The limit behavior of the solutions to a class of nonlinear dispersive wave equations

J. Math. Anal. Appl. 341 (2008) 1311–1333 www.elsevier.com/locate/jmaa

The limit behavior of the solutions to a class of nonlinear dispersive wave equations Lixin Tian a,∗ , Guilong Gui b , Boling Guo c a Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Jiangsu 212013, PR China b Department of Mathematics, Faculty of Science, Jiangsu University, Jiangsu 212013, PR China c Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China

Received 10 August 2006 Available online 22 November 2007 Submitted by R.E. Showalter

Abstract In this paper, we study the limit behavior of the solutions to a class of nonlinear dispersive wave equations. We also demonstrate that the solutions to Eq. (1.1) converge to the solution to the corresponding BBM equation as the parameter γ converges to zero. And the convergence of solutions to Eq. (1.1) as α 2 → 0 is studied in H s (R), s > 32 . Given the discussion of the parameters in the nonlinear dispersive wave equation (1.1), one will obtain some conditions in which compacton and peakon occur. © 2007 Elsevier Inc. All rights reserved. Keywords: Nonlinear dispersive wave equation; Bi-Hamiltonian structure; Well-posedness; Weak and strong limit; Peakon; Compacton

1. Introduction The nonlinear dispersive wave equation  ut − α 2 uxxt + kux + 3uux + βuxxx = γ α 2 (2ux uxx + uuxxx ), (1.1) u(0, x) = u0 (x), t > 0, x ∈ R, has recently been investigated with regard to well-posedness and singularity formation by Yin [1]. He also showed the existence of smooth solitary waves for certain values of the parameters k and γ . For k = 0, Eq. (1.1) was found by Dai and Huo [2] as a model for nonlinear waves in cylindrical hyperelastic rods with u(t, x) representing the radial stretch related to a pre-stressed state. The study of special compressible materials leads to values of γ ranging from −29.476 to 3.417. In this case, the equation is not a long-wave model, but rather a narrow-banded spectrum approximation. In [3], the dynamic stability of solitary wave solutions of (1.1) was considered by H. Kalisch. It shows that for all γ , the equation admits smooth stable solitary waves. When γ < 1, all solitary waves are stable. For γ = 1, β = 0 the equation is known as the Camassa–Holm equation ut − α 2 uxxt + kux + 3uux = α 2 (2ux uxx + uuxxx ). * Corresponding author.

E-mail addresses: [email protected] (L. Tian), [email protected] (G. Gui). 0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.11.030

(1.2)

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It was first found by Fokas and Fuchssteiner [4], and later recovered as a water wave model by Camassa, Holm and Hyman in [5,6]. They show that for all k, (1.2) has a Lax pair formulation, and for k = 0, (1.2) has traveling wave solutions to the form ce−|x−ct| , which is called peakon because they have a discontinuous first derivative at the wave peak. For every k, Eq. (1.2) is bi-Hamiltonian and thus possesses an infinite number of conservation law. Moreover, Eq. (1.2) has a simple multi-peakon which shows many fantastic properties. Recently, Johnson has given an account of the role of the Camassa–Holm equation as a long-wave model for water waves in a long uniform channel [7]. Besides being a model equation for water waves, the Camassa–Holm equation is also interesting due to its integrable bi-Hamiltonian structure. This property has led to many interesting developments, a sample of which can be found in [4–6,8–11]. One aspect of the integrability of the equation in case γ = 1 is that the solitary wave solutions are soliton (see [5,12]), and it is similar to the solitary wave solutions to the Korteweg–de Vries equation. In [22], the existence of global weak solutions to the Camassa–Holm equation is proved by using the method of compensated compactness. G. Rodriguez-Blanco demonstrated the Cauchy problem for the Camassa–Holm equation by the method of T. Kato (see [23]). In [24], Tang and Yang obtained the general explicit expressions of the two wave solutions to Eq. (1.2) by using the bifurcation phase portraits of the traveling wave system. Yet another application of the Camassa–Holm equation arises in the context of differential geometry, where it can be seen as a re-expression for geodesic flow on an infinite-dimensional Lie group (see [13,14]). In [34–36], the authors study the solution property of peakon in generalized Dullin–Gottwald–Holm equation and generalized Degasperis–Procesi equation. When γ = 0, the diffeomorphism transformation u(t, x) →

1 u(t, x) γ

will formally simplify Eq. (1.1) to the following equation ut − α 2 uxxt + kux +

3 uux + βuxxx = α 2 (2ux uxx + uuxxx ). γ

Considering the above transformation, one can see that the solutions to Eq. (1.1) are related in form to the solutions to the corresponding Camassa–Holm equation (1.2) as γ → 1. When γ → 0, however, the transformation cannot be used to Eq. (1.1). But, for γ = β = 0, Eq. (1.1) is related in form to become the BBM equation as follows ut − α 2 uxxt + 2ωux + 3uux = 0, where k = 2ω. It is a well-known model for surface waves in a channel [15], and the solitary waves are smooth. Despite having a Hamiltonian structure, the equation is not integrable and its solitary waves are not solitons [16]. With α = 0 in Eq. (1.1) we find the well-known Korteweg–de Vries equation, which describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. u(x, t) represents the wave height above a flat bottom. x is proportional to distance in the direction of propagation and t is proportional to the time elapsed. The Cauchy problem of the KdV equation has been studied extensively (see [21,25–30]). As soon as u0 ∈ (H 1 R), the solution to the KdV equation is global (see [30]). The equation is integrable (see [31]) and its solitary waves are solitons (see [32]). The traveling wave of the equation is linearly unstable (see [33]). And in [25], J.L. Bona and R. Scott demonstrated KdV equation as the limit of BBM equation. On the basis of the above researches, this paper studies the Cauchy problem, the behavior of solutions to the Cauchy problem for the nonlinear dispersive wave equation (1.1) as the dispersive parameter γ tends to zero. Given some a priori estimates, the locally strong or weak limit of the solution to the nonlinear dispersive wave equation (1.1) as the dispersive parameter γ tends to zero is obtained. By using the method in [17], we can demonstrate that the solutions to the nonlinear dispersive wave equation (1.1) as α 2 tends to zero converges to the solution of corresponding KdV equation. Finally, we can obtain some condition in which compactons and peakons do occur after the discussion of the parameters in the nonlinear dispersive wave equation (1.1). In the paper, main theorems are Theorems 3.2, 4.6, 5.3 and 5.4. Notations. We shall use the standard notation |•|p for the norm of the space Lp (R), 1  p < ∞, i.e., |f |p =  p ( R |f | dx)1/p . The space L∞ = L∞ (R) consists of all essentially bounded, Lebesgue measurable functions with the standard norm   |f |∞ = |f |L∞ = inf sup f (x). m(e)=0 x∈R\e

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

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And we denote the norm in the Sobolev space H s = H s (R) by  1/2 2   2 s ˆ  s f (ξ ) dξ 1 + |ξ | f s = f H = R

for s ∈ R. Here fˆ(ξ ) is the Fourier transform of f (x). We also define the operator Λsα for any α, s ∈ R by the formula s  s    respectively Λs := Λs1 = 1 − ∂x2 2 , Λ = Λ11 Λsα = 1 − α 2 ∂x2 2 and denote •,• s as the inner product on H s . In the paper, Section 1 gives the introduction. Section 2 deals with some a priori estimates. In Section 3, we research on the strong and weak limit behavior of solutions to (1.1) as γ → 0 and get the result that the solution converges to the solution of BBM equation under H s , s > 3/2. In Section 4, we study the behavior of solution of (1.1) as parameter α tends to zero and have that solution of (1.1) converges to the solution of KdV equation as ε → 0 under H s , s > 3, where ε denotes α 2 . In Section 5, we obtain some conditions in which compacton and peakon occur. 2. A priori estimates x

