New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 984791, 7 pages http://dx.doi.org/10.1155/2014/984791

Research Article New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup Equation Zhenshu Wen School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China Correspondence should be addressed to Zhenshu Wen; [email protected] Received 27 November 2013; Accepted 10 January 2014; Published 20 February 2014 Academic Editor: Yongkun Li Copyright © 2014 Zhenshu Wen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the nonlinear wave solutions for the Broer-Kaup equation. Many exact explicit expressions of the nonlinear wave solutions for the equation are obtained by exploiting the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which are new. Some previous results are extended.

1. Introduction

2. Bifurcation of Phase Portraits

In 1975, Broer [1] obtained a dispersive equation as follows:

In this section, we give the process of obtaining the bifurcation of phase portraits corresponding to (1). For given constant 𝑐, substituting 𝑢(𝑥, 𝑡) = 𝜑(𝜉), 𝜂(𝑥, 𝑡) = 𝜓(𝜉) with 𝜉 = 𝑥 − 𝑐𝑡 into (1), it follows

1 𝑢𝑡 + 𝜂𝑥 + (𝑢2 )𝑥 = 0, 2

(1)

𝜂𝑡 + (𝑢𝜂 + 𝑢 + 𝑢𝑥𝑥 )𝑥 = 0, which describes the evolution of horizontal velocity component 𝑢(𝑥, 𝑡) of water waves of height 𝜂(𝑥, 𝑡) propagating in both directions in an infinite narrow channel of finite constant depth. Equation (1) plays an important role in nonlinear physics and gains considerate attention [2–4]. The traveling wave solutions for (1) have been studied by many works, such as [5–10]. In this paper, we employ the bifurcation method and qualitative theory of dynamical systems [11–21] to investigate the nonlinear wave solutions for (1), and we obtain many exact explicit expressions of nonlinear wave solutions for (1). These nonlinear wave solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which, to our knowledge, are newly obtained. The remainder of this paper is organized as follows. In Section 2, we show the bifurcation of phase portraits corresponding to (1). We state our main results and the theoretical derivation for the main results in Section 3. A short conclusion will be given in Section 4.

󸀠 1 −𝑐𝜑󸀠 + 𝜓󸀠 + (𝜑2 ) = 0, 2 󸀠

󸀠󸀠 󸀠

(2)

−𝑐𝜓 + (𝜓𝜑 + 𝜑 + 𝜑 ) = 0. Integrating (2) once leads to 1 −𝑐𝜑 + 𝜓 + 𝜑2 = 𝑔, 2

(3)

−𝑐𝜓 + 𝜓𝜑 + 𝜑 + 𝜑󸀠󸀠 = 𝐺, where both 𝑔 and 𝐺 are integral constants, respectively. From the first equation of system (3), we obtain 1 𝜓 = 𝑔 + 𝑐𝜑 − 𝜑2 . 2

(4)

Substituting (4) into the second equation of system (3) leads to 1 3𝑐 𝜑󸀠󸀠 = 𝜑3 − 𝜑2 + (𝑐2 − 𝑔 − 1) 𝜑 + 𝑐𝑔 + 𝐺. 2 2

(5)

2

Journal of Applied Mathematics y

y

y

𝜙

𝜙

G < c − G0

𝜙

y 𝜙7

Γ3

y

Γ−5

𝜙6 𝜙

𝜙1

𝜙3 𝜙5

Γ+4

−𝜙0

c − G0 < G < c

G = c − G0

Γ−4

y + Γ1

Γ−2

𝜙0

𝜙

G=c y

Γ+5

𝜙8

Γ−1

Γ+2

𝜙

𝜙

𝜙4 𝜙2

c < G < G0 + c

G = G0 + c

G > G0 + c

Figure 1: The phase portraits of system (7).

