒬_1^0 x-x_1^0 ^ıγ_1^0 𝒬_2^0 x-x_2^0 ^ıγ_2^0≤ C_abs^-2θ_1 L ,

  where
  θ_1=min √(4ω_1^0-c_1^0^2)/8 , √(4ω_2^0-c_2^0^2)/8 ,
  and
  𝒬_j^0 denotes one of
  Q_j^0, ∂_xQ_j^0, .∂_ω_jQ_j|_=^0,
  and
  .∂_c_jQ_j|_=^0.
Inserting (<ref>) and (<ref>) into ϱ_j^k, we obtain

  
    ∂ϱ_j^1/∂ω_k^0,𝐐^0
        =
        
          -∂/∂ω_k^0 MQ_k^0,    j=k,
          
    ^-2θ_1 L,    j≠ k,
           ∂ϱ_j^1/∂ c_k^0,𝐐^0
        =
        
          -∂/∂ c_k^0 MQ_k^0,    j=k,
          
    ^-2θ_1 L    j≠ k,
        
       ∂ϱ_j^1/∂ x_k^0,𝐐^0
        =
        
          0,    j=k,
          
    ^ -2θ_1 L ,    j≠ k,
           ∂ϱ_j^1/∂γ_k^0,𝐐^0
        =
        
          0,    j=k,
          
    ^ -2θ_1 L ,    j≠ k,


  
    ∂ϱ_j^2/∂ω_k^0,𝐐^0
        =
        
          -∂/∂ω_k^0 PQ_k^0,    j=k,
          
    ^-2θ_1 L,    j≠ k,
           ∂ϱ_j^2/∂ c_k^0,𝐐^0
        =
        
          -∂/∂ c_k^0 PQ_k^0,    j=k,
          
    ^-2θ_1 L,    j≠ k,
        
       ∂ϱ_j^2/∂ x_k^0,𝐐^0
        =
        
          0,    j=k,
          
    ^ -2θ_1 L ,    j≠ k,
           ∂ϱ_j^2/∂γ_k^0,𝐐^0
        =
        
          0,    j=k,
          
    ^ -2θ_1 L ,    j≠ k,


  
    ∂ϱ_j^3/∂ω_k^0,𝐐^0
        =
        
          -∫∂/∂ω_k^0Q_k^0Q̅_k^0,    j=k,
          
    ^-2θ_2 L,    j≠ k,
              ∂ϱ_j^3/∂ c_k^0,𝐐^0
        =
        
          -∫∂/∂ c_k^0Q_k^0Q̅_k^0,    j=k,
          
    ^-2θ_2 L,    j≠ k,
        
       ∂ϱ_j^3/∂ x_k^0,𝐐^0
        =
        
          -2PQ_k^0 ,    j=k,
          
    ^ -2θ_2 L ,    j≠ k,
              ∂ϱ_j^3/∂γ_k^0,𝐐^0
        =
        
          -2M Q_k^0 ,    j=k,
          
    ^ -2θ_2 L ,    j≠ k,


  
    ∂ϱ_j^4/∂ω_k^0,𝐐^0
        =
        
          -∫∂/∂ω_k^0 Q_k^0∂_x Q̅_k^0,    j=k,
          
    ^-2θ_2 L,    j≠ k,
              ∂ϱ_j^4/∂ c_k^0,𝐐^0
        =
        
          -∫∂/∂ c_k^0 Q_k^0∂_x Q̅_k^0,    j=k,
          
    ^-2θ_2 L,    j≠ k,
        
       ∂ϱ_j^4/∂ x_k^0,𝐐^0
        =
        ∂_x Q_k^0_2^2,    j=k,
          
    ^ -2θ_2 L ,    j≠ k,
              ∂ϱ_j^4/∂γ_k^0,𝐐^0
        =
        
          2P Q_k^0 ,    j=k,
          
    ^ -2θ_2 L ,    j≠ k.


Hence we can decompose the Jacobian
.D P/D|_,u=^0,𝐐^0
into four 4× 4 submatrices,

    D P/D^0,𝐐^0
        =
       . [ D P_1,1/D D P_1,2/D; D P_2,1/D D P_2,2/D ]|_,u=^0,𝐐^0

where

    .D P_k,k/D     |_,u=^0,𝐐^0
      =
      [           -∂/∂ω_k^0 MQ_k^0          -∂/∂ c_k^0 MQ_k^0                          0                          0;           -∂/∂ω_k^0 PQ_k^0          -∂/∂ c_k^0 PQ_k^0                          0                          0;       ∫∂/∂ω_k^0Q_k^0Q̅_k^0      ∫∂/∂ c_k^0Q_k^0Q̅_k^0                   -2PQ_k^0                  -2M Q_k^0;  ∫∂/∂ω_k^0 Q_k^0∂_x Q̅_k^0 ∫∂/∂ c_k^0 Q_k^0∂_x Q̅_k^0              ∂_x Q_k^0_2^2                   2P Q_k^0 ].

By simple calculations, we have

    .D P_k,k/D     |_,u=^0,𝐐^0
      =
        d” ω_k^0 , c_k^0 × 2MQ_k^0∂_x Q_k^0_L^2^2 - 4[PQ_k^0 ]^2,

and

    .D P_j,k/D     |_,u=^0,𝐐^0
      =^ -2θ_1 L ,
      for  j≠ k.

Putting together, we obtain

    D P/D^0,𝐐^0
      =
      ∏_k=1^2{
        d”ω_k^0,c_k^0× 2MQ_k^0∂_x Q_k^0_L^2^2 - 4[PQ_k^0 ]^2}
      +
      ^ -2θ_1 L .

The fact that

    2MQ_k^0∂_x Q_k^0_L^2^2 - 4[PQ_k^0 ]^2>0

in Appendix A, together with the non-degenerate condition (<ref>) implies that

    D P/D^0,𝐐^0>0

for sufficiently large L. We can conclude the proof by the Implicit Function Theorem.




Let L_ and α_ be given by
Lemma <ref>. If u∈𝒞 [0,T^∗],H^1 is a solution to
(<ref>) with u0∈αω^0 c^0L, and

    ut∈αω^0 c^0L/2,
          for any   t∈(0,T^∗],

where α <α_ and L>2L_,
then there exist unqiue 𝒞^1 functions

    t:=ωt,𝐜t,𝐱t,γt
      :
      [0,T^∗]↦^8

with -2√(ω_j(t))<c_j(t)<2z_0√(ω_j(t)) for all t∈ [0, T^*], j=1,2 , such that

    (t) R_j(t) 
      =
      (t)ı∂_xR_j(t) 
      =
      (t)ı R_jt= (t)∂_xR_jt
      =0,

where R_jt,x=Q_ω_jt,c_jtx-x_jt^ıγ_jt, j=1,2, and

    t,x=ut,x-∑_j=1^2 R_jt,x.

Moreover, for t∈ [0, T^*], we have

    t_H^1+∑^2_j=1(ω_jt-ω_j^0  +  c_jt-c_j^0 )
      <C_α,
    
    1/2<√( 4ω_jt-c_jt^2 )/√( 4ω_j^0-c_j^0^2 )<2,
    
    ω̇_kt
        +
        ċ_kt+
        ẋ_kt-c_kt
        +
        γ̇_kt-ω_kt≤
        C_abst_H^1 +^ -θ_2  L+θ_2 t ,
    
    
        x_2t-x_1t>1/2 L + θ_2 t ,

where θ_2=min √(4ω_1^0-c_1^0^2)/8 , √(4ω_2^0-c_2^0^2)/8  , c_2^0-c_1^0 .


First, since ut∈αω^0 c^0L/2  for any t∈(0, T^∗], there exist 𝐱 ^0t
and
γ^0t such that

    ut-∑_j=1^2 Q_ω_j^0,c_j^0·-x_j^0t^ıγ_j^0t_H^1<α, with   x_2^0t- x_1^0t≥L/2 .

