Applied Mathematics and Computation 218 (2011) 1726–1734

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Lax pair, Bäcklund transformation and multi-soliton solutionsfor the Boussinesq–Burgers equations from shallow water waves

Pan Wang a, Bo Tian a,b,c,⇑, Wen-Jun Liu a, Xing Lü a, Yan Jiang a

a School of Science, P.O. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, Chinab State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, Chinac Key Laboratory of Information Photonics and Optical Communications (BUPT), Ministry of Education, P.O. Box 128,Beijing University of Posts and Telecommunications, Beijing 100876, China

a r t i c l e i n f o a b s t r a c t

Keywords:Boussinesq–Burgers equationsBinary Bell polynomialsLax pairBäcklund transformationHirota methodSoliton solutionsSymbolic computation

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.06.053

⇑ Corresponding author at: School of Science, P.OE-mail address: tian.bupt@yahoo.com.cn (B. Tian

Under investigation in this paper is the set of the Boussinesq–Burgers (BB) equations,which can be used to describe the propagation of shallow water waves. Based on the binaryBell polynomials, Hirota method and symbolic computation, the bilinear form and solitonsolutions for the BB equations are derived. Bäcklund transformations (BTs) in both the bin-ary-Bell-polynomial and bilinear forms are obtained. Through the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair for the BB equations are presented aswell. Propagation characteristics and interaction behaviors of the solitons are discussedthrough the graphical analysis. Shock wave and bell-shape solitons are respectivelyobtained for the horizontal velocity field u and height v of the water surface. In both thehead-on and overtaking collisions, the shock waves for the u profile change their shapes,which denotes that the collisions for the u profile are inelastic. However, the collisionsfor the v profile are proved to be elastic through the asymptotic analysis. Our results mighthave some potential applications for the harbor and coastal design.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

With the development of nonlinear science, nonlinear evolution equations (NLEEs) have been used as the models to de-scribe some physical phenomena in fluid mechanics, plasma waves, solid state physics, chemical physics, etc. [1–4]. In orderto understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties [5–10].Solutions for the NLEEs can not only describe the designated problems, but also give more insights on the physical aspects ofthe problems in the related fields [11–14]. For example, the nonlinear wave phenomena observed in fluid dynamics, plasmaand optical fibers can be illustrated by the bell-shape (sech profile) and kink-shape (tanh profile) solutions [15]. Methods toderive the solutions for the NLEEs have been proposed, such as the inverse scattering transformation (IST) method [16,17],Hirota method [18,19], Darboux transformation (DT) [20], Bäcklund transformation (BT) [21] and algebra-geometric method[22,23]. Among them, the Hirota method is a direct approach for deriving the soliton solutions through the dependent-var-iable transformation and formal parameter expansion [24–26]. Besides, this method is also helpful to investigate the inte-grable properties of the NLEEs, e.g., the BT and Lax pair [18,27]. Key step for the Hirota method is to derive the bilinearform through the proper dependent-variable transformations [28–30].

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. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China.).

P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734 1727

In this paper, the Boussinesq–Burgers (BB) equations [15,31–38],

ut ¼ �2uux þ12

vx; ð1aÞ

v t ¼12

uxxx � 2ðuvÞx; ð1bÞ

which describe the propagation of shallow water waves, will be considered, where x and t respectively represent the nor-malized space and time, the subscripts denote the derivatives, u(x, t) is the horizontal velocity field (at the leading orderit is the depth-averaged horizontal field) and v(x, t) denotes the height of the water surface above a horizontal bottom.Via the gauge transformation of the spectral problem, DT with multi-parameter for Eqs. (1) has been derived [31]. IST inte-grability for Eqs. (1) has been investigated [32]. The multi-phase periodic solutions for Eqs. (1) have been obtained [15,33],the Whitham theory of modulations has been applied to the problem of the decay of an initial discontinuity [34,35], and aquasiclassical description of soliton trains arising from a large initial pulse has also been developed [36]. The traveling wavesolutions for Eqs. (1) have been obtained via the extended homogeneous balance method [37].