By setting 𝜑 = 𝜙 + 𝑐, (5) becomes 1 1 𝜙󸀠󸀠 = 𝜙3 − (𝑐2 + 2𝑔 + 2) 𝜙 + 𝐺 − 𝑐. 2 2

Let 3

(6)

d𝜙 = 𝑦, d𝜉 (7)

1 1 1 𝐻 (𝜙, 𝑦) = 𝑦2 − 𝜙4 + (𝑐2 + 2𝑔 + 2) 𝜙2 − (𝐺 − 𝑐) 𝜙. 2 8 4 (8) Now, we study the bifurcation of phase portraits of system (7). Set

1 1 𝑓 (𝜙) = 𝜙3 − (𝑐2 + 2𝑔 + 2) 𝜙 + 𝐺 − 𝑐. 2 2

(9)

Obviously, 𝑓0 (𝜙) has three zero points, which can be expressed as 𝜙 = 0, ±𝜙0 ,

(10)

where 𝜙0 = √𝑐2 + 2𝑔 + 2, when 𝑐2 + 2𝑔 + 2 > 0. Additionally, it is easy to obtain the two extreme points of 𝑓(𝜙) as follows: 1 𝜙±∗ = ±√ (𝑐2 + 2𝑔 + 2). 3

𝜆 ± = ±√𝑓󸀠 (𝜙𝑖 ).

(13)

From the qualitative theory of dynamical systems, we therefore know that

with first integral

1 1 𝑓0 (𝜙) = 𝜙3 − (𝑐2 + 2𝑔 + 2) 𝜙, 2 2

(12)

which is the absolute value of extreme values of 𝑓0 (𝜙). Let (𝜙𝑖 , 0) be one of the singular points of system (7). Then the characteristic values of the linearized system of system (7) at the singular point (𝜙𝑖 , 0) are

Letting 𝑦 = 𝜙󸀠 , we obtain a planar system

d𝑦 1 3 1 2 = 𝜙 − (𝑐 + 2𝑔 + 2) 𝜙 + 𝐺 − 𝑐, d𝜉 2 2

2 󵄨 󵄨 √ (𝑐 + 2𝑔 + 2) , 𝑔0 = 󵄨󵄨󵄨𝑓0 (𝜙±∗ )󵄨󵄨󵄨 = 27

(11)

(i) if 𝑓󸀠 (𝜙𝑖 ) > 0, then (𝜙𝑖 , 0) is a saddle point; (ii) if 𝑓󸀠 (𝜙𝑖 ) < 0, then (𝜙𝑖 , 0) is a center point; (iii) if 𝑓󸀠 (𝜙𝑖 ) = 0, then (𝜙𝑖 , 0) is a degenerate saddle point. Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system (7) in Figure 1.

3. Main Results and the Theoretic Derivations of the Main Results In this section, we state our results about solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions for the first component of system (7). To relate conveniently, we omit 𝜑 = 𝜙 + 𝑐 and the expression of the second component of system (7), that is, 𝜓 = 𝑔 + 𝑐𝜑 − (1/2)𝜑2 , in the following theorem. Theorem 1. For given constants 𝑐 and 𝛼 (1 < 𝛼 < 3), which will be given later, the Broer-Kaup equation (1) has the following exact explicit nonlinear wave solutions. (1) When 𝐺 = 𝑐, one obtains two kink-shaped solutions 𝜙1± (𝑥, 𝑡) = ±√𝑐2 + 2𝑔 + 2 tanh (

√𝑐2 + 2𝑔 + 2 2

(𝑥 − 𝑐𝑡)) , (14)

Journal of Applied Mathematics

3

two singular solutions

+ √6 − 2𝛼 sinh (√

𝜙2± (𝑥, 𝑡) = ±√𝑐2 + 2𝑔 + 2 coth (

√𝑐2 + 2𝑔 + 2 2

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 𝛼 2

(𝑥 − 𝑐𝑡) ) , × |𝑥 − 𝑐𝑡| ) + 4 − 2𝛼)