By Lemma <ref>,  we have the decomposition (<ref>) with
the estimates (<ref>) and (<ref>). Moreover, by the proof of Lemma <ref>,
we can obtain the estimate on 𝐱(𝐭), i.e.

    x_j(t)-x_j^0(t) <C_α,

which together with  x_2^0(t) - x_1^0(t) ≥ L/2 implies that

    x_2(t)-x_1(t)
      >L/4.

for sufficiently small α and sufficiently large L.


Now, we turn to the proof of (<ref>).
The rigorous calculations for (<ref>) can be obtained by
Lemma 4 in <cit.>. Here, we only give the formally calculations.
On one hand, by the equation (<ref>) and the decomposition (<ref>), we have

    0=   ı∂_t +∂_xx
        - ∑_k=1^2ıẋ_k-c_k ∂_xR_k
        - ∑_k=1^2γ̇_k-ω_k R_k
        + ∑_k=1^2ıω̇_k∂_ω_kR_k
        
       
        + ∑_k=1^2ıċ_k∂_c_kR_k
        + ı∑_k=1^2 R_k + ^2σ∂_x∑_k=1^2 R_k + 
        - ∑_k=1^2ıR_k^2σ∂_xR_k
        
    
        =   ı∂_t +∂_xx
        - ∑_k=1^2ıẋ_k-c_k ∂_xR_k
        - ∑_k=1^2γ̇_k-ω_k R_k
        + ∑_k=1^2ıω̇_k∂_ω_kR_k
        
       
        + ∑_k=1^2ıċ_k∂_c_kR_k
        + ℛ_1ℛ_2
            +  + ∂_x,

where we used

    ı∂_tR_k + ∂_xxR_k
        =
           
        -ıR_k^2σ∂_xR_k - ω_k R_k + ıω̇_k∂_ω_kR_k
        
       
        + ıċ_k ∂_c_kR_k
        -ıẋ_k-c_k ∂_xR_k
        -γ̇_k -ω_k  R_k,

and + ℛ _1 + ℛ _2 ≲ 1
with ℛ_k is one of R_k and ∂_xR_k,
Then, by (<ref>) and the orthogonal condition (<ref>), we have

    ω̇_k (t) + ċ_k(t)  + ẋ_k(t)-c_k(t)  + γ̇_k(t)-ω_k (t)≤
         C_abs(_H^1 + ^ -2θ_2x_1(t)-x_2(t)),

where we used the fact:
  
    ∫ℛ_1ℛ_2≤ C_abs∫^ -√( 4ω_1(t)-c_1(t)^2 )/2x-x_1(t)^ -√( 4ω_2(t)-c_2(t)^2 )/2x-x_2(t)≤ C_abs^ -2θ_2x_1(t)-x_2(t).

Inserting (<ref>) into (<ref>),
we obtain the following "rough" estimate

    ω̇_k (t) + ċ_k(t)  + ẋ_k(t)-c_k(t)  + γ̇_k(t)-ω_k (t)≤
         C_abs(_H^1 + ^ -θ_2/2L).

On the other hand, combining (<ref>)
with (<ref>), we have
  
    ẋ_2(t)-ẋ_1(t)
        =   ẋ_2(t) - c_2(t)  - ẋ_1(t) - c_1(t)  +  c_2(t)-c_2^0  -  c_1(t)-c_1^0  +  c_2^0-c_1^0 
    ≥    c_2^0-c_1^0  - C_absα - C_abs^-θ_2/4 L
    ≥   1/2 c_2^0-c_1^0 ,

then integrating (<ref>), we obtain
  
    x_2(t)-x_1(t)
        ≥
        x_2(0)-x_1(0) + 1/2∫_0^t c_2^0-c_1^0 ṣ≥L/2+1/2 c_2^0-c_1^0 t,

which implies that
  
    ω̇_k (t) + ċ_k(t)  + ẋ_k(t)-c_k(t)  + γ̇_k(t)-ω_k (t)≤
        C_abs( _H^1 + ^ -θ_2 L+θ_2 t ).

This concludes the proof.




§ MONOTONICITY FORMULA



In <cit.>, under the non-degenerate condition

    d”ω, c <0,
Miao, Tang and Xu obtained the orbital stability of the single solitary wave of the equation (<ref>) with σ=1 in H^1() by the conservation laws of the energy, mass and momentum, these conservation laws were used to refine the estimates about the radiation term (t) and parameters variance ω(t)-ω(0) + c(t)-c(0).
In this section, because of the multi-dimension of parameters ω, 𝐜 in dealing with the multi-solitary waves, we will introduce the analogue monotonicity formulas as those in <cit.>,<cit.> instead of the conservation laws to refine the estimates (<ref>)
and (<ref>) about the radiation term (t) and parameters variance ω_k(t)-ω_k(0) and c_k(t)-c_k(0), k=1, 2. Those monotonicity formulas are  related to the localized mass and momentum.

We first give a Virial type identity.

Let g:↦ be a 𝒞^3 real-valued function such that
g', g” and g”' are bounded.
If u∈𝒞 [0, T^∗], H^1 is a solution of (<ref>),
then, for all t∈[0, T^*], we have

    /ṭ∫u^2 g
            =   
            2∫u̅u_x g' +1/σ∫u^2σ+2 g'.
            
    
            -/ṭ∫u̅u_x g
            =
               
            -2∫u_x^2 g'
            - ∫u^2σu̅u_x g'
            +1/2∫u^2 g”'.



It follows from simple computations.


Now suppose ω∈, c∈, x̅^0∈, μ∈, and a>0, then by Lemma <ref>, we have for any t∈ [0, T^*],
    
    /ṭω/2∫u^2 gx-x̅^0 -μ t /√(t+a)
            -
             c/2∫u̅u_x gx-x̅^0 -μ t /√(t+a)
    
          =   1/√(t+a)∫
                        -  cu_x^2
                        + ω + μ c / 2 u̅u_x
                        - μω/2u^2
                         g'x-x̅^0 -μ t /√(t+a)
        +  1 /t+a∫ c / 4 u̅u_x  - ω/4u^2
                Λ gx-x̅^0 -μ t /√(t+a)
            +  c / 4 (t+a)^3/2∫u^2 g”'x-x̅^0 -μ t /√(t+a)
       
            -  c/ 2 √(t+a)∫u^2σu̅u_x g'x-x̅^0 -μ t /√(t+a)
            + ω/ 2σ+1√(t+a)∫u^2σ+2 g'x-x̅^0 -μ t /√(t+a),

where Λ g x:=xg'x .

Now let
 
    x̅^0 = x^0_1 + x^0_2/2,  μ = 2ω_20-ω_10/c_20-c_10,

 and define the following functional

    𝒥_sumt
      =
         ω_10/2∫u(t,x)^2 φ -  x-x̅^0-μ t/√(t+a)
             -
              c_10/2∫u̅u_x φ -  x-x̅^0-μ t/√(t+a)
       
        +
             ω_20/2∫u(t,x)^2 φ x-x̅^0-μ t/√(t+a)
             -
              c_20/2∫u̅u_x φ x-x̅^0-μ t/√(t+a),

which is used to capture the localized mass and momentum around each solitary waves.  According to the weak interactions between the solitary waves, we have the following monotonicity properties.




 §.§ Monotone result for the line x̅^0 +μ t



  Let a=L^2/64 and u∈𝒞[0, T^∗], H^1
  be a solution satisfying the assumption of Lemma <ref>.
  Then, there exists C_abs such that
  
    /ṭ𝒥_sumt≤C_abs/(t+a)^3/2t_H^1^2 +   C_abs/(t+a)^3/2^-θ_3 L+θ_3 t ,

  where
    θ_3=min √(4ω_1^0-c_1^0^2)/64 , √(4ω_2^0-c_2^0^2)/64 , 4μ -c_1^0 , 4c_2^0- μ .