However, to our knowledge, the analytic properties such as the soliton solutions, BT and Lax pair for Eqs. (1) have notbeen studied via the binary Bell polynomials. With the help of the binary Bell polynomials [39–41], Hirota method[18,19] and symbolic computation [11–14], this paper will be organized as follows: In Section 2, concepts and formulaeabout the binary Bell polynomials will be introduced. In Section 3, the bilinear form and multi-soliton solutions for Eqs.(1) will be presented. Section 4 will give the BTs in both the binary-Bell-polynomial and bilinear forms. According to theBT in the binary-Bell-polynomial form, a type of solutions and Lax pair, different from that in Ref. [31], will be performedas well. Section 5 will concentrate on two types of interactions of the solitons, i.e., the head-on and overtaking interactions.Conclusions will be addressed in Section 6.

2. Binary Bell polynomials

With the assumption that w is a C1 function of x and wn ¼ @nx wðxÞ, the Bell polynomials presented in Refs. [39–41] are as

follows:

YnxðwÞ � Ynðwx; . . . ;wnxÞ ¼ e�wðxÞ@nx ewðxÞ ðn ¼ 1;2; . . .Þ: ð2Þ

Similarly, if w = w(x1, . . . ,xn) is a C1 function with multi-variables, the following polynomials [39–41]

Yn1x1 ;...;nlxlðwÞ � Yn1 ;...;nl

ðwr1x1 ;...;rlxlÞ ¼ e�w@n1

x1. . . @nl

xlew; ð3Þ

are the multi-dimensional Bell polynomials, in which we denote that wr1x1 ;...;rlxl¼ @r1

x1. . . @rl

xlw; ðr1 ¼ 0; . . . ;n1; � � � ; rl ¼ 0; . . . ;nlÞ.

The binary Bell polynomials take the following forms [39–41]

Ymx;nt v 0;u0Þ � Ymx;ntðw0Þð��w0

~px;~qt¼

v 0~px;~qt; if ~pþ ~q is odd

u0~px;~qt; if ~pþ ~q is even

( ;ð4Þ

where v0 and u0 are both the C1 functions of x and t, and w0~px;~qt � @~px@

~qt w0. Expression (4) can be rewritten as the recognizable

form

Yxðv 0Þ ¼ v 0x; Y2xðv 0;u0Þ ¼ v 02x þ u02x; Yx;tðv 0; u0Þ ¼ v 0xv0t þ u0xt;

Y3xðv 0;u0Þ ¼ v 03x þ 3v 0xu02x þ v 03x ; . . . :

Refs. [39–41] has pointed out the link between Y-polynomial and the Hirota D-operator [24–26]

Dr1x Dr2

t H � G � ð@x � @x0 Þr1 ð@t � @t0 Þr2 Hðx; tÞGðx0; t0Þjx0¼x;t0¼t;

by the identity

Ymx;nt v 0 ¼ lnðH=GÞ;u0 ¼ lnðHGÞ½ � � ðHGÞ�1Dmx Dn

t H � G: ð5Þ

In particular, if H = G, Expression (5) becomes

G�2Dmx Dn

t G � G � Ymx;ntð0;W ¼ 2 ln GÞ �0; if mþ n is odd;Pmx;ntðWÞ; if mþ n is even:

�ð6Þ

Moreover, Refs. [39–41] has given the relationship between the binary Bell polynomials and Lax pair through the followingexpression

Yr1x;r2tðv 0 ¼ ln X;u0 ¼ v 0 þ QÞ ¼ X�1Xr1

a1¼0

Xr2

a2¼0

r1

a1

� �r2

a2

� �Ya1x;a2tð0;QÞ@r1�a1

x @r2�a2t X; ð7Þ

where Q and X are functions of x and t.

1728 P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734

It should be noticed that the polynomial Yr1x;r2tðv 0;u0Þ, which is constructed with the derivatives of dimensionless vari-ables v0 and u0, is a homogeneous expression of weight r1 + r2s with s as the dimension of t (the dimension of x is equal to 1).

3. Binary-Bell-polynomial form and soliton solutions for Eqs. (1)

3.1. Binary-Bell-polynomial form

In this section, with the help of the binary Bell polynomials, the bilinear form for Eqs. (1) will be given. For Eqs. (1), itsinvariance under the scale transformations

x! kx; t ! k2t; u! k�1u; v ! k�2v

shows that u and v have the dimensions �1 and �2, respectively. Therefore, two dimensionless fields p and q can be intro-duced by setting u = c px and v = d qxx, where p and q are the functions of x and t with c and d as the dimensionless parametersto be determined. The equations for p and q can be derived from Eqs. (1) as

cpt ¼ �c2p2x þ

12

dqxx; ð8aÞ

dqxt ¼12

cpxxx � 2cdpxqxx; ð8bÞ

with d = �2c2. By virtue of Expressions (3), (4) and (8), we get

YtðpÞ ¼ �cY2xðp; qÞ; ð9aÞ

� 2c2Yxtðp; qÞ ¼c2Y3xðp; qÞ �

32

c � 4c3� �

pxqxx �12

cp3x � 2c2px �cp2

x � cqxx

� �: ð9bÞ

From Expressions (9), we give the binary-Bell-polynomial form for Eqs. (1) as follows:

Ytðp; qÞ þ cY2xðp; qÞ ¼ 0; ð10aÞ4cYxtðp; qÞ þ Y3xðp; qÞ ¼ 0; ð10bÞ

where c ¼ � 12. Substituting p = ln (f/g) and q = ln (f g) into Expressions (10), we obtain the bilinear form for Eqs. (1) as

Dt þ cD2x

� f � g ¼ 0; ð11aÞ

4cDxDt þ D3x

� f � g ¼ 0; ð11bÞ

where f and g are the functions of x and t. Via symbolic computation and Hirota method, f and g can be expanded as thepower series of a small parameter e:

f ¼ ef1ðx; tÞ þ e2f2ðx; tÞ þ e3f3ðx; tÞ þ � � � ; ð12aÞg ¼ 1þ eg1ðx; tÞ þ e2g2ðx; tÞ þ e3g3ðx; tÞ þ � � � : ð12bÞ

Substituting Expressions (12) into (11) and collecting the coefficients of the same power of e, we have

e : Dt þ cD2x

� ðf1 � 1Þ ¼ 0; ð13aÞ

e : 4cDxDt þ D3x

� ðf1 � 1Þ ¼ 0; ð13bÞ

e2 : Dt þ cD2x

� ðf2 � 1þ f1 � g1Þ ¼ 0; ð13cÞ

e2 : 4cDxDt þ D3x

� ðf2 � 1þ f1 � g1Þ ¼ 0; ð13dÞ

e3 : Dt þ cD2x

� ðf3 � 1þ f1 � g2 þ f2 � g1Þ ¼ 0; ð13eÞ

e3 : 4cDxDt þ D3x

� ðf3 � 1þ f1 � g2 þ f2 � g1Þ ¼ 0; ð13fÞ

..

.

3.2. Soliton solutions for Eqs. (1)

3.2.1. One-soliton solutionIn order to obtain the one-soliton solution for Eqs. (1), we choose

f1 ¼ aen and g1 ¼ ben with n ¼ kxþwt; ð14Þ

P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734 1729

where a, b and k are all arbitrary nonzero real constants, and w is a real constant to be determined. Truncating Expressions(12) with fl(x, t) = 0 and gl(x, t) = 0 (l = 2,3,4, . . .) and solving (13), we have w = �c k2.

Thus, the one-soliton solution for Eqs. (1) can be written as below:

u ¼ ckeck2t

eck2t þ bekx¼ ck

2þ ck

2Tanh

ck2t � kx� ln b2

!; ð15aÞ

v ¼ � bk2ekxþck2t

2ðeck2t þ bekxÞ2¼ � k2

8sech2 kx� ck2t þ ln b

2

!: ð15bÞ

It should be noticed that Solutions (15) are different from the traveling wave solutions obtained by the extended homoge-neous balance method in Ref. [37].

3.2.2. Two-soliton solutionFor constructing the two-soliton solution, we choose

f1 ¼ a1en1 þ a2en2 ; g1 ¼ b1en1 þ b2en2 ; ni ¼ kixþwit ði ¼ 1;2Þ; ð16Þ

where wi ¼ �ck2i . ai, bi and ki are all arbitrary nonzero real constants. Truncating Expressions (12) with fj(x, t) = 0 and

gj(x, t) = 0 (j = 3,4, . . .) and substituting Expressions (16) into (13), we get

f2 ¼ a3en1þn2 ; g2 ¼ b3en1þn2 ; a1 ¼a3k2

1

b2ðk1 � k2Þ2; b1 ¼

a3k22

a2ðk1 � k2Þ2; b3 ¼ 0:

Then the two-soliton solution for Eqs. (1) can be written as follows:

u ¼c a2k2ek2x�ck2

2t þ a3k31b�1

2 ðk1 � k2Þ�2ek1x�ck21t þ a3ðk1 þ k2Þek1xþk2x�ck2

1t�ck22t

h i1þ b2ek2x�ck2

2t þ a3k22a�1

2 ðk1 � k2Þ�2ek1x�ck21t

�c a2ek2x�ck2

2t þ a3k21b�1

2 ðk1 � k2Þ�2ek1x�ck21t þ a3ek1xþk2x�ck2

1t�ck22t

h i1þ b2ek2x�ck2

2t þ a3k22a�1

2 ðk1 � k2Þ�2ek1x�ck21t

� b2k2ek2x�ck22t þ a3k1k2

2a�12 ðk1 � k2Þ�2ek1x�ck2

1t

1þ b2ek2x�ck22t þ a3k2

2a�12 ðk1 � k2Þ�2ek1x�ck2

1t

8<:

9=;;ð17aÞ

v ¼ �12

a2k22ek2x�ck2

2t þ a3k41b�1

2 ðk1 � k2Þ�2ek1x�ck21t þ a3ðk1 þ k2Þ2ek1xþk2x�ck2

1t�ck22t

a2ek2x�ck22t þ a3k2

1b�12 ðk1 � k2Þ�2ek1x�ck2

1t þ a3ek1xþk2x�ck21t�ck2

2t

(

�a2k2ek2x�ck2

2t þ a3k31b�1

2 ðk1 � k2Þ�2ek1x�ck21t þ a3ðk1 þ k2Þek1xþk2x�ck2

1t�ck22t

h i2

a2ek2x�ck22t þ a3k2

1b�12 ðk1 � k2Þ�2ek1x�ck2

1t þ a3ek1xþk2x�ck21t�ck2

2th i2

9>=>;

� 12

b2k22ek2x�ck2

2t þ a3k21k2

2a�12 ðk1 � k2Þ�2ek1x�ck2

1t

1þ b2ek2x�ck22t þ a3k2

2a�12 ðk1 � k2Þ�2ek1x�ck2

1t�

b2k2ek2x�ck22t þ a3k1k2

2a�12 ðk1 � k2Þ�2ek1x�ck2

1th i2

1þ b2ek2x�ck22t þ a3k2

2a�12 ðk1 � k2Þ�2ek1x�ck2

1th i2

8><>:

9>=>;: ð17bÞ

4. BT and lax pair for Eqs. (1)

4.1. BT

The BT [24–26] provides a way of constructing new solutions from known ones for the soliton equations. In this section,based on Eqs. (10), we will obtain the BTs in both the binary-Bell-polynomial and bilinear forms for Eqs. (1).

We consider that

Q 1 ¼ Ytðp0; q0Þ þ cY2xðp0; q0Þ½ � � Ytðp; qÞ þ cY2xðp; qÞ½ �; ð18aÞQ 2 ¼ 4cYxtðp0; q0Þ þ Y3xðp0; q0Þ½ � � 4cYxtðp; qÞ þ Y3xðp; qÞ½ �; ð18bÞ

where (p0,q0) and (p,q) both satisfy Eqs. (10). Assuming p0 = ln (f0/g0) and q0 = ln(f0g0) and introducing the following relations:

v1 ¼ lnðg0=gÞ; v2 ¼ lnðf 0=f Þ; v3 ¼ lnðf 0=gÞ; v4 ¼ lnðg0=f Þ; ð19aÞw1 ¼ lnðg0gÞ; w2 ¼ lnðf 0f Þ; w3 ¼ lnðf 0gÞ; w4 ¼ lnðg0f Þ; ð19bÞp0 � p ¼ v2 � v1 ¼ w3 �w4; p0 þ p ¼ v3 � v4 ¼ w2 �w1; ð19cÞq0 � q ¼ v1 þ v2 ¼ v3 þ v4; q0 þ q ¼ w1 þw2 ¼ w3 þw4; ð19dÞ

1730 P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734

we derive the BT in the binary-Bell-polynomial form for Eqs. (8) as follows:

Ytðv1;w1Þ � cY2xðv1;w1Þ ¼ 0; ð20aÞYtðv2;w2Þ � cY2xðv2;w2Þ ¼ 0; ð20bÞYxðv3;w3Þ � lev1�v2 ¼ 0; ð20cÞY2xðv3;w3Þ � cYxðv3;w3Þ þ lev1�v2Yxðv4;w4Þ ¼ 0; ð20dÞ

with l and c as the arbitrary real constants.According to Expressions (5) and (20), the bilinear BT for Eqs. (8) can be given as