(15) and four periodic singular solutions

𝑐2 + 2𝑔 + 2 (3 − 𝛼) × (2 cosh (√ (𝑥 − 𝑐𝑡)) 𝛼 2

𝜙3± (𝑥, 𝑡) = ±√2 (𝑐2 + 2𝑔 + 2) × sec (√

𝑐2 + 2𝑔 + 2 (𝑥 − 𝑐𝑡)) , 2

+ √6 − 2𝛼 sinh (√ (16)

𝜙4± (𝑥, 𝑡) =

±√2 (𝑐2

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 𝛼 2

+ 2𝑔 + 2)

−1

× |𝑥 − 𝑐𝑡| ) − 2) ,

𝑐2 + 2𝑔 + 2 × csc (√ (𝑥 − 𝑐𝑡)) . 2

(18) and two periodic singular solutions

(2) When 𝑐 < 𝐺 < 𝐺0 + 𝑐, one gets two solitary wave solutions 𝜙5± (𝑥, 𝑡) = ±√

𝜙7± (𝑥, 𝑡) = ± (𝜙2 (𝜙6 − 𝜙7 ) sin (arcsin (

𝑐2 + 2𝑔 + 2 𝛼

−

𝑐2 + 2𝑔 + 2 (3 − 𝛼) × (√2𝛼 − 2 cosh (√ (𝑥 − 𝑐𝑡)) 𝛼 2

2𝜙2 − 𝜙6 − 𝜙7 ) 𝜙6 − 𝜙7

√− (𝜙2 − 𝜙6 ) (𝜙2 − 𝜙7 ) 2

× |𝑥 − 𝑐𝑡| ) (19) + 2𝜙6 𝜙7 − 𝜙2 𝜙6 − 𝜙2 𝜙7 )

+2𝛼 − 4)

× ((𝜙6 − 𝜙7 ) sin (arcsin (

𝑐2 + 2𝑔 + 2 (3 − 𝛼) × (√2𝛼 − 2 cosh (√ 𝛼 2

−

−1

× (𝑥 − 𝑐𝑡) ) + 2) ,

−1

where 𝜙2 , 𝜙6 , and 𝜙7 will be given in the proof of the theorem.

𝜙8± (𝑥, 𝑡) = ±√

2

𝑐 + 2𝑔 + 2 𝛼

× (2 cosh (√

2

(3) When 𝐺 = 𝐺0 + 𝑐, one obtains four singular solutions as follows:

𝜙6± (𝑥, 𝑡) = ±√

√− (𝜙2 − 𝜙6 ) (𝜙2 − 𝜙7 )

× |𝑥 − 𝑐𝑡|) − 4𝜙2 ) , (17)

two singular solutions

2𝜙2 − 𝜙6 − 𝜙7 ) 𝜙6 − 𝜙7

𝑐2 + 2𝑔 + 2 3 2

𝑐2 + 2𝑔 + 2 (3 − 𝛼) (𝑥 − 𝑐𝑡)) 𝛼 2

×

(1 + √(𝑐2 + 2𝑔 + 2) /3|𝑥 − 𝑐𝑡|) + 3 (1 +

√(𝑐2

2

+ 2𝑔 + 2) /3|𝑥 − 𝑐𝑡|) − 1

,

4

Journal of Applied Mathematics

×

At the same time, we note that if 𝜙 = 𝜙(𝜉) is a solution of system, then 𝜙 = 𝜙(𝜉+𝛾) is also a solution of system. Specially, we take 𝛾 = 𝜋/2; we obtain another two solutions

𝑐2 + 2𝑔 + 2 3

𝜙9± (𝑥, 𝑡) = ±√

(𝑐2 + 2𝑔 + 2) (𝑥 − 𝑐𝑡)2 + 9 (𝑐2 + 2𝑔 + 2) (𝑥 − 𝑐𝑡)2 − 3

.

