lijun zhang, li-qun chen and xuwen huo- new exact compacton, peakon and solitary solutions of the...

8/3/2019 Lijun Zhang, Li-Qun Chen and Xuwen Huo- New exact compacton, peakon and solitary solutions of the generalized

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Nonlinear Analysis 67 (2007) 32763282

www.elsevier.com/locate/na

New exact compacton, peakon and solitary solutions of thegeneralized Boussinesq-like B(m, n) equations with nonlinear

dispersion

Lijun Zhanga,b, Li-Qun Chenb,c,, Xuwen Huo d

a School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, PR Chinab Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China

cDepartment of Mechanics, Shanghai University, Shanghai 200444, PR Chinad School of Information and Electronic, Zhejiang Sci-Tech University, PR China

Received 22 September 2006; accepted 20 October 2006

Abstract

In this paper, exact solutions of the generalized Boussinesq-like B(m, n) equation with nonlinear dispersion utt + a(un )x x +b(um )x x x x = 0, m, n R are investigated. As a result, under different parameter conditions, abundant compactons, peakons andsolitary solutions, which include not only some known results but also some new ones, are obtained. The approach we used is

simple and also suitable for studying the traveling wave solutions of some other equations.

c 2006 Elsevier Ltd. All rights reserved.Keywords: Boussinesq-like B(m, n) equations; Nonlinear wave equation; Solitary solution; Compacton solution; Peakon solution; Periodic cusp

wave solution

1. Introduction

To understand the role of nonlinear dispersion in the formation of patterns in liquid drops, Rosenau and Hyman [7]

introduced and studied a family of fully nonlinear KdV equations K(m, n):

ut + (um )x + (un )xxx = 0, m > 0, 1 < n 3. (1)A class of solitary waves (which they named compactons) with the special property that after colliding with other

compacton solutions, they re-emerge with the same coherent shape was presented. From then on, compacton solutionsattracted a lot of interest [5,1016]. You can refer to [3,4,8] for example for more details about the properties of

compacton solutions. Compacton solutions and solitary solutions of other nonlinear evolution equations such as

Boussinesq equations and Boussinesq-like B(m, n) equations had been extensively investigated by many authors [11,

1315,17]. Recently, Yan [11] introduced a class of fully Boussinesq equations B(m, n)

utt = (un)x x + (um )xxxx , m, n R, (2)

Corresponding author at: Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China.E-mail addresses: [email protected] (L. Zhang), [email protected], [email protected] (L.-Q. Chen).

0362-546X/$ - see front matter c

2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.na.2006.10.009
http://www.elsevier.com/locate/namailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.na.2006.10.009http://dx.doi.org/10.1016/j.na.2006.10.009mailto:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/na


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L. Zhang et al. / Nonlinear Analysis 67 (2007) 32763282 3277

and presented some of its compacton solutions when m = n. Liu et al. [6] utilized the integral approach to investigateits compacton solutions. More recently, Zhu [1315] studied Boussinesq-like B(m, n) equations

utt + (un)x x (um )xxxx = 0, m, n > 1, (3)utt (un)x x + (um )xxxx = 0, m, n > 1 (4)

and

utt + (u2n )x x + (u2n)x x x x = 0, n 1, (5)by using the extended decomposition method.

In this paper, we will investigate a more generalized form of the above equations

utt + a(un)x x + b(um )x x x x = 0, m, n R. (6)Obviously, Eq. (6) is a very extensive equation. Eqs. (2)(5) are only the special cases of Eq. (6) when the absolute

values ofa and b are 1. When a = b = 1 and (m, n) = (1, 2), Eq. (6) becomes the well-known Boussinesq equation.When a = b = 1 and (m, n) = (1, 3), Eq. (6) becomes the modified Boussinesq equation which arises from thefamous FermiPastaUlam problem [1]. The integral approach has been successfully used to investigate traveling

wave solutions of integral nonlinear evolution equation by many authors [2,5,6,9,12,18]. We deduce Eq. (6) for aplanar integral dynamical system by a simple transformation and then obtain its explicit formulae for the compacton,

solitary solutions and so on by using the integral approach.The paper is organized as follows. In Section 2, we make a reduction of Eq. (6) and present the concrete scheme of

the approach for solving the equation. We obtain the exact compacton solitons and solitary solution of the equationsB(m, m) and B(1, n) in Section 3. Finally, some discussion and conclusions are given in Section 4.