We rewrite 𝒥_sumt
as the following identity

    𝒥_sumt
      =
         ω_10/2∫u(t,x)^2
             -
              c_10/2∫u̅u_xt,x
        +
        𝒥t

where

    𝒥t =     ω_20-ω_10/2∫u(t,x)^2 φx-x̅^0 -μ t /√(t+a)
        - c_20-c_10/2∫u̅u_xt,x φx-x̅^0 -μ t /√(t+a) .

 By the conservation of mass and momentum, it suffices to show

  Let a, θ_3 and u be as those in Proposition <ref>.
  Then, there exists C_abs such that
  
    /ṭ𝒥t≤C_abs/(t+a)^3/2t_H^1^2 +
                C_abs/(t+a)^3/2^-θ_3 L+θ_3 t .

Moreover, we have

    𝒥t - 𝒥0⩽C_abs/ L sup_0<s<ts_H^1^2
        +
        C_abs^-θ_3L.


Before the proof of Proposition <ref>, we fist give the following estimate.


    Let Ω_w:=x∈ x-x̅^0-μ t <√(t+a), then for any 2≤ p<∞, we have
    
    ∫_Ω_wu^p
          ≤
          C_abs^-θ_3 L+θ_3 t + C_abst_H^1^p
          ≤1/32,

    where
    θ_3 is given by Proposition <ref>.


By Lemma <ref>, the solution ut can be decomposed as
 
    ut= ∑_j=1^2 Q_j·-x_jt^ıγ_jt + t,

and (<ref>)-(<ref>) hold. Then we have

    ∫_Ω_wu^p
      ≤∑_j=1^2 C_abs∫_Ω_wQ_j·-x_jt^p
      +
      C_abs∫_Ω_w^p.

We first estimate the contribution from Q_1. If x∈Ω_w, we obtain x-x^0-μ
t<√(t+a)⩽√(t) + L/8, and

    x-x_1t   
    =
         x-x^0-μ t  - x_1t -x^0-μ t 
       ⩾ x_1t -x^0-μ t  -  x-x^0-μ t 
       ⩾ x_1t -x^0-μ t  - √(t) - L/8.

By (<ref>) and (<ref>), we have for
sufficiently small α and sufficiently large L that

    d/dtx^0+μ t - x_1t    =
        μ - ẋ_1t - c_1t - c_1t
       ⩾μ - C_abs(C_α +
                    e^ -θ_2  L + θ_2 t ) - c_1^0 - C_ α
       ⩾μ - c_1^0/2,

and so,

    x^0+μ t - x_1t⩾x^0 - x_10 + μ - c_1^0/2 t
    ⩾L/4 + μ - c_1^0/2 t.

Now inserting the above estimate into (<ref>),
 we obtain for sufficiently small α and sufficiently large L that

    x-x_1t   ⩾x^0  +μ t - x_1t   - √(t) -
         L/8
       ⩾L/16 + μ - c_1^0/4 t
        +
        μ - c_1^0/4 t - √(t) + L/16
       ⩾L/16 + μ - c_1^0/4 t.

Then, it follows from the explicit expression of Q_ω,c that

    ∫_Ω_w Q_1·-x_1t^p
      ≤ C_abs^-θ_3 L+θ_3 t ,

where we used the fact that p≥ 2. By the similar argument, we have

    ∫_Ω_w Q_2·-x_2t^p
      ≤ C_abs^-θ_3 L+θ_3 t .

By (<ref>) and (<ref>) and the Sobolev inequality, we have

    ∫_Ω_wu^p
        ≤   
        C_abs^-θ_3 L+θ_3 t  + C_abst_H^1^p.

This concludes the proof.

Now, let us prove Proposition <ref>.

First of all, let

    vt,x:=ut,x^-ı1 / 2 μ x,
 then we have

    -u_x^2 +   μu̅u_x - μ^2/4u^2
      =
        1/4v̅v_x,
    
    
        -1/2u^2σu̅u_x 
      =
       -1/2v^2σv̅v_x  -μ/4 v ^2σ+2 .

Simple calculations yield that

    1/c_2(0)-c_1(0)/ṭ𝒥t
    
      =      -1/√(t+a)∫v_x^2φ' x-x̅^0-μ t/√(t+a)
            +1/ 4(t+a)^3/2∫v^2 φ”' x-x̅^0-μ t/√(t+a)
       
            + 1/4t+a∫v̅v_x Λφ x-x̅^0-μ t/√(t+a)
       
            -1/2√(t+a)∫v^2σv̅v_x φ' x-x̅^0-μ t/√(t+a)
       
            -σμ/4√(t+a)∫v^2σ+2 φ' x-x̅^0-μ t/√(t+a).




Next, we estimate (<ref>)-(<ref>) separately.

  Estimate for (<ref>). The definition of Λφ immediately implies that
  Λφ≤φ' , which together with the Cauchy-Schwarz inequality, yields that
    
    1/4t+a∫v̅v_x Λφ x-x̅^0-μ t/√(t+a)
    ≤   1/4t+a√(∫v_x^2 φ' x-x̅^0-μ t/√(t+a)·∫v^2 φ' x-x̅^0-μ t/√(t+a))
    ≤   1/4√(t+a)∫v_x^2 φ' x-x̅^0-μ t/√(t+a)
            +
            1/4(t+a)^3/2∫v^2 φ' x-x̅^0-μ t/√(t+a)


    Estimate for (<ref>).
    Applying the Cauchy-Schwarz inequality, we have
    
    1/2√(t+a)∫v^2σv̅v_x φ' x-x̅^0-μ t/√(t+a)
    ≤   1/2√(t+a)√(∫v^4σ+2 φ' x-x̅^0-μ t/√(t+a)·∫v_x^2  φ' x-x̅^0-μ t/√(t+a))
    ≤   1/4√(t+a)∫v_x^2  φ' x-x̅^0-μ t/√(t+a)
          +
          1/√(t+a)∫v^4σ+2 φ' x-x̅^0-μ t/√(t+a).

    By the Hölder inequality, we have
    
    ∫v^4σ+2 φ' x-x̅^0-μ t/√(t+a)≤v^2√(φ' x-x̅^0-μ t/√(t+a))_L^∞^2
                ∫_φ'v^4σ-2.

    By the  Sobolev inequality in Lemma 5.2 in <cit.> and Lemma <ref>, we have
    
    v^2√(φ'( x-x̅^0-μ t/√(t+a)) )_L^∞^2
              ≤   
                8∫v_x^2 φ'∫_φ'u^2
                +
                1/2(t+a)∫v^2φ”^2 /φ' ∫_φ'v^2
              
    ≤   
                  8∫_φ'v^2    ∫v_x^2 φ'
                +
                  1/2(t+a)∫_φ'v^2
                ∫v^2φ”^2 /φ' 
    ≤      8∫_φ'v^2    ∫v_x^2 φ'
                +1/2(t+a)∫_φ'v^2 ^2
                
    ≤   1/4∫v_x^2 φ'
                + C_abs/t+a,

    which implies that
    
    1/√(t+a)∫v^4σ+2 φ' x-x̅^0-μ t/√(t+a)
    ≤1/4√(t+a)∫v_x^2 φ' x-x̅^0-μ t/√(t+a) + C_abs/ (t+a)^3/2^-θ_3 L+θ_3 t + t_H^1^4σ-2

    Now inserting the above estimate into (<ref>), we have
    
    1/2√(t+a)∫v^2σv̅v_x φ' x-x̅^0-μ t/√(t+a)
    ≤1/2√(t+a)∫v_x^2  φ' x-x̅^0-μ t/√(t+a)
          +
          C_abs/ (t+a)^3/2^-θ_3 L+θ_3 t + t_H^1^4σ-2.



    Estimate for (<ref>). By μ >0, φ'>0, we have
    
    -σμ/4√(t+a)∫v^2σ+2 φ' x-x̅^0-μ t/√(t+a)≤ 0.