𝜙 = ±√2 (𝑐2 + 2𝑔 + 2)csc (√ (20)

Proof. (1) When 𝐺 = 𝑐, we consider the following two kinds of orbits. (i) First, we see that there are two heteroclinic orbits Γ1± connected as two saddle points (𝜙0 , 0) and (−𝜙0 , 0) from Figure 1. In (𝜙, 𝑦)-plane, from (8), the expressions of the heteroclinic orbits are given as 1 𝑦 = ± (𝜙02 − 𝜙2 ) . 2

𝜙

𝑓(𝜙) = 0 to be 𝜙1 = √(𝑐2 + 2𝑔 + 2)/𝛼 (1 < 𝛼 < 3), and then we can obtain another two solutions of 𝑓(𝜙) = 0 as follows: 1 𝑐2 + 2𝑔 + 2 √ 𝜙2 = √ ( 4𝛼 − 3 − 1) , 2 𝛼

(21)

0

∫

d𝑠 1󵄨 󵄨 = 󵄨󵄨𝜉󵄨󵄨 , (𝜙02 − 𝑠2 ) 2 󵄨 󵄨

+∞

𝜙

(22)

d𝑠 1󵄨 󵄨 = 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 . 2 2 (𝑠 − 𝜙0 ) 2

(27)

Noting that 𝜙 = 𝜙(𝜉) and 𝜉 = 𝑥 − 𝑐𝑡, we get four periodic singular solutions 𝜙3± (𝑥, 𝑡) and 𝜙4± (𝑥, 𝑡) as (16). (2) When 𝑐 < 𝐺 < 𝐺0 + 𝑐, we set the largest solution of

(28)

2

1 𝑐 + 2𝑔 + 2 √ 𝜙3 = − √ ( 4𝛼 − 3 + 1) . 2 𝛼

Substituting (21) into the first equation of system (7) and integrating along the heterclinic orbits, it follows that ∫

𝑐2 + 2𝑔 + 2 𝜉) . 2

(i) First, we see that there is a homoclinic orbit Γ3 , which passes the saddle point (𝜙1 , 0). In (𝜙, 𝑦)-plane, from (8), the expressions of the homoclinic orbit are given as 1 2 𝑦 = ± √(𝜙1 − 𝜙) (𝜙 − 𝜙4 ) (𝜙 − 𝜙5 ), 2

From (22), we have

(29)

where 𝜙 = ±√𝑐2 + 2𝑔 + 2 tanh (

√𝑐2 + 2𝑔 + 2 2

𝜉) ,

𝜙4 = √ (23)

𝜙 = ±√𝑐2 + 2𝑔 + 2 coth (

√𝑐2 + 2𝑔 + 2 2

𝜙5 = −√

𝜉) .

Noting that 𝜙 = 𝜙(𝜉) and 𝜉 = 𝑥 − 𝑐𝑡, we get two kinkshaped solutions 𝜙1± (𝑥, 𝑡) and two singular solutions 𝜙2± (𝑥, 𝑡) as (14) and (15). (ii) Second, from the phase portrait in Figure 1, we note that there are two special orbits Γ2± , which have the same Hamiltonian as that of the center point (0, 0). In (𝜙, 𝑦)-plane, from (8), the expressions of these two orbits are given as 1 𝑦 = ± 𝜙√𝜙2 − 2𝜙02 . 2

+∞

𝜙

d𝑠 𝑠√𝑠2 −

2𝜙02

=

1 󵄨󵄨 󵄨󵄨 󵄨𝜉󵄨 . 2󵄨 󵄨

𝑐 + 2𝑔 + 2 (√2𝛼 − 2 + 1) . 𝛼

𝜙

d𝑠

𝜙4

(𝜙1 − 𝑠) √(𝑠 − 𝜙4 ) (𝑠 − 𝜙5 )

∫

+∞

∫

(24)

=

1 󵄨󵄨 󵄨󵄨 󵄨𝜉󵄨 , 2󵄨 󵄨

1󵄨 󵄨 = 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 . (𝑠 − 𝜙1 ) √(𝑠 − 𝜙4 ) (𝑠 − 𝜙5 ) 2 d𝑠