1 −| α | We define p(x) = 2α e , x ∈ R, then (1 − α 2 ∂x2 )−1 f = p ∗ f for all f ∈ L2 (R), where ∗ denotes convolution. Using this identity, we can rewrite Eq. (1.1) as the following nonlocal form 

 β 3 − γ 2 γ α2 2 β (2.1) u + u + k + 2 u = 0. ut + γ uux − 2 ux + ∂x p ∗ 2 2 x α α

Eq. (1.1) can be written in Hamiltonian form and has the following invariants   1 1 2 2 2 E(u) = u + α ux dx, F (u) = u3 + γ uu2x + ku2 − βu2x dx. 2 2 R

R

Similar to the discussion of Proposition 2.1 in [17], one can see that Eq. (1.1) is suitable for applying Kato’s theory (see [18]). Lemma 2.1. Given u0 ∈ H s (R), s > 32 , there exists a maximal time T (α, ω, γ , u0 ) > 0, and a unique solution u to Eq. (1.1), such that     u = u(·, u0 ) ∈ C [0, T ); H s (R) ∩ C 1 [0, T ); H s−1 (R) . Moreover, the solution u to (1.1) depends continuously on the initial data u0 . According to the inequalities   u2 + u2x dx  u2 + α 2 u2x dx = 2E(u) R

R



2

u R

+ u2x

1 dx  2 α

 u2 + α 2 u2x dx = R

(α  1),

2 E(u) α2

(α  1),

the invariance of E(u) guarantees that all solutions to Eq. (1.1) are uniformly bounded as long as they exist. Considering the above, we can obtain the following results by applying the method of Theorem 2.2’s proof in [17]. Theorem 2.2. Given u0 ∈ H s (R), s > 32 , the solution u = u(·, u0 ) to Eq. (1.1) is uniformly bounded on [0, T ). Moreover, T < ∞ if and only if  lim inf inf γ ux (t, x) = −∞. t↑T

x∈R

For later estimation of Sobolev norms of solutions, we will require the following basic inequalities, due to Y.A. Li and P.J. Olver [19].

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Lemma 2.3. Given q  0, let u = u(x) ∈ H q be any function so ux L∞ < ∞. Then there is a constant cq depending only on q, such that the following inequalities hold:      Λq uΛq (uux ) dx   cq ux L∞ u2 q , (2.2) H   R

       Λq uΛq u2 dx   cq uL∞ u2 q . H  

(2.3)

R

Moreover, if u and f are functions in H q+1 ∩ {ux L∞ < ∞}, then      Λq uΛq (uf )x dx    R

 

cq f H q+1 u2H q , cq (f H q+1 uL∞ uH q + f H q ux L∞ uH q + fx L∞ u2H q ),

q ∈ ( 12 , 1], q ∈ (1, ∞).

(2.4)

Now we study norms of solution to (1.1) using energy estimates. Without loss of generality, we always assume the parameter α = 1 in the remainder of this section. The proof of the following theorem is motivated for Theorem 4.1 in [19]. Theorem 2.4. Suppose that for some s > 32 , the function u(x, t) is a solution of Eq. (1.1) corresponding to the initial data u0 ∈ H s . Then for any real number q ∈ (0, s − 1], there exists a constant c depending only on q, such that 

t u2H q

+ ux 2H q

 u0 2H q

+ u0x 2H q

+c

ux 

L∞

 q 2  q 2 Λ u + Λ ux dx dt.

(2.5)

R

0

For any q ∈ (1/2, s − 1] and any r ∈ (1/2, q], there exists a constant c depending only on r and q, such that t   u2H q

+ ux 2H q

 u0 2H q

+ u0x 2H q

+c 0

 r+1 2 Λ u dx

1/2  

R



2 2  Λq u + Λq ux dx dt.

(2.6)

R

Moreover, for any q ∈ (0, s − 1], there exists a constant c depending only on q, such that   ut H q  c uH 1 + 1 uH q+1 .

(2.7)

Proof. For any q ∈ (0, s − 1], applying (Λq u)Λq to both sides of Eq. (1.1) and integrating on R with respect to x again, one obtains the equation     q 2  q 2      1 d γ Λ u + Λ ux dx = −3 Λq uΛq (uux ) dx + Λq uΛq ∂x3 u2 + Λq ux Λq u2x dx 2 dt 2 R R R   q q = (−3 + γ ) Λ uΛ (uux ) dx − γ Λq+1 uΛq+1 (uux ) dx γ + 2



R

  Λ ux Λ u2x dx q

R

q

R

using integration by parts. It follows from the inequalities (2.2) and (2.3) that there is a constant cq so that  1 d 3 u2H q + ux 2H q  4cq ux L∞ u2H q + γ cq ux L∞ ux 2H q 2 dt 2      4cq ux L∞ u2H q + γ ux 2H q  4cq ux L∞ u2H q + ux 2H q .

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

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Integrating with respect to t on both sides of the above inequality leads to inequality (2.5). Applying the inequality 1/2   r+1 2 ux L∞  cr+1 uH r+1  cr+1 Λ u dx R

for r > to the right-hand side of (2.5) yields the estimate (2.6). For any q ∈ (0, s − 1], applying (Λq ut )Λq to both sides of Eq. (2.1)  3−γ 2 γ 2 ut = −γ uux + βux − ∂x p ∗ u + ux + (k + β)u 2 2 1 2

and integrating on R with respect to x again, one obtains the equation    1  q 2 q q Λ ut dx = −γ Λ ut Λ (uux ) dx − 2ω Λq ut Λq−2 (ux ) dx 2 R R R     2    3−γ γ q q−2 − ∂x u dx − Λ ut Λ Λq ut Λq−2 u2x x dx. 2 2 R

(2.8)

R

By using the inequalities (2.2), (2.3), one may obtain the following inequalities     2 1/2       2 q+1  Λq ut Λq (uux ) dx   Λq ut  2 1+ξ dξ u(ξ ˆ − η)u(η) ˆ dη   L R

R

R

 cq ut H q u ˆ L1 uH q+1  cq ut H q uH 1 uH q+1 . Using the above inequalities and integration by parts, we have                Λq ut Λq−2 ∂x u2 dx  =  Λq−1 ut Λq−1 ∂x u2 dx      R

R

   Λq−1 ut 



L2

 q−1 1 + ξ2 dξ



R

2 1/2 u(ξ ˆ − η)u(η) ˆ dη

R

 cq ut H q−1 u ˆ L1 uH q  cq ut H q−1 uH 1 uH q and

     2 1/2       2 q−1 γ Λq ut Λq−2 u2 dx   cq γ ut H q 1+ξ dξ uˆ x (ξ − η)uˆ x (η) dη x x   R

  cq γ

R

R

 −1 1 + ξ 2 ux 2L2 ux 2H q dξ

1/2

R

 cq γ ut H q uH 1 uH q+1 . And we also obtain the following inequalities by using integration by parts      Λq ut Λq−2 (ux ) dx   cq ut H q ux  q−2 . H   R

Applying the above six inequalities to (1.1) yields the inequality   ut 2H q  cq uH 1 + 1 uH q+1 ut H q , so,

  ut H q  cq uH 1 + 1 uH q+1 .