Inserting (<ref>), (<ref>) and (<ref>) into (<ref>), we can obtain the result.




 §.§ Monotone result for different lines


Now let 
    θ_4:=min √(4ω_1^0-c_1^0^2)/64, √(4ω_2^0-c_2^0^2)/64, μ-2c_1^0, 4c_2^0-2μ ≤θ_3,


    μ_+,0= μ_0,-=4ω_2(0)-ω_1(0)/c_2(0)-c_1(0),  μ_-,0=μ_0,+=ω_2(0)-ω_1(0)/c_2(0)-c_1(0),



    ϕ_±,0t,x:=φx-x̅^0 -μ_±,0 t /√(t+a),
             ϕ_0,±t,x:=φx-x̅^0 -μ_0,± t /√(t+a),

    and define

    𝒥_+,0(t)    = (ω_2(0)-ω_1(0))∫u(t,x)^2 ϕ_+,0(t,x) - c_2(0)-c_1(0)/2∫u̅u_x ϕ_+,0(t,x),
      
    𝒥_-,0(t)    = (ω_2(0)-ω_1(0))/4∫u(t,x)^2 ϕ_-,0(t,x) - c_2(0)-c_1(0)/2∫u̅u_x ϕ_-,0(t,x),
      
    𝒥_0,+(t)    = (ω_2(0)-ω_1(0))/2∫u(t,x)^2 ϕ_0,+(t,x) - (c_2(0)-c_1(0))∫u̅u_x ϕ_0,+(t,x),
      
    𝒥_0,-(t)    = (ω_2(0)-ω_1(0))/2∫u(t,x)^2 ϕ_0,-(t,x) - c_2(0)-c_1(0)/4∫u̅u_x ϕ_0,-(t,x).

By the analogue proof as that in Proposition <ref>, we have


  Let u∈𝒞[0,T^*], H^1
  be a solution satisfying the assumption of Lemma <ref>.
  Then, there exists C_abs such that
  
    𝒥_±,0t-𝒥_±,00   ⩽C_abs/ L sup_0<s<ts_H^1^2
            +
            C_abs^-θ_4L,
          
    𝒥_0,±t-𝒥_0,±0   ⩽C_abs/ L sup_0<s<ts_H^1^2
            +
            C_abs^-θ_4L.





§ PROOF OF THEOREM <REF>

Let σ∈1,2 and z_0=z_0(σ)∈0,1 satisfy F(z_0;σ) =0,
    where F(z;σ) is defined by (<ref>). Let ω^0_k and c^0_k satisfy the assumptions in Theorem
<ref>. Let α_0 be defined by Lemma <ref>, and
A_0>2, δ_0=δ_0(A_0), L_0=L_0(A_0) be chosen later.
Suppose that ut is the solution of (<ref>) with initial
data
u_0∈αω^0 c^0L,
and define

    T^∗
      :=
      sup t≥ 0 sup_τ∈[0,t]inf_x_2^0-x_1^0 > L/2
    γ^0_1, γ^0_2∈ u(τ,·)-
                ∑_j=1^2 Q_ω_j^0,c_j^0·-x_j^0^ıγ_j^0_H^1≤  A_0δ + ^-θ_0/2 L,

where

    θ_0=min√(4ω_1^0-c_1^0^2)/128,
         √(4ω_2^0-c_2^0^2)/128,
         c_2^0-c_1^0,
         2c_2^0- 2μ,
         μ-2c_1^0
      .

By the continuity of u(t) in H^1, we know that T^*>0. In order
to prove Theorem <ref>, it suffices to show
T^*=+∞ for some A_0>2, δ_0>0, and L_0. We argue with contradiction. Suppose that T^*<+∞, we know
that for any t∈ [0, T^*], there exist (x^0_k(t),
γ^0_k(t))∈^2, k=1, 2 such that x^0_2(t)≥
x^0_1(t)+L/2 and

    u(t,·)-∑^2_k=1Q_ω^0_k,c^0_k·-x^0_k(t)e^ıγ^0_k(t)≤ A_0(δ +e^-θ_0
          L/2).






  * Decomposition of ut.
    Let L_0>0 be
determined by Lemma <ref>, and L_2, L_3 be
determined by Proposition <ref> and Corollary
<ref>, and choose δ_0>0 small enough and L_0
large enough, such that for δ<δ_0 and
L>L_0(A_0)>max 2L_0, L_2, L_3,

    A_0 δ + e^ - θ_6L/2<α_0.


By Lemma <ref> and Lemma <ref>, we have
    
    ut, x
            =
            t, x
            +
            ∑_j=1^2R_jt, x

  where
    R_jt, x = Q_ω_jt,c_jt x-x_jt^ıγ_jt, and the orthogonality
    
    (t) R_j
          =
          (t)ı∂_x R_j
          =
          (t)ı R_j
          =
          (t) ∂_x R_j=0,

    hold for any t∈[ 0 , T^∗ ]. Moreover, we have
    
    t_H^1+∑^2_j=1(ω_jt-ω_j^0  +  c_jt-c_j^0 )
          <C_ A_0δ + ^-θ_0 L/2,
          
    ω̇_kt
            +
            ċ_kt+
            ẋ_kt-c_kt
            +
            γ̇_kt-ω_kt≤
            C_abst_H^1 +^ -θ_0  L+θ_0 t 
    
          x_2t-x_1t>1/2 L + θ_0 t .

  In particular, we have
     
    (0)_H^1+∑^2_j=1(ω_j0-ω_j^0  +  c_j0-c_j^0 )
          <C_δ.



     Refined estimate on _H^1^2 and 𝒥t - 𝒥0. In order to do so, we first introduce the functional

    t:=Eut + _sumut.

  and expand it as following
    
    
    
    t
             =
                ∑_k=1^2S_ω_k0,c_k0 R_k0
             +
             ℋtt
       
             +
            ∑_k=1^2(ω_kt-ω_k0^2 + c_kt-c_k0^2) 
       
             +
             t_H^1^2t_H^1
             +
             ^-θ_0L + θ_0 t .

     where
     
    ℋtt
           =
              1/2∫_xt^2
           +
           ω_1t/2∫t^2 1-ϕ
           +
           ω_2t/2∫t^2 ϕ
       
           +
           c_1t/2∫_x 1-ϕ
           +
           c_2t/2∫_x ϕ
           +
           1/2
           Nt,

  and
     
    N
      =
                 ∑_k=1^2∫R_k^2σ_x
              +
              σ∑_k=1^2∫R_k^2σ-2R̅_k∂_xR_k^2 + R_k∂_xR_k^2 .


 Please refer to the proof in Appendix A.


  
    ℋ≥κ_H^1^2.




Please refer to Lemma 6.2 in <cit.> and Lemma 6.2 in <cit.>.



    By Lemma <ref>, we have for all t∈[0, T^∗],
        
    t
                =
                   ∑_k=1^2S_ω_k0,c_k0 R_k0
                +
                ℋtt
       
                +
               ∑_k=1^2(ω_kt-ω_k0^2 + c_kt-c_k0^2) 
       
                +
                t_H^1^2t_H^1
                +
                ^-θ_0L + θ_0 t .

   In particular, we have
        
    0
                =
                   ∑_k=1^2S_ω_k0,c_k0 R_k0
                +
                ℋ00
                +
                0_H^1^20_H^1
                +
                ^-θ_0 L ,

        which implies that
        
    ∑_k=1^2S_ω_k0,c_k0 R_k0
                =
                   0
                -
                ℋ00
                +
                0_H^1^20_H^1
                +
                ^-θ_0 L .