From (31), we have 𝜙 = ±√

𝑐2 + 2𝑔 + 2 𝛼

(25) × (√2𝛼 − 2 cosh (√

From (25), we have 𝑐2 + 2𝑔 + 2 𝜙 = ±√2 (𝑐2 + 2𝑔 + 2)sec (√ 𝜉) . 2

(26)

(30)

2

Substituting (29) into the first equation of system (7) and integrating along the homoclinic orbit, it follows that

𝜙

Substituting (24) into the first equation of system (7) and integrating along the two orbits Γ2± , it follows that ∫

𝑐2 + 2𝑔 + 2 (√2𝛼 − 2 − 1) , 𝛼

+ 2𝛼 − 4)

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 𝜉) 𝛼 2

(31)

Journal of Applied Mathematics

5 30

−1

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 𝜉) + 2) , × (√2𝛼 − 2 cosh (√ 𝛼 2 𝜙 = ±√

𝑐2 + 2𝑔 + 2 𝛼

× (2 cosh (√

20 10 −15

−10

−5

5

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 𝜉) 𝛼 2

−30

Figure 2: The graphic of 𝜙8+ (𝜉) when 𝑐 = 2, 𝑔 = 1, and 𝛼 = 2.

From (35), we have 𝜙 = ± (𝜙2 (𝜙6 − 𝜙7 ) sin (arcsin (

+ 4 − 2𝛼) 𝑐2 + 2𝑔 + 2 (3 − 𝛼) × (2 cosh (√ 𝜉) 𝛼 2 + √6 − 2𝛼 sinh (√

−

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 󵄨󵄨 󵄨󵄨 󵄨𝜉󵄨) 𝛼 2 󵄨 󵄨

(32) Noting that 𝜙 = 𝜙(𝜉) and 𝜉 = 𝑥 − 𝑐𝑡, we get two solitary solutions 𝜙5± (𝑥, 𝑡) and two singular solutions 𝜙6± (𝑥, 𝑡) as (17) and (18). (ii) Second, from the phase portrait in Figure 1, we note that there are another two special orbits Γ4± , which have the same Hamiltonian as that of the center point (𝜙2 , 0). In (𝜙, 𝑦)plane, from (8), the expressions of these two orbits are given as (33)

where 𝜙6 = −𝜙2 + √2 (𝑐2 + 2𝑔 + 2) − 2𝜙22 ,

(34)

𝜙7 = −𝜙2 − √2 (𝑐2 + 2𝑔 + 2) − 2𝜙22 . Substituting (33) into the first equation of system (7) and integrating along these two special orbits Γ4± , it follows that

𝜙

√− (𝜙2 − 𝜙6 ) (𝜙2 − 𝜙7 )

× ((𝜙6 − 𝜙7 ) sin ( arcsin (

− 2) .

1 2 𝑦 = ± √(𝜙 − 𝜙2 ) (𝜙 − 𝜙6 ) (𝜙 − 𝜙7 ), 2

d𝑠 (𝑠 − 𝜙2 ) √(𝑠 − 𝜙6 ) (𝑠 − 𝜙7 )

=

1 󵄨󵄨 󵄨󵄨 󵄨𝜉󵄨 . 2󵄨 󵄨

2𝜙2 − 𝜙6 − 𝜙7 ) 𝜙6 − 𝜙7

2

󵄨󵄨 󵄨󵄨 󵄨󵄨𝜉󵄨󵄨)

+ 2𝜙6 𝜙7 − 𝜙2 𝜙6 − 𝜙2 𝜙7 )

−1

+∞

15

−20

𝑐2 + 2𝑔 + 2 (3 − 𝛼) 󵄨󵄨 󵄨󵄨 + √6 − 2𝛼 sinh (√ 󵄨𝜉󵄨) 𝛼 2 󵄨 󵄨

∫

10

−10

(35)

−

2𝜙2 − 𝜙6 − 𝜙7 ) 𝜙6 − 𝜙7

√− (𝜙2 − 𝜙6 ) (𝜙2 − 𝜙7 ) 2

󵄨󵄨 󵄨󵄨 󵄨󵄨𝜉󵄨󵄨)