2

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Lemma 2.5. Let uε0 be the convolution uε0 = φε ∗ u0 of the functions φε (x) = ε −1/4 φ(ε −1/4 x) and u0 such that the )  0, and φ(ξ ) = 1 for any ξ ∈ (−1, 1), the following Fourier transform φˆ of φ satisfies φˆ ∈ C0∞ ([0, ∞); H ∞ ), φ(ξ estimates hold for any ε with 0 < ε < 1/4: uε0 H q  c uε0 H q  cε

(q  s), s−q 4

uε0 − u0 H q  cε

(2.9)

(q > s),

(2.10)

s−q 4

(2.11)

(q  s),

uε0 − u0 H s = o(1).

(2.12)

Combining the estimates in Lemma 2.3, we shall evaluate norms of the function uε in the following theorem, which will be used to show the convergence of {uε }. For convenience, we shall always omit ε in the solution uε of Eq. (1.1) in the remainder of the paper. Lemma 2.6. There exist constants c1 and c2 independent of ε such that the following inequalities hold 4Mr

u2H r 



1/2 (2 − cMr t)2

3 < r  s, 2

M , (2 − Mt)2

(2.13)

1/2

where Mr = u0 2H r , M = max{4Mr , cMr }, uH s 

c1 , (2 − Mt)c2

uH s+p

c1 ε − 4  , (2 − Mt)c2

(2.14)

p

p > 0,

(2.15)

and p+1

ut H s+p

c1 ε − 4  , (2 − Mt)c2

p > −1,

hold for any ε sufficiently small and t <

(2.16)

2 M.

Proof. Choose a fixed number r with 3/2 < r  s. It follows from (2.6) that 

 r 2 Λ u dx 

R



 r 2 Λ u0 dx + c

R

t   0

 r 2 Λ u dx

3/2 (2.17)

dt.

R

 Multiplying (2.17) with ( R (Λr u)2 dx)1/2 , we obtain the inequality 

 r 2 Λ u dx

3/2

 

R

 r 2 Λ u dx

1/2  

R

 r 2 Λ u0 dx + c

R

t   0

 r 2 Λ u dx

3/2

 dt .

(2.18)

R

Denote that  V (t) :=

 r 2 Λ u0 dx + c

R

and we have d V (t)  dt

t   0



 r 2 Λ u dx

R

from the inequality (2.17).

 r 2 Λ u dx

3/2 dt,

R

1/2 V (t)  V (t)

(2.19)

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

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It follows from Gronwall’s inequality and (2.17) that u2H r 

4Mr



1/2 (2 − cMr t)2

M , (2 − Mt)2

3/2 < r  s,

holds for any t ∈ [0, 2/M). Substituting the inequality (2.13) into (2.6) with q = s and u = uε , one obtains the estimate t u2H s

 u0 2H s

+c 0

√   s 2 M Λ u dx dt (2 − Mt) R

for any t ∈ [0, 2/M). It follows from Gronwall’s inequality and (2.9), (2.10) that there are constants c1 , c2 depending on Msuch that c1 uH s  . (2 − Mt)c2 According to the inequalities (2.9), (2.10), one can see that Ms = u0 2H s  c02  c1 , where c0 , c is independent of ε. So M, c1 , c2 are independent of ε. In a similar way, one may obtain uH s+p  the inequality

−

p

c1 ε 4 (2−Mt)c2 , p

> 0. And by using the inequalities (2.7), (2.15), we obtain

p+1

ut H s+p 

c1 ε − 4 , (2 − Mt)c2

p > −1.

2

Remark. According to the inequality (2.14) in Lemma 2.4, one can see that uH s is uniformly bounded and the boundedness is independent of ε. 3. The strong and weak limit behavior of solutions as γ → 0 In this section, we shall demonstrate the behavior of solutions of the Cauchy problem (1.1) as the parameter γ tends to zero. Firstly, we shall prove that {uε } is a Cauchy sequence in the space C([0, T ); H s (R)). We consider Eq. (1.1) as the following formula  3−γ 2 γ 2 ut + γ uux − βux + ∂x p ∗ u + ux + (k + β)u = 0, 2 2 where we assume α = 1. Let u = uε and v = uγ be solutions of (2.1), corresponding to the parameters γ = ε and γ , respectively, and let w = u − v and f = u + v. Then w satisfies the problem (γ − ε)  2  ε u x + (wf )x − βwx 2 2  ε − γ  2 3−ε γ − ε  2 ε u x+ (wf )x + ux x + (wx fx )x + (k + β)wx . = −p ∗ 2 2 2 2

wt +

(3.1)

Without loss of generality, we always assume 0 < ε  γ  2ε  1/8. Theorem 3.1. Given the initial data u0 ∈ H s (R) (s > 32 ). There exists T = T (u0 H s ) > 0, independent of ε, such that {uε } and {uεt } are Cauchy sequences in the space C([0, T ); H s (R)) and C([0, T ); H s−1 (R)), respectively. Proof. First, for a constant q with 1/2 < q  min{1, s − 1}, multiplying to Λ2q w both sides of Eq. (3.1) and integrating with respect to x again, we may obtain the equation

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1 d 2 dt





  q  q   ε  q  q Λ w Λ (uux ) dx − Λ w Λ (wf )x dx 2 R R     q  q−2 ε − γ  2  3−ε γ − ε  2 u x+ ux x + Λ w Λ (wf )x + 2 2 2 R ε dx. + (wx fx )x + (k + β)wx 2

 q 2 Λ w dx = (ε − γ )

R

Consider the following inequalities:    2    q    q  q     2 q+1  (ε − γ )   Λ w Λ (uux ) dx   ε Λ w L2 1+ξ dξ u(ξ ˆ − η)u(η) ˆ dη  R

R

R

 εuH q uH q+1 wH q  c1 εwH q ,     ε  q  q   ε    ε   Λ w Λ (wf )x dx   c3 wx L∞ + vH q w2H q + c3 vH q+1 + wH q wH q 2 2 2 R

    R

   

R

   

R

   

R



  ε  ε   c3 w 3 +β + vH q w2H q + c3 vH q+1 + wH q wH q , 2 2 2  H     q  q−2 ε − γ  2  dx   εuH q uH q−1 wH q−1 , u x Λ w Λ 2      q  q−2 3 − ε dx   cf H q w2H q , (wf )x Λ w Λ 2      q  q−2 γ − ε  2  dx   εuH q uH q+1 wH q , ux x Λ w Λ 2      q  q−2 ε dx   εc6 f H q w2H q , Λ w Λ (wx fx )x 2

 q  q−2  Λ w Λ (wx ) dx = 0,

R

and 1 d 2 dt

 R

 q 2    ε  Λ w dx  ε uH q uH q+1 wH q + c3 wx L∞ + vH q w2H q 2    ε  + c3 vH q+1 + wH q wH q + ε uH q uH q−1 wH q−1 2   + ε uH q uH q+1 wH q + εc6 f H q w2H q    c εuH q uH q+1 wH q + uH q w2H q .