        Inserting (<ref>) into (<ref>), we obtain by  Lemma <ref> and the conservation laws of mass, momentum and energy that
        
    κt_H^1^2
              
    ≤   ℋtt
    
              =
                 t - ∑_k=1^2S_ω_k0,c_k0 R_k0
              +
             ∑_k=1^2( ω_kt-ω_k0^2 + c_kt-c_k0^2) 
       
              +
              t_H^1^2t_H^1
              +
              ^-θ_0L + θ_0 t 
    
              =
                 t - 0
              +
              ℋ00
              +
              0_H^1^20_H^1
              +
              ^-θ_0 L 
       
              +
             ∑_k=1^2(ω_kt-ω_k0^2 + c_kt-c_k0^2 ) 
              +
              ^-θ_0L + θ_0 t 
    
              =
                 _sumt - _sum0
              +
              ℋ00
              +
              0_H^1^20_H^1
              +
              ^-θ_0 L 
       
              +
    ∑_k=1^2(ω_kt-ω_k0^2 + c_kt-c_k0^2 )
              +
              ^-θ_0L + θ_0 t .

    By Proposition <ref>, we obtain
        
    t_H^1^2
              ≤   C_abs/ L sup_0<s<ts_H^1^2
              +
              C_abs^-θ_0 L 
              +
              C_abs0_H^1^2
       
              +
           C_abs∑_k=1^2( ω_kt-ω_k0^2 + c_kt-c_k0^2 ).

Moreover, by (<ref>), we have
    
    𝒥 0 
            -
          𝒥 t 
    
          =
             𝒥_sum 0 
            -
          𝒥_sum t 
    
          =
             ℋ00
          -
          ℋtt
          +
          0_H^1^20_H^1
          +
          ^-θ_0 L 
       
          +
    ∑_k=1^2(ω_kt-ω_k0^2 + c_kt-c_k0^2 )
                +
                ^-θ_0L + θ_0 t 
    ≤   
            C_abs0_H^1^2
            +
            C_abs^-θ_0 L 
          +
                  C_abs∑_k=1^2ω_kt-ω_k0^2
                +
                c_kt-c_k0^2
                ,

which together with (<ref>) implies that

    𝒥t - 𝒥0≤   C_abs/ L sup_0≤ s≤ ts_H^1^2
        +
        C_abs^-θ_0 L 
        +
        C_abs0_H^1^2 
       
      +
      ∑_k=1^2
        C_absω_kt-ω_k0^2
            +
            c_kt-c_k0^2
            .



  * Refined estimates of ω_kt-ω_k0 and c_kt-c_k0.
Recall that

    ϕt,x=φx-x̅^0 -μ  t /√(t+a),  ϕ_±,0t,x=φx-x̅^0 -μ_±,0 t /√(t+a),
        ϕ_0,±t,x=φx-x̅^0 -μ_0,± t /√(t+a),

we have


  
    ∫u(t,x)^2 ϕ(t,x) - ∫R_2t^2 ≤
        C_abs^-θ_0 L + θ_0 t  + C_abs_L^2^2.
        
    ∫u(t,x)^2 1-ϕ(t,x) - ∫R_1t^2 ≤
        C_abs^-θ_0 L + θ_0 t  + C_abs_L^2^2.
        
    ∫u̅u_x ϕ(t,x) - ∫R̅_2∂_xR_2t≤
        C_abs^-θ_0 L + θ_0 t  + C_abs_H^1^2.
        
    ∫u̅u_x 1-ϕ(t,x) - ∫R̅_1∂_xR_1t≤
        C_abs^-θ_0 L + θ_0 t  + C_abs_H^1^2.



By the definition of ϕ and the exponential decay estimate of R_k, it is easy to check that
  
    ∫u(t,x)^2 ϕ(t,x) - ∫R_2^2 
    
        =
           ∫R_1+R_2+^2 ϕ - ∫R_2^2 
    
        =
           ∫R_1^2 ϕ - R_2^2 1-ϕ + ^2 ϕ
        +
        2[
            R̅_1R_2ϕ + R̅_1ϕ - 2R̅_21-ϕ]
        
    ≤   
        2∫R_1^2 ϕ
            +   R_2^2 1-ϕ
            +   R̅_1R_2
            +   ^2
        .

First, by inserting the estimate (<ref>) into (<ref>), from the definition of θ_0 in (<ref>), we have

    ∫R̅_1R_2
      <
      C_abs^-θ_0 L + θ_0 t .

Now, by
(<ref>) and (<ref>), we obtain

    /ṭ[ x̅^0+μ t -√(t+a) - x_1t]
       =
         μ - ẋ_1(t) - 1/2√(t+a)
    
      =
         μ - ẋ_1(t)-c_1(t)  -  c_1(t)-c_1^0  -c_1^0 - 1/2√(t+a)
    ≥   μ -c_1^0 - C_abst_H^1
        - C_abs^-θ_2 L+θ_2 t  - C_α
    ≥   μ -c_1^0 - C_absC_α
        - C_abs^-θ_2 L  - C_α
    ≥   1/2μ -c_1^0 ,
Integrating from 0 to t, we have

    x̅^0+μ t -√(t+a) - x_1t≥L/4 + 1/2μ -c_1^0  t.

In the similar way, we have

    x_2t - x̅^0+μ t + √(t+a)≥L/4 + 1/2 c_2^0 - μ t.

By (<ref>), (<ref>), the definition of ϕ and  the explicit expression of R_1 and R_2, we obtain

    ∫[ R_1^2 ϕ + R_2^2 1-ϕ]<C_abs^-θ_0 L + θ_0 t .

Inserting (<ref>) and (<ref>) into (<ref>), it is easy to check that (<ref>) holds. The estimates
(<ref>)-(<ref>)
can be proved in the similar way.




 
   
    ∫u(t,x)^2 ϕ - ϕ_0,-(t,x)
        +
        ∫u̅u_xϕ - ϕ_0,-≤
        C_abs^-θ_0 L + θ_0 t 
        +
        C_abs_H^1^2,
    
       ∫u(t,x)^2 ϕ_-,0 - ϕ(t,x)
      +
      ∫u̅u_xϕ_-,0 - ϕ≤
        C_abs^-θ_0 L + θ_0 t 
        +
        C_abs_H^1^2,
        
       ∫u(t,x)^2 ϕ - ϕ_+,0(t,x)
        +
        ∫u̅u_xϕ - ϕ_+,0≤
        C_abs^-θ_0 L + θ_0 t 
        +
        C_abs_H^1^2,
        
       ∫u(t,x)^2 ϕ_0,+ - ϕ(t,x)
        +
        ∫u̅u_xϕ_0,+ - ϕ≤
        C_abs^-θ_0 L + θ_0 t 
        +
    C_abs_H^1^2.


We only give the proof of (<ref>). The estimates (<ref>)-(<ref>) can be shown in the similar way. Now by the definition of ϕ and ϕ_0,-, it is easy to check that for any time t>0

    ϕ - ϕ_0,-(t,·)⊂ x̅^0 + μ/2t - √(t+a) , x̅^0 + μ t + √(t+a) .

Then, it follows from (<ref>) that,

    ∫u(t,x)^2 ϕ - ϕ_0,-(t,x)
      ≤   
      C_abs∫_ϕ - ϕ_0,-[
        R_1^2 + R_2^2 + ^2
      ].

Now, by
(<ref>) and (<ref>), we obtain

    /ṭ[ x̅^0+μ/2t -√(t+a) - x_1t]
       =
         μ/2 - ẋ_1(t) - 1/2√(t+a)
    
      =
         μ/2 - ẋ_1(t)-c_1(t)  -  c_1(t)-c_1^0  -c_1^0 - 1/2√(t+a)
    ≥   μ/2 -c_1^0 - C_abst_H^1
        - C_abs^-θ_2 L+θ_2 t  - C_α
    ≥   μ/2 -c_1^0 - C_absC_α
        - C_abs^-θ_2 L  - C_α
    ≥   1/4μ -2c_1^0 .

Integrating from 0 to t, we have

    x̅^0+μ/2 t -√(t+a) - x_1t≥L/4 + 1/4μ -c_1^0  t.