2. Reductions of Eq. (6)

Under the traveling wave transformation u(x, t) = u(), = x ct, where c is the wave speed, Eq. (6) is reducedto the following nonlinear ordinary equation:

c2 d

2

ud 2

+ a d2

(u

n

)d 2

+ b d4

(u

m

)d 4

= 0. (7)

Integrating Eq. (7) twice, we have

g + c2u + au n + b(um ) = 0 (8)where g is an integral constant and denotes the derivative with respect to . Letting z = un1 (when n = 1),Eq. (8) becomes

zmn+1

n1

gz

m+n1n1 + c2z nmn1 + az nmn1 +1 + bm

n 1

z + m n + 1

n 1z2

z

= 0. (9)

Eq. (9) is equivalent to a two dimensional system as follows:

z = y, y = m n + 1n 1

y2

z n 1

bmz

nmn1 (az + c2) n 1

bmgz 1

mn1 (10)

which has first integral

H(z, y) = z2 mn+1n1

y2 + 2 (n 1)2

bmz

nmn1

a

1

m + nz2 + c2 1

m + 1z +1

mgz

n2n1

= h. (11)

From (11) we get

y= hz2

mn+1n

1

2

(n 1)2bm

znmn

1 a 1

m + nz2

+c2

1

m + 1z

+1

mgz

n2n

1. (12)


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3278 L. Zhang et al. / Nonlinear Analysis 67 (2007) 32763282

Inserting Eq. (12) into the first equation of (10), it follows that

dzhz2

mn+1n1 2 (n1)2

bmz

nmn1

a 1

m+nz2 + c2 1

m+1z + 1m gzn2n1 = d. (13)

It is easy to see that the traveling wave solutions of the generalized B(m, n) equation (6) can be obtained by usingthe integral of Eq. (13). We will give the exact compactons, peakons and solitary wave solutions of Eq. (6) for concretem and n by integrating Eq. (13) and solving z in the following section.

3. Exact solutions of generalized B(n, n) and B(1, n)

In this section, for the case m = n or the case m = 1, the exact compactons, peakons and solitary wave solutionsof Eq. (6) are investigated.

3.1. Exact solutions of generalized B(n, n)

When m = n = 1 Eq. (13) becomesdz

hz21

n1 2 (n1)2bn

a 1

2nz2 + c2 1

n+1z + 1n gzn2n1 = d. (14)

Case 1. The B(2, 2) equation

When m = n = 2, letting h = 0, Eq. (14) is reduced todz

a

4b z +2c2

3a 2 4c418ag

9a2

= d. (15)

Notice that z = 0 is a straight line of the system (10) when m n + 1 = 0. For different values of the parameters a,b, c and g, we can derive the traveling wave solutions of Eq. (10) from (15) as follows:

1. For ab > 0 and 4c4 > 18ag > 0, there exists an explicit periodic wave solution

z = 2c2 +

4c4 18ag cos

ab

2

3a

. (16)

2. For ab > 0 and g = 0, there exists an explicit compacton solution

z = 4c2

3acos2 a

b

4 a

b

4

20 otherwise.

(17)

3. For ab < 0 and g = 0, there exists an explicit peakon solution

z = 2c2

3a

1 exp

a

b

| |2

. (18)

4. For ab < 0 and ag > 0, there exists an explicit periodic cusp wave solution

z = 2c2

3a+

4c4 18ag cosh | 2kT|3|a| (2k 1)T 2kT (19)

where k is an integer and T = arch(2c2

4c418ag ).