By using the inequalities (2.14), (2.15), one can see that   q 2   d Λ w dx  c εwH q + w2H q . dt R

Combining with the inequality (2.10), we obtain the following inequality   s−q wH q  c1 ε 4 ect + ε ect − 1 . In a similar way, combining with the inequalities (2.10) and (2.9), one can obtain   wH s  c1 w0 H s ect + ε ect − 1

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

and wH s+p  c1 ε

−p 4

ect + ε

3−p 4

 ct  e −1 ,

1319

(3.2)

for all p > 0. So wH s → 0, as ε, γ → 0. Next, we consider convergence of the sequence {uεt }. Multiplying to Λ2s−2 wt both sides of Eq. (3.1) and integrating with respect to x again, we may obtain the equation     s−1 2  s−1  s−3   s−1  s−1  1 d Λ wt dx = −2ω Λ wt Λ wx dx − (γ − ε) Λ wt Λ (uux ) dx 2 dt R R R     s−1  s−3  ε  s−1  s−1 − Λ wt Λ (wf )x dx − (γ − ε) Λ wt Λ (uux ) dx 2 R R     s−1  s−3  2   ε  s−1  s−3 ε−γ − Λ wt Λ (wx fx )x dx − Λ wt Λ ux x dx 2 2 R R   3 − ε  s−1  s−3 − Λ wt Λ (wf )x dx. 2 R

By using the following inequalities        2ω Λs−1 wt Λs−3 wx dx   2ωwx  s−3 wt  s−1 , H H   R

       s−1  s−1 (γ − ε) Λ wt Λ (uux ) dx   εv2H s wt H s−1 ,  R

    ε  s−1  s−1      Λ wt Λ (wf )x dx   cε w2H s + wH s uH s wt H s−1 , 2 R      s−1  s−3   (γ − ε) Λ wt Λ (uux ) dx   εu2H s−2 wt H s−1 ,  R

    3 − ε  s−1  s−3       c w2 s−2 + w s−2 u s−2 wt  s−1 ,  w (wf ) Λ Λ dx t x H H H   2 H R     ε  s−1  s−3      Λ wt Λ (wx fx )x dx   cε w2H s−1 + wH s−1 uH s−1 wt H s−1 , 2 R

and

   ε − γ  s−1  s−3  2     Λ wt Λ ux x dx   cεu2H s wt H s−1 ,  2 R

and combining with (2.9), (2.10), (2.14), (2.15), (2.21), we have   wt H s−1  c εwH s + ε → 0, as ε, γ → 0.

2

From Theorem 3.1, one can see that Theorem 3.2. Under the assumption ε > 0, let uγ be the solution of (1.1) with the initial data uγ 0 in H s (R), s > 32 . Then uγ converges to the solution to the following BBM equation (3.3) with the initial data u0 in H s (R), s > 32 , as γ → 0 and k = 2ω, β = O(γ ):  ut − uxxt + 2ωux + 3uux = 0, (3.3) u(0, x) = u0 (x).

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4. The strong and weak limit behavior of solutions as α → 0 Now let us study the behavior of solutions to the Cauchy problem (1.1) as the parameter α tends to zero. For the convenience, we denote ε as α 2 , and we shall demonstrate that {uε } is a Cauchy sequence as ε → 0. 4.1. A priori estimates Firstly, we shall give the following estimates about norm in H s (R), s  0. Theorem 4.1. Suppose that for some s > 32 , the function u(x, t) is a solution to Eq. (1.1) corresponding to the initial data u0 ∈ H s (R). Then for any real number q ∈ (0, s], there exists a constant c depending only on q, so that 

t u2H q

+ εux 2H q

 u0 2H q

+ εu0x 2H q

+c

ux L∞

 q 2 2  Λ u + ε Λq ux dx dt.

(4.1)

R

0

For any q ∈ (1/2, s] and any r ∈ (1/2, q], there exists a constant c depending only on r and q, such that t   u2H q

+ εux 2H q

 u0 2H q  ×

+ εu0x 2H q

+c 0

  r+1 2 2 Λ u + ε Λr+1 ux dx

1/2

R

 q 2  q 2 Λ u + ε Λ ux dx dt.

(4.2)

R

Moreover, for any q ∈ (0, s) there exists a constant c depending only on q, such that   ut H q + εut H q+1  cut H q+3 c(k, γ ) + uH 1+θ

(4.3)

for all θ > 1/2. And under the assumption β = O(ε), one can see that   ut H q + εut H q+1  cuH q+2 c(k) + uH 1+θ . Proof. For any q ∈ (0, s], applying (Λq u)Λq to both sides of Eq. (4.1) and integrating on R with respect to x again, one obtains the equation     q 2       q 2 1 d ε q q Λ u + ε Λ ux dx = −3 Λ uΛ (uux ) dx + Λq uΛq ∂x3 u2 + Λq ux Λq u2x dx 2 dt 2 R R R   q q = (−3 + ε) Λ uΛ (uux ) dx − ε Λq+1 uΛq+1 (uux ) dx ε + 2



R

  Λq ux Λq u2x dx

R

R

using integration by parts. It follows from the inequalities (4.2) and (4.3) that there is a constant cq such that  1 d 3 u2H q + εux 2H q  4cq ux L∞ u2H q + εcq ux L∞ ux 2H q 2 dt 2    4cq ux L∞ u2H q + εux 2H q . Integrating with respect to t on both sides of the above inequality leads to inequality (4.1). Applying the inequality  1/2  r+1 2  r+1 2 ux L∞  cr+1 uH r+1  cr+1 Λ u + ε Λ ux dx R

for r > to the right-hand side of (4.1) yields the estimate (4.2). For any q ∈ (0, s], applying (Λq ut )Λq to both sides of Eq. (4.1) and integrating with respect to x again, one obtains the equation 1 2

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

1 d 2 dt



 q 2 2  Λ ut + ε Λq uxt dx

R





= −3



Λq ut Λq (uux ) dx − k R

ε + 2



ε + 2

R

R





Λq ut Λq (uxxx ) dx R

 Λ ut Λ (uux ) dx − ε q

q

R

  Λq uxt Λq u2x dx − k

R



Λq+1 ut Λq+1 (uux ) dx R

Λq ut Λq (ux ) dx R

 −γ

Λq ut Λq (ux ) dx − γ

     Λq ut Λq ∂x3 u2 − Λq ut Λq u2x x dx

= (−3 + ε)

Λq ut Λq (uxxx ) dx.

(4.4)

R

By using the inequalities (4.2), (4.3), one may obtain the following inequalities:      Λq ut Λq (uux ) dx   cq ut H q u ˆ L1 uH q+1  cq ut H q uH 1 uH q+1 ,   R

     ε Λq+1 ut Λq+1 (uux ) dx   cq εut  q+1 u ˆ L1 uH q+2  cq εut H q+1 uH 1 uH q+2 , H   R

       ε Λq ut Λq u2 dx   cq εuxt H q uˆ x  1 ux H q  cq εuxt H q u 1+θ ux H q L H x x   R

 cq εut H q+1 uH 1+θ uH q+1    ε    q q 2    cq εuxt H q u 1+θ ux H q u dx Λ u Λ xt H x 2  R      Λq ut Λq (ux ) dx   cq ut H q ux H q ,  

 1 ∀θ > , 2  1 ∀θ > , 2

R

and

     γ Λq ut Λq (uxxx ) dx   cq γ ut H q uxxx H q .   R

Applying the above six inequalities to (4.4) yields the inequality     ut 2H q + εuxt 2H q  c ut H q + εuxt H q uH q+3 k + β + uH 1+θ , where c = c(q) is constant independent of ε, i.e.,   ut H q + εut H q+1  cuH q+3 k + β + uH 1+θ or

  ut H q + εut H q+1  cuH q+2 k + uH 1+θ

So we have

1321

  with β = O(ε) .

  ut H q + εut H q+1  cuH q+3 k + β + uH 1+θ

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L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

for all θ > 12 . And under the assumption β = O(ε), one can see that   ut H q + εut H q+1  cuH q+2 k + uH 1+θ for some constant c independent of ε. The proof of Theorem 4.1 is complete.