A similar argument implies that

    x_2t -
      [ x̅^0+ μ t + √(t+a)]
      ≥L/4 + 1/4μ -2c_1^0  t.

Hence, we have

    ∫u(t,x)^2 ϕ - ϕ_0,-(t,x)
    ≤   
      C_abs∫_ϕ - ϕ_0,-[
        R_1^2 + R_2^2 + ^2
      ].
      
    ≤   
      C_abs^-2θ_2x̅^0 + μ/2t - √(t+a) - x_1 (t)
      +
      C_abs^ -2θ_2 x_2(t) - x̅^0 - μ t - √(t+a)
      +
      C_abs_L^2^2
      
    ≤   
      C_abs^- θ_0  L + θ_0 t 
      +
      C_abs_L^2^2.

By a similar argument as above, it is not hard to see that

    ∫u̅u_xϕ - ϕ_0,-≤   
      C_abs^- θ_0  L + θ_0 t 
      +
      C_abs_H^1^2.

This gives (<ref>).



By Lemma <ref> and <ref>, we are able to show the following result.

  
    ∑_k=1^2M(R_k(t)) - M(R_k(0))
      +    ∑_k=1^2P(R_k(t)) - P(R_k(0))
    ≤   C_abs/ L sup_0<s<ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L 
       
           +
          C_abs∑_k=1^2ω_kt-ω_k0^2
                 +
                 c_kt-c_k0^2
                 .


    
On one hand, from the expression of 𝒥_+,0t and 𝒥t, we have
         
    𝒥_+,0t - 𝒥t
              -    ω_20-ω_10/2∫R_2t^2 
    
             =
                 ω_20-ω_10/2∫u^2 ϕ - ∫R_2t^2 
       
              +
              ω_20-ω_10∫u^2 ϕ_+,0 - ϕ
       
              -
              c_20-c_10/2∫u̅u_xϕ_+,0 - ϕ
              .

    Combining (<ref>) with (<ref>), we have for any t∈ [0, T^*]
        
    𝒥_+,0t - 𝒥t - ω_20-ω_10/2∫R_2t^2 ≤
              C_abst_H^1^2 + C_abs^-θ_0L + θ_0 t .

Thus,
        
    -𝒥_+,0t - 𝒥t - ω_20-ω_10/2∫R_2t^2    ≤
              C_abst_H^1^2 + C_abs^-θ_0L + θ_0 t ,
              
    𝒥_+,00 - 𝒥0 - ω_20-ω_10/2∫R_20^2
                 ≤
              C_abs0_H^1^2 + C_abs^-θ_0 L,

        which implies that
        
    ω_20-ω_10/2[ ∫R_2t^2 - ∫R_20^2 ]
              +
                𝒥t - 𝒥0
              -
                𝒥_+,0t -𝒥_+,00
    ≤   
              C_abs0_H^1^2
              +
              C_abst_H^1^2
              +
              C_abs^-θ_0 L,

   which together with (<ref>) and (<ref>) implies that
        
    ω_20-ω_10/2[
                ∫R_2t^2 - ∫R_20^2
              ]
              
    ≤   𝒥t - 𝒥0
                +
              [ 𝒥_+,0t -𝒥_+,00]
              +
              C_abs0_H^1^2
              +
              C_abst_H^1^2
              +
              C_abs^-θ_0 L
    ≤   C_abs/ L sup_0<s<ts_H^1^2
                +
                C_abs0_H^1^2
                +
                C_abst_H^1^2
                +
                C_abs^-θ_0 L 
       
              +    C_abs∑_k=1^2[
                    ω_kt-ω_k0^2
                    +
                    c_kt-c_k0^2
                ]
              
    ≤   C_abs/ L sup_0<s<ts_H^1^2
                +
                C_abs0_H^1^2
                +
                C_abs^-θ_0 L 
       
              +  C_abs∑_k=1^2[
                    ω_kt-ω_k0^2
                    +
                    c_kt-c_k0^2
                ].

On the other hand, we have
        
    𝒥t - 𝒥_-,0t -    ω_20-ω_10/4∫R_2t^2
              
    
             =
                 ω_20-ω_10/4∫u^2 ϕ - ∫R_2t^2 
       
              +
              ω_20-ω_10/4∫u^2 ϕ - ϕ_-,0
       
              -
              c_20-c_10/2∫u̅u_x ϕ - ϕ_-,0.

Thus, by (<ref>) and (<ref>), we have for any t∈ [0, T^*]
        
    𝒥t - 𝒥_-,0t - ω_20-ω_10/4∫R_2t^2
              ≤
              C_abst_H^1^2 + C_abs^-θ_0L + θ_0 t ,

        which implies that,
        
    𝒥t - 𝒥_-,0t - ω_20-ω_10/4∫R_2t^2
              ≤
              C_abst_H^1^2 + C_abs^-θ_0L + θ_0 t ,
              
    
              -𝒥0 - 𝒥_-,00 - ω_20-ω_10/4∫R_20^2 ≤
              C_abs0_H^1^2 + C_abs^-θ_0 L.

        Therefore,
        
    ω_20-ω_10/4[
                    ∫R_20^2 - ∫R_2t^2
                 ]
              +
                𝒥t - 𝒥0
              -
                𝒥_-,0t -𝒥_-,00
    ≤   
              C_abssup_0≤ s≤ t0_H^1^2
              + C_abs^-θ_0 L,

   which together with (<ref>) implies that
        
    ω_20-ω_10/4[
                    ∫R_20^2 - ∫R_2t^2
                 ]
              
    ≤   𝒥t - 𝒥0
                +
              𝒥_-,0t -𝒥_-,00
              +
                C_abs0_H^1^2
              + C_abst_H^1^2
              + C_abs^-θ_0 L
    ≤   C_abs/ L sup_0<s<ts_H^1
                +
                C_abs0_H^1^2
                +
                C_abs^-θ_0 L 
       
              +   C_abs∑_k=1^2[
                    ω_kt-ω_k0^2
                    +
                    c_kt-c_k0^2
                ].

Combining (<ref>) with (<ref>), we have

    ∫R_2t^2 - ∫R_20^2 ≤   C_abs/ L sup_0<s<ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L 
       
           +    C_abs∑_k=1^2ω_kt-ω_k0^2
                 +
                 c_kt-c_k0^2
                 .

Similar argument implies that

    ∫R̅_2∂_xR_2t
            -
            ∫R̅_2∂_xR_20≤   C_abs/ L sup_0<s<ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L 
       
           +   C_abs∑_k=1^2ω_kt-ω_k0^2
                 +
                 c_kt-c_k0^2
                 .



This concludes the estimates of the solitary wave R_2. In order to obtain the estimates of the solitary wave R_1, we will make use of the conservation laws of mass and momentum and the orthogonality condition (<ref>).
Firstly, by the mass conservation and the orthogonality condition (<ref>),

    ∫R_1t+R_2t+t^2
        -
        ∫R_10+R_20+0^2
      
    
      =
         ∑_k=1^2∫R_kt^2-∫R_k0^2
        +
        2∫R̅_1R_2t - R̅_1R_20
        +
        ∫t^2 - 0^2 
    ≥   ∑_k=1^2∫R_kt^2-∫R_k0^2
        -
        2∫R̅_1R_2t
            +
            R̅_1R_20
        -
        ∫t^2 + 0^2 ,

which together with (<ref>) implies that

    ∑_k=1^2∫R_kt^2-∫R_k0^2≤
      C_abs^-θ_0 L
      +
      C_abs0_L^2^2 + t_L^2^2.