ðDt � cD2x Þg0 � g ¼ 0; ð21aÞ

ðDt � cD2x Þf 0 � f ¼ 0; ð21bÞ

Dxf 0 � g � lfg0 ¼ 0; ð21cÞ

D2x � cDx

� f 0 � g þ lDxg0 � f ¼ 0: ð21dÞ

Choosing f = 1 and g = 1 that correspond to the solutions u = 0 and v = 0, and substituting them into Eqs. (21), we obtain

f 0 ¼ s2 �l1

e�1xþ12ct�s1 ; g0 ¼ e�1xþ12ct�s1 ; c ¼ �21; ð22Þ

with s1 and s2 as two arbitrary real constants and 1 as a nonzero one. Based on BT (21), the solution for Eqs. (1) can be pre-sented as below:

u ¼ s2c12e1xþs1

1s2e1xþs1 � le12ct; v ¼ 13s2le1xþ12ctþs1

2ð�1s2e1xþs1 þ le12ctÞ2: ð23Þ

4.2. Lax pair

Through Expressions (19), we can get the following relations:

v2 ¼ v3 � p; w2 ¼ w3 þ p; v4 ¼ v1 � p; w4 ¼ w1 þ p: ð24Þ

Setting that

wi ¼ v i þ Q i; v i ¼ ln Wi; ði ¼ 1;3Þ; ð25Þ

we can get

Q1 ¼ w1 � v1 ¼ lnðg0gÞ � lnðg0=gÞ ¼ q� p; ð26aÞQ3 ¼ w3 � v3 ¼ lnðf 0gÞ � lnðf 0=gÞ ¼ q� p: ð26bÞ

Via Expressions (20) and (24)–(26), after some calculations, we give the Lax pair for Eqs. (8) in the form:

Ux ¼MU; Ut ¼ NU; ð27Þ

where

U ¼W1

W3

� �; N ¼

A B

C D

� �; M ¼

c2 � qxx�pxx

2lep

lep 0

!;

with

A ¼ cc2 þ 2cqxx � 2cpxx

4;

B ¼ 2cpxqxx � 2cpxpxx � 2cqxxx þ 2cpxxx � ccqxx þ ccpxx

4lep;

C ¼ clcep � 2cleppx

2; D ¼ 2pt þ 2cp2

x þ cqxx þ cpxx

2:

We can verify that Eqs. (1) can be recovered by the compatibility condition Mt �N x þMN �NM ¼ 0.

5. Analysis and discussions

In this section, Figs. 1–4 are depicted to describe the propagation characteristics and interaction behaviors of the solitonsvia Solutions (15) and (17).

P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734 1731

Fig. 1(a) shows the motion of the shock wave for the horizontal velocity field u, and Fig. 1(b) displays the bell-shape sol-iton for the height v of the water surface. We also observe that the shock wave and bell-shape soliton maintain their veloc-ities and directions during the propagation.

Seen from Figs. 2–4, solitons maintain their original directions after the collisions. Figs. 2 and 3 show the head-on colli-sions of the shock waves and bell-shape solitons, while the overtaking cases are depicted in Fig. 4. As the the shapes of shockwaves have changed after the collisions in Figs. 2(a), 3(a) and 4(a), the collision for the horizontal velocity field u is inelastic.In Figs. 2(b), 3(b) and 4(b), the collisions of the bell-shape solitons are depicted for the height v of the water surface above ahorizontal bottom. In Fig. 2(b), the two solitons move toward each other before the collision, merge into one with the sta-tionary wave at the moment of t = 4, then separate and travel along their original directions after the collision. Fig. 3(b)shows that the bell-shape soliton and the wave of the maximum amplitude travel along the opposite directions with anotherbell-shape soliton. The difference between Figs. 2(b) and 3(b) is that whether the wave of the maximum amplitude travels ornot. It should be noticed that the soliton, which travels toward the same direction as the wave of the maximum amplitude,catches up with the latter after the collision in Fig. 3(b). The similar analysis can be performed on the overtaking collisionsdisplayed in Figs. 4. Moreover, from Figs. 2–4, we observe that the head-on collision between two solitons occurs whenk1k2 < 0; the overtaking collision happens as k1k2 > 0.

Fig. 2. Head-on collision of two solitons expressed by Solutions (17) with k1 = 1, k2 = �1, b2 = 1, a2 = 1, a3 = 2 and c ¼ 12.