2

Now let us apply some lemma as follows: Lemma 4.2. (See [17].) Under the above assumptions, the following estimates hold for any ε with 0 < ε < 1/8 uε0 H q  c uε0 H q  cε

(q  s), s−q 8

uε0 − u0 H q  cε

(4.5)

(q > s),

(4.6)

s−q 8

(4.7)

(q  s),

uε0 − u0 H s = o(1),

(4.8)

here uε0 is the convolution uε0 = φε ∗ u0 of the functions φε (x) = ε −1/8 φ(ε −1/8 x) and u0 such that the Fourier ˆ )  0, and φ(ξ ˆ ) = 1 for any ξ ∈ (−1, 1). transform φˆ of φ satisfies φˆ ∈ Cc∞ ([0, ∞); H ∞ ), φ(ξ Lemma 4.3. There exist constants c1 and c2 independent of ε such that the following inequalities hold u2H r + εux 2H r 

4Mr 1/2 (2 − cMr t)2



M , (2 − Mt)2

3 < r  s, 2

(4.9)

1/2

where Mr = u0 2H r + εu0x 2H r , M = max{4Mr , cMr },  

u2H s + εux 2H s u2H s+p

1/2



c1 , (2 − Mt)c2

1/2 + εux 2H s+p

(4.10)

p

c1 ε − 8  , (2 − Mt)c2

p > 0,

(4.11)

and 

ut 2H s+p + εuxt 2H s+p

1/2

p+1



c1 ε − 8 , (2 − Mt)c2

p > −1,

(4.12)

for any ε sufficiently small. Proof. Choose a fixed number r with 3/2 < r  s. It follows from (4.2) that 

  r 2 2 Λ u + ε Λr ux dx 

R



  r 2 2 Λ u0 + ε Λr u0x dx + c

R

t   0

  r 2 2 Λ u + ε Λr ux dx

3/2 dt.

(4.13)

R

 Multiplying (4.13) with ( R (Λr u)2 + ε(Λr ux )2 dx)1/2 , we obtain the inequality  R

 r 2 2  Λ u + ε Λr ux dx 



3/2

 r 2 2  Λ u + ε Λr ux dx

1/2

R

 × R

  r 2 2 Λ u0 + ε Λr u0x dx + c

t   0

R

  r 2 2 Λ u + ε Λr ux dx

3/2

 dt .

(4.14)

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

1323

Denote that  V (t) :=

 r 2 2  Λ u0 + ε Λr u0x dx + c

R

and we have d V (t)  dt

t  

 r 2 2  Λ u + ε Λr ux dx

3/2 dt,

R

0



 r 2 2  Λ u + ε Λr ux dx

1/2 (4.15)

V (t)

R

from the inequality (4.13). It follows from Gronwall’s inequality and (4.13) that 4Mr M  , 3/2 < r  s, u2H r + εux 2H r  1/2 2 (2 − Mt)2 (2 − cMr t) holds for any t ∈ [0, 2/M). Substituting the inequality (4.9) into (4.2) with q = s and u = uε , one obtains the estimate  t √  s 2  s 2 M 2 2 2 2 uH s + εux H s  u0 H s + εu0x H s + c Λ u + ε Λ ux dx dt (2 − Mt) R

0

for any t ∈ [0, 2/M). It follows from Gronwall’s inequality and (4.5), (4.6) that there are constants c1 , c2 depending on M such that 1/2  c1  . u2H s + εux 2H s (2 − Mt)c2 According to the inequalities (4.5), (4.6), one can see that   Ms = u0 2H s + εu0x 2H s  c0 1 + ε 3/4  c, where c0 , c is independent of ε. So M, c1 , c2 are independent of ε. In a similar way, one may obtain p

c1 ε − 8 , p > 0. (2 − Mt)c2 And by using the inequalities (4.3), (4.11), we obtain the inequality 1/2   u2H s+p + εux 2H s+p

p+1

1/2  c1 ε − 8  , ut 2H s+p + εuxt 2H s+p (2 − Mt)c2

p > −1.

2

Remark. According to the inequality (4.10) in Lemma 4.3, one can see that uH s is uniformly bounded and the boundedness is independent of ε. 4.2. The convergence of solutions as ε → 0 We shall demonstrate that {uε } is a Cauchy sequence. We rewrite Eq. (1.1) as ut − εuxxt + kux + (3 − 3εγ )uux + βuxxx = εγ (−3uux + 2ux uxx + uuxxx ), i.e., ut − εuxxt + kux + (3 − 3γ ε)uux + βuxxx

   1 2 2 2 = −γ ε 1 − ∂x (uux ) − γ ε∂x u + ux . 2

Let uε and uδ be solutions of (1.1), corresponding to the parameters ε and δ, respectively, and let w = uε − v δ and f = uε + v δ . Then w satisfies the problem for βε = β = O(ε) and βδ = β = O(δ): 1 wt − εwxxt + (δ − ε)vxxt + kwx + (3 − 3ε)(wf )x + (δ − ε)vvx + βε wxxx + (βε − βδ )vxxx 2      ε  1 2 1 2 2 2 = − γ 1 − ∂x (wf )x + γ (δ − ε) 1 − ∂x (vvx ) − εγ ∂x wf + wx fx + (δ − ε)γ ∂x v + vx . 2 2 2 (4.16)

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L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

Without loss of generality, we always assume that 0 < ε  δ  2ε  1/8. And considering the different properties under the assumption β = O(ε) or without the assumption β = O(ε), one can see Theorem 4.4. Under the assumption β = O(ε), there exists T > 0, independent of ε, such that {uε } and {uεt } are Cauchy sequences in the space C([0, T ); H s (R)) and C([0, T ); H s−1 (R)), respectively. Proof. First, for a constant q with 1/2 < q  min{1, s − 1}, multiplying to Λ2q w both sides of Eq. (4.16) and integrating with respect to x again, we may obtain the equation   q 2 2  1 d Λ w + ε Λq wx dx 2 dt R    q  q   q  q  = −(δ − ε) Λ w Λ vxxt dx − k Λ w Λ wx dx R



R

   q  q 1 − (3 − 3ε) Λ w Λ (vvx ) dx 2 R R      q  q  q  q −γ Λ w Λ (wxxx ) dx − (γε − γδ ) Λ w Λ (vxxx ) dx R



ε − 2

R

 −ε

R

 q  q+2  Λ w Λ (wf )x dx + (δ − ε)  q  q  1 Λ w Λ (wf )x dx − ε 2 

+ (δ − ε) R

 = −(δ − ε)

R



− (δ − ε) −



ε 2

R

 −ε

R

 q  q  Λ w Λ (wf )x dx − (δ − ε)



R

R

 q  q  2   Λ w Λ v x dx + (δ − ε)

R





   q  q 1 2 dx v Λ w Λ 2 x x



R

 q  q  Λ w Λ (vvx ) dx − (γε − γδ )

 q  q  1 Λ w Λ (wf )x dx − ε 2

R



 q  q  1 Λ w Λ vxxt dx − (3 − 3ε) 2

R

+ (δ − ε)

  q  q+2 Λ w Λ (vvx ) dx

 q  q  Λ w Λ (wx fx )x dx

 q  q+2  Λ w Λ (wf )x dx + (δ − ε)



R



 q  q  Λ w Λ (wf )x dx 

 q  q  Λ w Λ (vxxx ) dx

R

 q  q+2  Λ w Λ (vvx ) dx

R

 q  q  Λ w Λ (wx fx )x dx

R

 q  q  2   Λ w Λ v x dx + (δ − ε)