Secondly, by (<ref>) and (<ref>), we have

    ∫R_1t^2-∫R_10^2
      ≤   ∫R_2t^2-∫R_20^2
      
      +
      C_abs^-θ_0 L
      +
      C_abssup_0≤ s≤ ts_L^2^2
      
    ≤   C_abs/ L sup_0<s<ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L 
       
           + C_abs∑_k=1^2ω_kt-ω_k0^2
                 +
                 c_kt-c_k0^2

In a similar way, we have

    ∫R̅_1t∂_xR_1t
        -
        ∫R̅_10∂_xR_10
    ≤   C_abs/ L sup_0<s<ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L 
       
           +   C_abs∑_k=1^2ω_kt-ω_k0^2
                 +
                 c_kt-c_k0^2
                 .

This ends the proof.


Next, by the nondegenerate condition [ d”ω_k^0,c_k^0 ]<0, for k=1,2,  we have for sufficiently small α and sufficiently large L,

    [ d”ω_k0,c_k0]
      <
      1/2[ d”ω_k^0,c_k^0]<0.

By the smallness of ω_kt-ω_k0 and c_kt-c_k0, we have the following expression,
    
    [  M(R_k(t))- M(R_k(0)); P(R_k(t)) - P(R_k(0)) ]
            =
               
            d”ω_k0, c_k 0 [ ω_kt-ω_k0; c_kt-c_k0 ]
       
            +
            ω_kt-ω_k0^2 +  c_kt-c_k0^2 ,

    it follows that, for k=1,2,
    
    ω_kt-ω_k0
          +
           c_kt-c_k0
    
          <
             
          C_abs∫R_kt^2-∫R_k0^2 
            +
          C_abs∫R̅_k∂_xR_kt
                -
                ∫R̅_k∂_xR_k0.

By Lemma <ref>, we have

    ω_kt-ω_k0
      +
       c_kt-c_k0
      <
                C_abs/ L sup_0≤ s≤ ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L 
       
           +
             C_abs∑_k=1^2ω_kt-ω_k0^2
                 +
                 c_kt-c_k0^2
                          ,

which together with the smallness of  ω_kt-ω_k0 and c_kt-c_k0 implies

    ω_kt-ω_k0
      +
       c_kt-c_k0
      <
                C_abs/ L sup_0≤ s≤ ts_H^1^2
             +
             C_abs0_H^1^2
             +
             C_abs^-θ_0 L .




  *  Conclusion.  By (<ref>) and (<ref>), we obtain

    t_H^1^2
        ≤   C_abs/ L sup_0≤ s≤ ts_H^1^2
        +
        C_abs0_H^1^2
        +
        C_abs^-θ_0 L 
       
      +
       C_abs/L^2sup_0≤ s≤ ts_H^1^4
      +
      C_abs ^-2θ_0 L  + 0_H^1^20_H^1
    ≤   C_abs/ L sup_0≤ s≤ ts_H^1^2
        +
        C_abs0_H^1^2
        +
        C_abs^-θ_0 L .

Taking L is large enough, we have

    sup_0≤ s≤ ts_H^1^2
        ≤
        C_abs0_H^1^2
        +
        C_abs^-θ_0 L ,

which together with (<ref>) implies that

    sup_0≤ s≤ ts^2_H^1 + ∑^2_k=1[ ω_kt-ω_k0
      +
       c_kt-c_k0]
        ≤
        C_abs0^2_H^1
        +
        C_abs^-θ_0  L .

Therefore, we have

    inf_ x_2^0-x_1^0>L/2,
    γ_1^0, γ_2^0∈ ut,·-∑^2_k=1Q_ω_k^0,c_k^0·-x_k^0^ıγ_k^0
    ⩽    ut,·-∑^2_k=1Q_ω_k^0,c_k^0·-x_kt^ıγ_kt
    ⩽    ut, ·-∑^2_k=1Q_ω_kt,c_kt·-x_kt^ıγ_kt
        +
        C_abs∑^2_k=1(ω_k(t)-ω_k^0+c_k(t)-c_k^0)
      
    ⩽   εt
        +
        C_abs∑^2_k=1(ω_kt-ω_k0 + c_kt-c_k0+ω_k0-ω_k^0+ c_k0-c_k^0)
        
    ⩽    
      C_absε0
        +∑^2_k=1(ω_k0-ω_k^0+ c_k0-c_k^0) + C_abs e^-θ_0 L/2
    ⩽       C_absC_(δ + e^-θ_0 L/2).

By choosing A_0>2C_absC_, we obtain a contradiction with the definition of T^∗. Thus, T^∗=∞. This concludes the proof.




§ APPENDIX A


By the explicit expression of the solitary wave, we have

    ∂_xQ_ω,c
      =
      ∂_xΦ_ω,c
        +
        ıc/2Φ_ω,c
        -
        ı1/2σ+2Φ_ω,c^2σ+1expı{c/2x-1/2σ +
          2∫^x_-∞Φ_ω,c^2σ(y)ỵ}.

Hence we have

    ∫Q_ω,c^2 = ∫Φ_ω,c^2,

and

    ∫∂_xQ_ω,c^2
      =
         ∫∂_xΦ_ω,c^2
      +
      ∫c/2Φ_ω,c - 1/2σ+2Φ_ω,c^2σ+1^2 
    
      =
         ∫∂_xΦ_ω,c^2
      +
      c^2/4∫Φ_ω,c^2
      +
      1/2σ+2^2∫Φ_ω,c^4σ+2
      -
      c/2σ+2∫Φ_ω,c^2σ+2,

and

    ∫Q̅_ω,c∂_xQ_ω,c
      =
         ∫Φ_ω,c∂_xΦ_ω,c
        +
        ıc/2Φ_ω,c
        -
        ı1/2σ+2Φ_ω,c^2σ+1
    
      =
         ∫c/2Φ_ω,c^2 - 1/2σ+2Φ_ω,c^2σ+2.

Therefore,
 we have

    2MQ_ω,c∂_x Q_ω,c_L^2^2 - 4[PQ_ω,c]^2
      
    
      =
         ∫Φ_ω,c^2
      ∫∂_xΦ_ω,c^2
          +
          c^2/4∫Φ_ω,c^2
          +
          1/2σ+2^2∫Φ_ω,c^4σ+2
          -
          c/2σ+2∫Φ_ω,c^2σ+2
       
      -
      ∫c/2Φ_ω,c^2 - 1/2σ+2Φ_ω,c^2σ+2^2
      
    
      =
         ∫Φ_ω,c^2∫∂_xΦ_ω,c^2
      +
      1/2σ+2^2∫Φ_ω,c^2∫Φ_ω,c^4σ+2
      -
      1/2σ+2^2∫Φ_ω,c^2σ+2^2
      
    ≥   ∫Φ_ω,c^2∫∂_xΦ_ω,c^2>0.

where we use the explicit expression (<ref>) of Φ_ω,c in the last inequality.



§ APPENDIX B


In this appendix, we prove Lemma <ref>.
First note that

    t
      =
         
      Eut + _sumut
    
      =
         1/2∫ u_xt^2
      +
      1/2(σ+1)∫_-∞^∞
      |u|^2σu̅u_x
      
       
      +
        ω_10/2∫u^2 ϕ
             -
              c_10/2∫u̅u_x ϕ
       
        +
        ω_20/2∫u^2 1-ϕ
             -
              c_20/2∫u̅u_x 1-ϕ.

Now we will expand the right hand side of the above equality.