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Case 2. The B(n, n) equation (n = 1, 2)Letting h = g = 0, Eq. (14) becomes

dz

a(n1)2

bn2 z2 + c2 2n

a(n+1)z= d. (20)

From (20), we get the solutions of Eq. (10) given by

z =

2c2n

a(n + 1) cos2

a

b

n 12n

a

b

n 12n

20 otherwise

(21)

when ab > 0 and

z = 2c2n

a(n + 1) cosh2

a

b

n 12n

(22)

when ab < 0. It is easy to see that Eq. (21) is a compacton solution and Eq. (22) is an unbounded solution of Eq. (10).Exact traveling wave solutions of the generalized B(m, n)equation when m

=n can be obtained by using the above

results. We describe this with the following theorems.

Theorem 3.1. When m = n = 2, Eq. (6) has the following exact solutions:1. For ab > 0 and 4c4 > 18ag > 0, there exists an explicit periodic wave solution

u(x, t) = 2c2 +

4c4 18ag cos

12

ab

(x ct)

3a. (23)

2. For ab > 0 and g = 0, there exists an explicit compacton solution

u(x, t) =

4c

2

3acos2

1

4

a

b(x ct)

1

4

a

b(x ct)

20 otherwise.

(24)

3. For ab < 0 and g = 0, there exists an explicit peakon solution

u(x, t) = 2c2

3a

1 exp

1

2

a

b|x ct|

. (25)

4. For ab < 0 and 4c4 > 18ag > 0, there exists an explicit periodic cusp wave solution

u(x, t) = 2c2

3a+

4c4 18ag cosh |x ct 2kT|3|a| (2k 1)T x ct 2kT (26)

where k is an integer and T = arch( 2c24c418ag

).

Theorem 3.2. When m=

n=

1 or 1, Eq. (6) has the following exact solutions:1. For ab > 0, and n > 1, there exists the following explicit compacton solution:

u(x, t) =

2c

2n

a(n + 1) cos2

a

b

n 12n

(x ct) 1

n1

a

b

n 12n

(x ct) 2

0 otherwise.

(27)

In particular, when n is a positive odd number and a < 0,

u(x, t) =

2c

2n

a(n + 1) cos2

a

b

n 12n

(x ct) 1

n1

a

b

n 12n

(x ct)

20 otherwise

(28)

is also a compacton solution of Eq. (6).


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2. For ab < 0 and n < 1, there exists the following explicit solitary wave solution:

u(x, t) = 2c

2n

a(n + 1) cosh2

a

b

n 12n

(x ct) 1

n1. (29)

In particular, when n is a negative odd number and a < 0,

u(x, t) = 2c

2n

a(n + 1) cosh2

a

b

n 12n

(x ct) 1

n1(30)

is also a solitary wave solution of Eq. (6).

Remark 1. Letting a = 1, b = 1, (27) is (50) in [11]. (35) and (36) in [15] are only special cases of(27) and (28)when a = 1 and b = 1. Letting a = 1, b = 1, (29) is just the result (57) in [13]. Letting a = 1, b = 1, (29) and(30) is just the result (54)(56) in [14].

Remark 2. We can also obtain the solution (53) in [13], which is an unbounded solution. Therefore we neglect it.

3.2. Exact solutions of generalized B(1, n)

When m = 1, Eq. (13) becomesdz

hz2n2n1 2 (n1)2

bz

a 11+nz

2 + c22

z + gz n2n1 = d. (31)

Case 1. The B(1, 2) equationWhen n = 2, Eq. (31) is reduced to

dz

h

2

bz a

3z

2

+c2

2 z + g= d. (32)

Define f(z) = h 2bz( a3z

2 + c22 z + g). Let z(z+ > z) be the two real roots of the equation f(z) = 0 when

c4 4ag > 0, i.e. z = c2

2a

c44ag2|a| .