 R

   q  q 1 2 dx. vx Λ w Λ 2 x

Consider the following inequalities:      q  q   (δ − ε) Λ w Λ vxxt dx   δvt H q+2 wH q ,  R

    q  q        Λ w Λ (wf )x dx   c3 wx L∞ + vH q w2H q + c3 vH q+1 + wH q wH q ,  R

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

1325

     q  q     (δ − ε) Λ w Λ (vvx ) dx   δ vH q vH q+1 wH q ,  R

        ε  q  q+2  q  q     (γε − γδ )   Λ w Λ (vxxx ) dx   cδvH q+3 wH q  Λ w Λ (wf )x dx   2 R R

   c3 wH q εf H q+3 f H q+3 + vH q+3      q  q+2   1/4  q  c3 ε wH (δ − ε) Λ w Λ (vvx ) dx  R

   δ vH q+2 vH q+3 wH q ,       q  q    ε Λ w Λ (wf )x dx   ε 2vH q+1 f H q+1 wH q ,  R

    ε  q  q      c4 εf  q+1 w2 q+1 , Λ dx w Λ (w f ) x x x H 2  H R      q  q  2    (δ − ε)   δv2 q+1 wH q , Λ v dx w Λ   x H R

and

        q  q 1 2 (δ − ε) dx   δv2H q+2 wH q . v Λ w Λ x  2 x R

By using the inequalities (4.10), (4.11), one can see that   q 2 2    d Λ w + ε Λq wx dx  c δ ρ wH q + δ 1/4 wH q + w2H q , dt R

where ρ=



s  3 + q, s < 3 + q.

1, 1+s−q 4 ,

Combining with the inequality (4.6), we obtain the following inequality     s−q wH q  c1 δ 4 ect + δ ρ ect − 1 + δ 1/4 ect − 1 . In a similar way, combining with the inequalities (4.6) and (4.5), one can obtain    3 wH s  c1 w0 H s + δ 4 ect + δ m ect − 1 , where m = min( 14 , s−q−1 4 ) and wH s+p  c1 δ

−p 4

ect + δ

1−p 4

 ct  e −1 ,

(4.17)

for all p > 0. So wH s → 0, as ε, δ → 0. Next, we consider convergence of the sequence {uεt }. Multiplying both sides of Eq. (4.16) by Λ2s−2 wt and integrating on R with respect to x again, we may obtain the equation   s−1 2 2  1 d Λ wt + ε Λs−1 wxt dx 2 dt R    s−1  s−1   s−1  s−1  = −(δ − ε) Λ wt Λ vxxt dx − k Λ wt Λ wx dx R

R

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L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

   s−1  s−1   s−1  s−1  1 Λ wt Λ (wf )x dx − (δ − ε) Λ wt Λ (vvx ) dx − (3 − 3ε) 2 R R    s−1  s−1  s−1  s−1   − Λ wt Λ (wxxx ) dx − (γε − γδ ) Λ wt Λ (vxxx ) dx R



ε − 2

R

 −ε

R

 s−1  s+1  Λ wt Λ (wf )x dx + (δ − ε)  s−1  s−1  1 Λ wt Λ (wf )x dx − ε 2



R



  s−1  s+1 Λ wt Λ (vvx ) dx

R

 s−1  s−1  Λ wt Λ (wx fx )x dx

R

     s−1  s−1  2    s−1  s−1 vx2 + (δ − ε) dx . Λ wt Λ v x dx + Λ wt Λ 2 x R

R

By using the following inequalities       s−1  s−1    (δ − ε) Λ wt Λ vxxt dx   δ vxxt H s−1 wt H s−1 ,  R

      s−1  s−1    k Λ wt Λ wx dx   k wx H s−1 wt H s−1 ,  R

    s−1  s−1       c w2 s + wH s vH s wt  s−1 ,  Λ Λ dx w (wf ) t x H H   R

      s−1  s−1    (δ − ε)   δ v2 s wt  s−1 , Λ Λ w (vv ) dx t x H H   R

     s−1  s−1     γε Λ wt Λ (wxxx ) dx   γε wH s+2 wt H s−1 ,  R

     s−1  s−1     (γε − γδ ) Λ wt Λ (vxxx ) dx   γδ vH s+2 wt H s−1 ,  R

and

    ε  s−1  s+1        c ε 1/2 w s+2 ε 1/2 f  s+2 wt  s−1 , Λ Λ dx w (wf ) t x H H H 2  R

and combining with (4.5), (4.6), (4.10), (4.11), (4.17), we have   wt H s−1  c δ 1/2 wH s+2 + δ 1/2 → 0 as ε, δ → 0.

2

From Theorem 4.4, one can see that Corollary 4.5. Under the assumption ε > 0, let uε be the solution to (1.1) with the initial data uε0 ∈ H s , s > 32 . Then uε converges to the solution to the following equation (4.18) with the initial data u0 ∈ H s , s > 32 , as ε → 0,  ut + kux + 3uux = 0, (4.18) u(0, x) = u0 (x). From the proof of Theorem 4.2 and a priori estimates in the proof of Theorem 4.1, one can see that, without the assumption β = O(ε), we have

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

1327

Theorem 4.6. If u = uε and v = uδ are the solutions to the problem (1.1) in C([0, T ), H s ), s > 32 , with ε and ε = δ, respectively, then u − vH s−3 converges to zero as ε, δ → 0. Moreover, uε converges to the solution to the following KdV equation (4.19) with the initial data u0 in H s , s > 3, as ε → 0,  ut + kux + 3uux + βuxxx = 0, (4.19) u(0, x) = u0 (x). Remark. According to some a priori estimates in the proof of Theorem 4.2, and analogue with the proof of Theorem 3.1 in Section 3 of [17], we can obtain that the solutions to Eq. (1.1) ut − α 2 uxxt + kux + 3uux + βuxxx = γ α 2 (2ux uxx + uuxxx ) converge to the solution of the following equation ut − α 2 uxxt + kux + 3uux = γ α 2 (2ux uxx + uuxxx ) as β → 0, γ = 0, particularly, to the solution of the Camassa–Holm equation as β → 0, γ = 1. And for γ = 0, the above convergence is also evident. 5. Convergence of solitary wave solutions Substituting the traveling wave solution u = ϕ(x − V t), for constant wave speed V , into Eq. (1.1), one can obtain the ordinary differential equation −V ϕ  + α 2 V ϕ  + 3ϕϕ  + kϕ  + βϕ  = γ α 2 (2ϕ  ϕ  + ϕϕ  ), i.e.,

  2 α V + β − γ α 2 ϕ ϕ  = −3ϕϕ  + (V − k)ϕ  − 2γ α 2 ϕ  ϕ  .

Considering the transformation ϕ = ψ + a, we have the following ordinary differential equation  2  α V + β − γ α 2 a − γ α 2 ψ ψ  = −3ψψ  + (V − k − 3a)ψ  − 2γ α 2 ψψ  .