−1

− 4𝜙2 ) . (36) Noting that 𝜙 = 𝜙(𝜉) and 𝜉 = 𝑥 − 𝑐𝑡, we get two periodic singular solutions 𝜙7± (𝑥, 𝑡) as (19). To illustrate, we give the graphic of 𝜙8+ (𝜉) in Figure 2 by taking 𝑐 = 2, 𝑔 = 1, and 𝛼 = 2. (3) When 𝐺 = 𝐺0 + 𝑐, from the phase portrait in Figure 1, we note that there are two orbits Γ5± , which have the same Hamiltonian as the degenerate saddle point (𝜙8 , 0). In (𝜙, 𝑦)plane, from (8), the expressions of these two orbits are given as 1 3 (37) 𝑦 = ± √(𝜙 − 𝜙8 ) (𝜙 + 3𝜙8 ), 2 where 1 (38) 𝜙8 = √ (𝑐2 + 2𝑔 + 2). 3 Substituting (37) into the first equation of system (7) and integrating along these two orbits Γ5± , it follows that ∫

+∞

𝜙

d𝑠 (𝑠 − 𝜙8 ) √(𝑠 − 𝜙8 ) (𝑠 + 3𝜙8 )

=

1 󵄨󵄨 󵄨󵄨 󵄨𝜉󵄨 , 2󵄨 󵄨

6

Journal of Applied Mathematics ∫

−3𝜙8

d𝑠 (𝜙8 − 𝑠) √(𝜙8 − 𝑠) (−3𝜙8 − 𝑠)

𝜙

=

𝜑 = 𝜙 + 𝑐,

1 󵄨󵄨 󵄨󵄨 󵄨𝜉󵄨 . 2󵄨 󵄨 (39)

From (3), we have 2 √(𝑐2 + 2𝑔 + 2) /3 󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨) + 3 (1 + 𝑐 + 2𝑔 + 2 𝜙 = ±√ , 3 󵄨󵄨 󵄨󵄨 2 2 √ (1 + (𝑐 + 2𝑔 + 2) /3 󵄨󵄨𝜉󵄨󵄨) − 1

𝜙 = ±√

(41) (40)

𝑐 + 2𝑔 + 2 (𝑐 + 2𝑔 + 2) (𝑥 − 𝑐𝑡) + 9 . 3 (𝑐2 + 2𝑔 + 2) (𝑥 − 𝑐𝑡)2 − 3

Noting that 𝜙 = 𝜙(𝜉) and 𝜉 = 𝑥 − 𝑐𝑡, we get four singular solutions 𝜙8± (𝑥, 𝑡) and 𝜙9± (𝑥, 𝑡) as (20). Remark 2. One may find that we only consider some special orbits in Figure 1 when 𝐺 ≥ 𝑐. In fact, we may obtain exactly the same results when 𝐺 < 𝑐. Remark 3. We employ the software Mathematica to check the correctness of the above nonlinear wave solutions. To illustrate, we show the commands of verifying 𝜙8+ (𝑥, 𝑡), 1 𝑐2 + 2𝑔 + 2 √ 𝜙2 = √ ( 4𝛼 − 3 − 1) , 2 𝛼

2

𝜙7 = −𝜙2 − √2 (𝑐2 + 2𝑔 + 2) − 2(𝜙2 ) ,

[1] L. J. F. Broer, “Approximate equations for long water waves,” Applied Scientific Research, vol. 31, no. 5, pp. 377–395, 1975.

] (𝑥 − 𝑐𝑡)] ]

+ 2𝜙6 𝜙7 − 𝜙2 𝜙6 − 𝜙2 𝜙7 )

− 4𝜙2 ) ,

[2] D. J. Kaup, “A higher-order water-wave equation and the method for solving it,” Progress of Theoretical Physics, vol. 54, no. 2, pp. 396–408, 1975. [3] B. A. Kupershmidt, “Mathematics of dispersive water waves,” Communications in Mathematical Physics, vol. 99, no. 1, pp. 51– 73, 1985. [4] M. Ablowitz and H. Segur, “Solitons, nonlinear evolution equations and inverse scattering,” Journal of Fluid Mechanics, vol. 244, pp. 721–725, 1992. [5] M. Wang, J. Zhang, and X. Li, “Application of the 𝐺󸀠 /𝐺expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 321–326, 2008.