1. For the case ofab < 0, let h = z2+( 3c2

2a+ 2z+); then

f(z) = 2a3b

(z z+)2

z + 2z+ +3c2

2a

. (33)

From (32), we get the following valley-shaped solitary wave solution of Eq. (10):

z = c22a

+

c4 4ag2|a|

3

c4 4ag2|a| sec h

2c4 4ag

b2

14

2

. (34)2. For the case ofab > 0, let h = z2( 3c

2

2a+ 2z); then

f(z) = 2a3b

(z z)2

z + 2z +3c2

2a

. (35)

From (32), we get the following peak-shaped solitary wave solution of Eq. (10):

z

=

c2

2a c4 4ag

2|a| +

3

c4 4ag

2|a|sec h2

c4 4ag

b2

14

2 . (36)


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Case 2. The B(1, n) equation (n = 1, 1)Let h = g = 0; then Eq. (31) is reduced to

dz

z2a(n1)2

(1+n)b z +(n+1)c2

2a = d. (37)

From (37), we get the following solitary wave solution of Eq. (10):

z = (n + 1)c2

2asec h2

(n 1)c

2b

(38)

when b < 0 and the following unbounded solution:

z = (n + 1)c2

2asec2

(n 1)c

2

b

(n 1)c2

b

< 2 (39)when b > 0.

Exact traveling wave solutions of generalized B(m, n) when m = 1 can be obtained by using the above results. Wedescribe this with the following two theorems.Theorem 3.3. When m = 1 and n = 2, Eq. (6) has the following exact solutions.1. When ab < 0 and c4 4ag > 0, Eq. (6) has a solitary wave solution of valley form as follows:

u(x, t) = c2

2a+

c4 4ag2|a|

3

c4 4ag2|a| sec h

2

c4 4ag

b2

14 x ct

2

. (40)

2. When ab > 0 and c4 4ag > 0, Eq. (6) has a solitary wave solution of peak form as follows:

u(x, t) = c2

2a

c4 4ag2|a| +

3

c4 4ag2|a| sec h

2

c4 4agb2

14 x ct

2

. (41)

Theorem 3.4. When m = 1 and n = 1, 1, Eq. (6) has the following exact solutions.1. When b < 0 and n > 1, Eq. (6) has a solitary wave solution as follows:

u(x, t) = (n + 1)c

2

2asec h2

(n 1)c

2b (x ct)

1n1

. (42)

In particular, when n is an odd number and a < 0,

u(x, t) = (n + 1)c

2

2asec h2

(n 1)c

2b (x ct)

1n1

(43)

is also a solitary wave solution of Eq. (6).

2. When b > 0 and n < 1, Eq. (6) has a compacton solution as follows:

u(x, t) =

(n + 1)c

2

2acos2

(n 1)c

2

b(x ct)

11n (n 1)c

2

b(x ct)

20 otherwise.

(44)

In particular, when n is a negative odd number and a > 0,

u(x, t) =

(n + 1)c

2

2acos2

(n 1)c

2

b(x ct)

11n (n 1)c

2

b(x ct)

20 otherwise

(45)

is also a compacton solution of Eq. (6).


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4. Conclusions

In summary, we have presented several compactons, peakons, solitary and periodic cusp wave solutions of the

generalized B(m, n) equation by using the integral approach. When the absolute values ofa and b are 1, some of our

results are identical to those in [6,11,1315]. In addition, to the best of our knowledge we are the first to present the

exact expressions for the peakons and the periodic cusp wave solutions and some new types of compacton solution of

the generalized B(m, n) equation when m = n = 2. It is worth pointing out that this approach can also be used toaddress some other equations.

Acknowledgements

This work was supported by the Natural Science Foundation of Zhejiang Province (No Y604359). Zhang L. would

like to express sincere gratitude to Prof. Jibin Li for his kindly encouragement.

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