(5.1)

(5.2)

Definition 5.1. A solitary wave ϕ(x) with undisturbed depth a = lim|x|→∞ φ(x) is a weak solution of the ordinary differential equation (5.1) if and only if ψ = φ − a ∈ H 1 , and      2  3 2 γ α2  2 γ α 2 2   (5.3) φ − (φ ) + (k − V )φ, g + α V + β φ − φ , g = 0, 2 2 2 for any g ∈ Cc∞ (R). For (5.1), integrating on R with respect to η, and letting integrate constants be zero, and integrating by parts, we have  2  3 γ α2  2 (ϕ ) . α V + β − γ α 2 ϕ ϕ  = − ϕ 2 + (V − k)ϕ + (5.4) 2 2 Let z = ϕ  , and we can reduce (5.4) to the dynamical system ⎧ dϕ ⎪ ⎪ = z, ⎨ dη (5.5) dz 2(V − k)ϕ − 3ϕ 2 + γ α 2 z2 ⎪ ⎪ ⎩ = , dη 2(α 2 V + β − γ α 2 ϕ) where ϕ = γ1 (V + αβ2 ) is a singular line. The transformation dη = (α 2 V + β − γ α 2 ϕ) dτ reduce (5.5) to the Hamilton system ⎧ dϕ ⎪ ⎨ = z(α 2 V + β − γ α 2 ϕ), dτ 2 ⎪ ⎩ dz = (V − k)ϕ − 3 ϕ 2 + γ α z2 . dτ 2 2

(5.6)

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L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

The first integral is z2 (α 2 V + β − γ α 2 ϕ) ϕ 2 (ϕ + k − V ) + = h, 2 2 where h is a constant. Noting H (ϕ, z) =

   χ = γ V α 2 + β (3 − 2γ )V α 2 + 3β + 2kγ α 2 ,  β 2 1 ϕ1 = (V − k), ϕ2 = V+ 2 , 3 γ α

0 z± =±

1 √ χ, |γ 2 α 3 |

one can see that (i) (ii) (iii) (iv) (v)

when V = k, there is only a singular point O(0, 0) on ϕ-axis of the system (5.5); when V = k, there are two singular points (i.e., O(0, 0) and (ϕ1 , 0)) on ϕ-axis of the system (5.5); 0 ) and (ϕ , z0 )) on the line ϕ = ϕ of the system (5.5); when χ > 0, there are two singular points (i.e., (ϕ2 , z+ 2 − 2 when χ = 0, there is only a singular point (ϕ2 , 0) on the line ϕ = ϕ2 of the system (5.5); when χ < 0, the singular point does not exist on the line ϕ = ϕ2 .

Considering Hamilton function H (ϕ, z), we have h0 = H (0, 0) = 0,   0 = h2 = H ϕ2 , z±

h1 = H (ϕ1 , 0) =

2 (k − V )3 , 27

 2 1  2 V α + β (1 − γ )V α 2 + β + kγ α 2 . 2γ 3 α 6

Letting λ(ϕ, z) be an eigenvalue of the point (ϕ, z), and fλ be its corresponding eigenfunction, we have    λ + γ α2z γ α 2 ϕ − (V α 2 + β)  fλ =  , 3ϕ − (V − k) λ − γ α2z i.e.,

So

i.e.,

    λ2 − γ 2 α 4 z2 + γ α 2 ϕ − V α 2 + β 3ϕ − (V − k) = 0.

   λ2 (0, 0) − V α 2 + β (V − k) = 0,   V α 2 + β (V − k),

    λ2 (ϕ1 , 0) − γ α 2 ϕ1 − V α 2 + β 3ϕ1 − (V − k) = 0,

λ(0, 0) = ±

i.e.,



  2 2 2 2 λ(ϕ1 , 0) = ± (V − k) γ − 1 V α − γ kα − β 3 3 and   2 0  1 √ 0 = γ α z± = ± χ. λ ϕ2 , z± |γ α| According to the proof of Proposition 1 in [20], we obtain Lemma 5.2. If (ϕ(η), z(η)) is a homoclinic orbit representation of system (5.6), then u = ϕ(x − V t) is a solitary solution to Eq. (1.1).

L. Tian et al. / J. Math. Anal. Appl. 341 (2008) 1311–1333

1329

The transformation     ξ1 ψ ξ = ξ2 = ψ  ξ3 ψ  reduces (5.2) to the dynamical system ⎞   ⎛ ξ2 ψ ξ3 ⎠. ξ  = ψ  = ⎝ −3ξ1 ξ2 +(V −k−3a)ξ2 −2γ α 2 ξ2 ξ3  ψ α 2 V +β−γ α 2 a−γ α 2 ξ 1

Let

⎛

0 D = ∇f (0) = ⎝ 0 0

1 0

V −k−3a α 2 V +β−γ α 2 a

⎞ 0 1⎠, 0

using the transformation   ξ1 + a  y = ξ + a = ξ2 ξ3 as before yields the system of equations ⎞   ⎛ ξ2 ψ ξ3 ⎠ = D ξ + g(ξ ), ξ  = ψ  = ⎝ −3ξ1 ξ2 +(V −k−3a)ξ2 −2γ α 2 ξ2 ξ3  ψ α 2 V +β−γ α 2 a−γ α 2 ξ

(5.7)

1

where

⎛

0 D = ∇f (0) = ⎝ 0 0

φ = ψ + a,

1 0

V −k−3a α 2 V +β−γ α 2 a

⎞ 0 1⎠. 0

The set of fixed points and singularities of system (5.3) consists of all points of the y1 -axis. Next, we discuss properties of each fixed point or singularity (a, 0, 0) of system (5.1) in different cases. Considering ⎛ ⎞ λ −1 0 λ −1 ⎠ , λI − D = ⎝ 0 −k−3a 0 − α 2 VV+β−γ λ α2 a one can see that

 fA (λ) = λ λ2 +

The constants γ1 (V +

3a + k − V 2 α V + β − γ α2a

β ) α2

and

V −k 3



 a − V 3−k 3 = λ λ2 − γ α 2 a − 1 (V + γ

β ) α2

.

divide the y1 -axis into three intervals.

Case I. When γ < 0 and V 3−k > γ1 (V + αβ2 ), the intervals are      1 1 V −k β β V −k −∞, , V+ 2 V+ 2 , and ,∞ . γ γ 3 3 α α If a ∈ (−∞, γ1 (V +

β )) ∪ ( V 3−k , ∞), α2

one can see that

a − V 3−k 1 3a + k − V = α 2 V + β − γ α 2 a −γ α 2 a − 1 (V + γ

β ) α2

> 0,

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the system (5.3) shows that (5.1) has a three-dimensional center manifold near the point (a, 0, 0). On the other hand, for any a ∈ ( γ1 (V + αβ2 ), V 3−k ), one can see that a − V 3−k 1 3a + k − V = α 2 V + β − γ α 2 a −γ α 2 a − 1 (V + γ

β ) α2

< 0.

Then there is a one-dimensional center manifold, a one-dimensional stable manifold, and a one-dimensional unstable manifold at the point (a, 0, 0), near which there is a unique, analytic, homoclinic orbit represented by the solution φ(x − V t) = ξ(x − V t) + a. We consider the periodic solutions φδ [x − (V + δ)t] = a + the differential equations   2 [ζδ − H (δ)]2 [ζδ + 2H (δ) − B + ζδ = γ α 2 ζδ − A

3−γ γ

δ γ

+ ζδ [x − (V + δ)t] to Eq. (5.2), the function ζδ satisfy

δ] (5.8)

,

δ where H = H (δ) → 0 (as δ → 0), A := α 2 V + β − γ α 2 a and B := V − k − 3a. In particular, letting H (δ) := −γ , 1 we can see that ζδ ∈ H . The solution φ(x − V t) = ξ(x − V t) + a is obtained as the limit, as δ → 0, of periodic solutions



 δ φδ x − (V + δ)t = a + + ζδ x − (V + δ)t , (5.9) γ where the function ξ satisfies the respective differential equations

(ζ  )2 =

ζ 2 (ζ − B) . γ α2ζ − A

The point ( γ1 (V +

(5.10)

β ), 0, 0) α2

φ0 [x − V t] =

"

forms a singular point of system, providing the compacton solution √ β β x 1 3 2 √ γ (V + α 2 ) + [V − k − γ (V + α 2 )] cos 2|α| −γ , |x|  |α| −γ π, 1 γ (V

+

β ), α2

otherwise.