 Term 1/2∫ u_xt^2 :
By the weak interaction (<ref>) between the solitary waves, we have

    ∫ u_xt^2
      =
         ∫∂_xR_1 + ∂_xR_2 + ∂_x^2
      
    
      =
         ∫∂_xR_1 ^2 + ∂_xR_2 ^2 + ∂_x^2
      
      +
      2∫∂_xR̅_1∂_xR_2
      
       
      -
      2∫∂_x,xR_1 + ∂_x,xR_2
    
      =
         ∫∂_xR_1 ^2 + ∂_xR_2 ^2 + ∂_x^2
      
    
      =
         ∫∂_xR_1 ^2 + ∂_xR_2 ^2 + ∂_x^2
      
       
      -
      2∫∂_x,xR_1 + ∂_x,xR_2
      +
      ^ -θ_0 L + θ_0 t



    
        Term ω_20/2∫u^2ϕ
    :
 By (<ref>) and the definition of ϕ,
the simple calculations give

    ω_20/2∫u^2ϕ
      =
         ω_20/2∫R_1+R_2+^2ϕ
    
      =
         ω_20/2∫R_1^2ϕ
        + R_2^2ϕ
        + ^2ϕ
        + ω_20∫
            R_1R̅_2ϕ
            +
            R_1ϕ
            +
            R_2ϕ
    
      =
         ω_20/2∫R_2^2
            +
            2 R_2
            +
            ^2ϕ
        +
       ω_20/2∫R_1^2ϕ
        +
        R_2^2ϕ - 1 
       
        + ω_20∫
            R_1R̅_2ϕ
            +
            R_1ϕ
            +
            R_2ϕ - 1 
    
       =
          ω_20/2∫R_2^2
            +
            2 R_2
            +
            ^2ϕ
        +
        ^ -θ_0 L + θ_0 t 
        +
        _L^2^2_L^2
    
       =
          ω_20/2∫R_2^2
      +
      ω_2t/2∫^2ϕ
      +
      ω_20∫ R_2
       
        +
        ^ -θ_0 L + θ_0 t 
        +
        _L^2^2_L^2
        +
        ω_2t-ω_20^2 
    
       =
          ω_20/2∫R_2^2
      +
      ω_2t/2∫^2ϕ
      +
      ω_2t∫ R_2
       
        +
        ^ -θ_0 L + θ_0 t 
        +
        _L^2^2_L^2
        +
        ω_2t-ω_20^2 
        ,

where we used the orthogonality condition R_2=0 in the last equality,

In the similar way, we can obtain


    
        Term
        ω_10/2∫u^21-ϕ
    :


    ω_10/2∫u^2 1-ϕ
       =
          ω_10/2∫R_1^2
      +
      ω_1t/2∫^2 1-ϕ
      +
      ω_1t∫ R_1
       
        +
        ^ -θ_0 L + θ_0 t 
        +
        _L^2^2_L^2
        +
        ω_1t-ω_10^2 
        .



    
        Term
        c_20/2∫u̅u_xϕ
    :


    c_20/2∫u̅u_x ϕ
       =
           c_20/2∫R̅_2∂_xR_2
      +
       c_2t/2∫_x ϕ
      +
      c_2t∫ R_2
       
        +
        ^ -θ_0 L + θ_0 t 
        +
        _H^1^2_H^1
        +
         c_2t-c_20^2 
        .



    
        Term
        c_10/2∫u̅u_x1-ϕ
    :


    c_10/2∫u̅u_x1-ϕ
       =
           c_10/2∫R̅_1∂_xR_1
      +
       c_1t/2∫_x ϕ
      +
      c_1t∫ R_1
       +
        ^ -θ_0 L + θ_0 t 
        +
        _H^1^2_H^1
        +
         c_1t-c_10^2 
        .


Term Nu:=1/2(σ+1)∫ |u|^2σu̅u_x: In order to expand it, we introduce the following cut-off functions around each solitary waves,

    g_1t,x :=
      
        1,     x<x_1 + 1/16 L + θ_2 t, 
    
        0,     x>x_1 + 1/8 L + θ_2 t,
      
    
      g_2t,x :=
      
        1,     x>x_2 - 1/16 L + θ_2 t, 
    
        0,     x<x_2 - 1/8 L + θ_2 t,
      
    g̃ := 1-g_1-g_2

where θ_2 is given by Lemma <ref>.
Now, we decompose Nu as following,

    Nu=N_1u + Ñu + N_2u,

where

    N_1u = 1/2(σ+1)∫ |u|^2σu̅u_x g_1,
      
    
      N_2u = 1/2(σ+1)∫ |u|^2σu̅u_x g_2,
      
    Ñu = Nu-N_1u-N_2u.

Note that
R_2t,x <C_abs^-4θ_0x-x_2(t), we have

    ∫R_2(t,x) g_1(t,x)  + ∫R_2(t,x) 1-g_2(t,x) =^ -θ_0 L+θ_0 t ,

which implies that

    N_1u
      =
         1/2(σ+1)∫ R_1 + R_1 + ^2σR̅_1 + R̅_2 + ∂_xR_1 + ∂_xR_2 + _x g_1 
    
      =
         
      N_1 R_1 
      +
      N_1'R_1 R_2+
      +
      1/2N_1”R_1R_2+R_2+
       
      +
      ∫R_2^3g_1    + _H^1^2_H^1,

where

    N_1' R_1  R_2+
      =
         1/2(σ+1)∫σR_1^2σ-2R̅_1^2∂_xR_1 R_2+g_1
      
       
      +
      1/2(σ+1)∫σ+1R_1^2σ∂_xR_1R̅_2+g_1
      
       
      +
      1/2(σ+1)∫R_1^2σR̅_1∂_xR_2+_x g_1,

and

    N_1” R_1  R_2+ R_2+
      =
         1/2(σ+1)∫2σR_1^2σ-2R̅_1^2 R_2+∂_xR_2 +_x g_1
      
       
      +
      1/2(σ+1)∫2σ+1R_1^2σR̅_2+∂_xR_2 +_x g_1
      
       
      +
      1/2(σ+1)∫ 2σσ+1R_1^2σ-2R̅_1∂_xR_1 R_2+^2 g_1
      
       
      +
      1/2(σ+1)∫σσ-1R_1^2σ-4R̅_1^3∂_xR_1 R_2 +^2 g_1
      
       
      +
      1/2(σ+1)∫σσ+1R_1^2σ-2R_1∂_xR_1R̅_2 +^2 g_1.

Therefore, the decay estimates (<ref>) implies that

    N_1' R_1  R_2+
      =
      N_1' R_1 
      +
      ^ -θ_0 L+θ_0 t 
     N_1” R_1  R_2+ R_2+
      =
       N_1” R_1 
      +
      ^ -θ_0 L+θ_0 t 
      .

Now, note that
R_1t,x <C_abs^-2θ_2x-x_1(t), we have

    ∫R_1  g_2  + ∫R_1 1-g_1 =^ -θ_0 L+θ_0 t ,

which yields that

    N R_1  - N_1 R_1 
      =
      ^ -θ_0 L+θ_0 t ,
      
    N_1' R_1 -N' R_1 
      =
      ^ -θ_0 L+θ_0 t ,
      
     N_1” R_1  -  N” R_1 
      =
      ^ -θ_0 L+θ_0 t .

Inserting (<ref>)-(<ref>) and (<ref>)-(<ref>) into (<ref>), we obtain that

    N_1u
      =
      NR_1 + N' R_1  + 1/2 N” R_1 
      +
      ^ -θ_0 L+θ_0 t .

In the similar way, we have

    N_2u
      =
      NR_2 + N' R_2  + 1/2 N” R_2 
      +
      ^ -θ_0 L+θ_0 t .

As for the term Ñu, we have

    Ñu
      =
         1/2(σ+1)∫ |u|^2σu̅u_x 1-g_1-g_2
    ≤   
      C_absu_L^∞^2σ∫u_x^2 1-g_1-g_2 ∫u^2 1-g_1-g_2^1/2.

which together with the smallness of _H^1, (<ref>) and (<ref>) implies that

    Ñu
       =
       ^ -θ_0 L+θ_0 t 
       +_H^1^2_H^1.

Combining (<ref>) with (<ref>) and (<ref>), we have

    Nu
      =
         ∑_k=1^2
      NR_k
      +
      ∑_k=1^2N' R_k 
      +
      ∑_k=1^21/2 N” R_k 
       
      +
      ^ -θ_0 L+θ_0 t 
       +_H^1^2_H^1.

Summing up (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we finish the proof.









10

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