Fig. 1. Propagation of one soliton via Solutions (15) with parameters as k = �1, a = 1, b = 2 and c ¼ 12.

Fig. 3. Head-on collision of two solitons expressed by Solutions (17) with k1 ¼ 1; k2 ¼ �1:5; b2 ¼ 1; a2 ¼ 1; a3 ¼ 2; c ¼ 12.

Fig. 4. Overtaking collision of two solitons via Solutions (17) with k1 = �1, k2 = �2, b2 = 1, a2 = 1, a3 = 1 and c ¼ 12.

1732 P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734

In order to check whether the interaction between solitons for v is elastic, we make an asymptotic analysis on Solutions(17) as follows:

(1) Before collision (t ? �1):

S�1 ¼ �18

k21sech2

k1x� ck21t þ ln a3a�1

2 k22ðk1 � k2Þ�2

h i2

8<:

9=;; ðn1 þ n�1 0; n2 þ n�2 ! �1Þ; ð29aÞ

S�2 ¼ �18

k22sech2 k2x� ck2

2t þ ln b2

2

!; ðn2 þ n�2 0; n1 þ n�1 ! þ1Þ; ð29bÞ

(2) After collision (t ? +1):

Sþ1 ¼ �18

k21sech2 k1x� ck2

1t þ ln a3a�12

� �2

" #; ðn1 þ n�1 0; n2 þ n�2 ! þ1Þ; ð30aÞ

Sþ2 ¼ �18

k22sech2

k2x� ck22t � ln k2

1b�12 ðk1 � k2Þ�2

h i2

8<:

9=;; ðn2 þ n�2 0; n1 þ n�1 ! �1Þ: ð30bÞ

From Expressions (29) and (30), we can see the collision of bell-shape solitons for v is elastic. Relevant issues are seen inRefs. [42–47].

6. Conclusions

In this paper, with the help of the binary Bell polynomials, Hirota method and symbolic computation, the BB equations[Eqs. (1)], which describe the propagation of the shallow water waves, have been investigated. We have obtained the bilinearform, multi-soliton solutions, BT and Lax pair for Eqs. (1).

Based on the above results, we conclude that:

1. In virtue of the binary Bell polynomials, Binary-Bell-Polynomial Form (10) and Bilinear Form (11b) have been obtained forEqs. (1). Via Expressions (19), BT (20) in the binary-Bell-polynomial form has been derived. According to Expression (5),BT (21) in the bilinear form and Solution (23) have been presented. Moreover, on the basis of Binary-Bell-Polynomial BT(20), Lax Pair (27) for Eqs. (1) has been constructed, which is different from that in Ref. [31]. Construction of the BT andLax pair can be applied to other similar NLEEs.

2. With the help of the Hirota method and symbolic computation, we have got the one- and two-soliton solutions [see Solu-tions (15) and (17)] for Eqs. (1) and given the graphical analysis. The shock wave and bell-shape soliton have been respec-tively obtained for the horizontal velocity field u (at the leading order it is the depth-averaged horizontal field) and theheight v of the water surface above a horizontal bottom (see Figs. 1). After the collisions of both the head-on and over-taking types, the shock waves change their shapes, which denotes the collision for u is inelastic [see Figs. 2(a), 3(a) and4(a)]. Through Expressions (29) and (30), we have known the collision for v is elastic with the help of asymptotic analysis[see Figs. 2(b), 3(b) and 4(b)]. Observing Figs. 2–4, we have also obtained that the head-on collision appears as k1k2 < 0;the overtaking collision occurs when k1k2 > 0.

3. Analytic solutions for Eqs. (1) may be of some use for the study of the nonlinear water phenomena taking place in theharbor and coast [15].

P. Wang et al. / Applied Mathematics and Computation 218 (2011) 1726–1734 1733

Acknowledgments

We express our sincere thanks to Dr. H. Q. Zhang and other members of our discussion group for their valuablesuggestions. This work has been supported by the National Natural Science Foundation of China under GrantNo. 60772023, by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02,by the Supported Project (No. SKLSDE-2010ZX-07) and Open Fund (No. SKLSDE-2011KF-03) of the State Key Laboratory ofSoftware Development Environment, Beijing University of Aeronautics and Astronautics, by the National High TechnologyResearch and Development Program of China (863 Program) under Grant No. 2009AA043303, and by the SpecializedResearch Fund for the Doctoral Program of Higher Education (No. 200800130006), Chinese Ministry of Education.

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