(5.11)

For each a ∈ ( γ1 (V + αβ2 ), V 3−k ), we substitute φ = ψ + a into (5.1), integrating the resulting equation once and setting the integral constant to zero, we obtain 

 3 γ α2  2 (ψ ) . α 2 V + β − γ α 2 a − γ α 2 ψ ψ  = − ψ 2 + (V − k − 3a)ψ + 2 2

(5.12)

The system (5.12) has two fixed points—the origin O(0, 0) and ( 2(V3−k) − 2a, 0). The origin is a saddle point whose unique homoclinic orbit represents an analytic solitary wave solution. Near the point ( 23 (V − k) − 2a, 0), then exists a two-dimensional center manifold having periodic orbits converging to the homoclinic orbit at the origin. Substitute ψ = ξ + 23 (V − k) − 2a into (5.12) and comparing (5.1) with the resulting equation  2 2 3 γ α2  2 2 2 2 α V + β − γ α (V − k) + γ α a − γ α ξ ξ  = − ξ 2 + (V − k − 3a)ξ + (ξ ) , (5.13) 3 2 2 one may recognize that periodic orbits near the fixed point ( 2(V3−k) − 2a, 0) of (5.2) come from the center manifold of the fixed point ( 2(V3−k) − a, 0, 0) in system (5.1). Therefore the homeomorphism 2 Ψ (a) = (V − k) − a 3

(5.14)

of ( γ1 (V + αβ2 ), V 3−k ) onto ( V 3−k , 23 (V − k) − γ1 (V + αβ2 )) determines a one-to-one mapping from the set {(a, 0, 0); a ∈ ( γ1 (V + αβ2 ), V 3−k )} of quasi-hyperbolic points to the set of points {(Ψ (a), 0, 0); a ∈ ( γ1 (V + β ), V 3−k )} whose center manifolds contain periodic orbits converging to homoclinic orbits at the corresponding α2 quasi-hyperbolic fixed points. One may also notice that the mapping Ψ is defined in such a way that the points O(0, 0) and (Ψ (a) − a, 0) always appear as a pair of fixed points in (5.12).

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Case II. γ > 0. (II)1 When γ > 3 and V 3−k > γ1 (V + αβ2 ), the intervals are      1 V −k β β V −k 1 , V+ 2 V+ 2 , and −∞, ,∞ . γ γ 3 3 α α If a ∈ ( γ1 (V +

β ), V 3−k ), α2

one can see that

a − V 3−k 3a + k − V 1 = α 2 V + β − γ α 2 a −γ α 2 a − 1 (V + γ

β ) α2

> 0,

then there is three-dimensional center manifold near the point (a, 0, 0). On the other hand, for any a ∈ (−∞, γ1 (V + β )) ∪ ( V 3−k , ∞), α2

one can see that

a − V 3−k 1 3a + k − V = α 2 V + β − γ α 2 a −γ α 2 a − 1 (V + γ

β ) α2

< 0,

the system (5.3) shows that (5.1) has a one-dimensional center manifold, a one-dimensional stable manifold, and a one-dimensional unstable manifold at the point (a, 0, 0), near which there is a unique, analytic, homoclinic orbit represented by the solution φ(x − V t) = ξ(x − V t) + a. (II)2 When 0 < γ < 3 and V 3−k < γ1 (V + αβ2 ), the intervals are      1 V −k 1 β β V −k and , , V+ 2 V + 2 ,∞ . −∞, 3 3 γ γ α α When γ > 0, we have Theorem 5.3. If the coefficients of Eq. (1.1) satisfy the inequalities  1 β V −k > V+ 2 , α = 0, γ > 0, and 3 γ α then there exists an orbitally unique and analytic solitary wave solution a + ζa (x − V t) for each a ∈ ( γ1 (V + 2 β ), V 3−k ). Moreover, as an approaches (V 2−k) − (α2γVα+β) 2 , the sequence of the solitary wave solutions {a α2

converges to the peakon solution 

(V − k) (α 2 V + β) 1/2 − Ae−γ |αx| , − φ(x) = 2 2 2γ α

+ ζa (x − V t)}

(5.15)

where A = α 2 V + β − γ α 2 a. Theorem 5.4. When β = 0, α → 0, φp (x) ≡ − k2 − A is the unique stationary solution to the following KdV equation ut + kux + 3uux + βuxxx = 0, and compactons and peakons do not occur. And when γ < 0, one can see that Theorem 5.5. If the coefficients of Eq. (1.1) satisfy the inequalities  1 β V −k > V+ 2 , α = 0, γ < 0, and 3 γ α

(5.16)

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then there exists an orbitally unique and analytic solitary wave solution a + ζa (x − V t) for each a ∈ ( γ1 (V + β ), V 3−k ). α2

Moreover, as an approaches γ1 (V + αβ2 ), the sequence of the solitary wave solutions {a + ζa (x − V t)} converges to the compacton solution given in (5.16): "1 √ β β x 3 2 √ γ (V + α 2 ) + [V − k − γ (V + α 2 )] cos 2|α| −γ , |x|  |α| −γ π, (5.17) φ0 [x − V t] = 1 β otherwise. γ (V + α 2 ), Proof. We consider the periodic solutions



 δ φδ x − (V + δ)t = a + + ζδ x − (V + δ)t γ to Eq. (5.2). Let A = α 2 V + β − γ α 2 a, B = V − k − 3a and B0 = V − k − γ3 (V + wave solution reduces to   2 ζa2 (ζa − B) ζa = . γ α 2 ζa − A

β ). Then Eq. (5.2) for the solitary α2

(5.18)

 Using the inequality 0  ζδ  B −2H (δ)− 3−γ γ δ valid for all x ∈ R, one may show that sequence of functions {ζa } and

{ζa } are uniformly bounded on the real axis. Therefore, the Ascoli–Arzela Theorem shows that, as a → γ1 (V + αβ2 ), there exist subsequences of the families {ζa } and {ζa }, without loss of generality still denoted by {ζa } and {ζa }, which are uniformly convergent to a function and its derivative, respectively, on any compact set of R. Here we are relying on the fact that each is an even function, since it is symmetric with respect to its elevation and translation invariant. Taking the limit on both sides of (5.9) as a → γ1 (V + αβ2 ), or as A → 0 leads to the equation γ α 2 ζ (ζ  )2 = ζ 2 (ζ − B0 )

(5.19)

satisfied by the function ζ . Since limA→0 maxx∈R ζa (x) = limA→0 B = B0 > 0 and each ζa is even, monotone on each side of the origin and exponentially decaying to zero at infinity, the limiting function ζ is a nontrivial solution to (5.11). Thus, function ζ satisfies the equation γ α 2 (ζ  )2 = ζ (ζ − B0 ). Therefore, as an even and monotone decreasing function on the positive real axis, φ = ζ + γ1 (V + αβ2 ) is the compacton solution (5.11). This completes the proof of Theorem 5.5. 2 Acknowledgments The authors would like to express their sincere thanks to the referee for his valuable suggestions. Research was supported by the National Nature Science Foundation of China (No. 10771088) and Nature Science Foundation of Jiangsu Province (No. BK2007098) and Outstanding Personal Program in six fields (No. 6-A-029).

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