2𝜙 − 𝜙6 − 𝜙7 [ × ((𝜙6 − 𝜙7 ) Sin [ArcSin [ 2 ] 𝜙6 − 𝜙7 [

−1

The author declares that there is no conflict of interests regarding the publication of this paper.

References

2𝜙 − 𝜙6 − 𝜙7 [ 𝜙 = (𝜙2 (𝜙6 − 𝜙7 ) Sin [ArcSin [ 2 ] 𝜙6 − 𝜙7 [

2

Conflict of Interests

This research was supported by Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (no. ZQN-PY119) and the Foundation of Huaqiao University (no. 12BS223).

2

√− (𝜙2 − 𝜙6 ) (𝜙2 − 𝜙7 )

In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the nonlinear wave solutions for the Broer-Kaup equation (1) and obtain exact explicit expressions of the various kinds of nonlinear wave solutions, which include solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions. To the best of our knowledge, most of the nonlinear wave solutions are newly obtained.

Acknowledgments

𝜙6 = −𝜙2 + √2 (𝑐2 + 2𝑔 + 2) − 2(𝜙2 ) ,

−

4. Conclusions

2

2

−

1 Simplify [𝜕𝑡 𝜑 + 𝜕𝑥 𝜓 + 𝜕𝑥 (𝜑2 )] , 2 Simplify [𝜕𝑡 𝜓 + 𝜕𝑥 (𝜑𝜓 + 𝜑 + 𝜕𝑥,𝑥 𝜑)] .

2

2

1 𝜓 = 𝑔 + 𝑐𝜑 − 𝜑2 , 2

√− (𝜙2 − 𝜙6 ) (𝜙2 − 𝜙7 ) 2

] (𝑥 − 𝑐𝑡)] ]

[6] D. Ganji, H. B. Rokni, M. Sfahani, and S. Ganji, “Approximate traveling wave solutions for coupled whitham-broer-kaup shallow water,” Advances in Engineering Software, vol. 41, no. 7, pp. 956–961, 2010. [7] S. Guo, Y. Zhou, and C. Zhao, “The improved 𝐺󸀠 /𝐺-expansion method and its applications to the Broer-Kaup equations and approximate long water wave equations,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 1965–1971, 2010.

Journal of Applied Mathematics [8] M. Alquran, K. Al-Khaled, and H. Ananbeh, “New soliton solutions for systems of nonlinear evolution equations by the rational sine-cosine method,” Studies in Mathematical Sciences, vol. 3, no. 1, pp. 1–9, 2011. [9] M. F. El-Sabbagh and S. I. El-Ganaini, “The He’s variational principle to the Broer-Kaup (BK) and Whitham Broer-Kaup (WBK) systems,” International Mathematical Forum, vol. 7, no. 41–44, pp. 2131–2141, 2012. [10] M. F. El-Sabbagh and S. I. El-Ganaini, “New exact solutions of Broer-Kaup (BK) and Whitham Broer-Kaup (WBK)systems via the first integral method,” International Journal of Mathematical Analysis, vol. 6, no. 45–48, pp. 2287–2298, 2012. [11] M. Song, S. Li, and J. Cao, “New exact solutions for the (2 + 1)-dimensional Broer-Kaup-Kupershmidt equations,” Abstract and Applied Analysis, vol. 2010, Article ID 652649, 9 pages, 2010. [12] Z. Wen, Z. Liu, and M. Song, “New exact solutions for the classical drinfel’d-sokolov-Wilson equation,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2349–2358, 2009. [13] Z. Wen and Z. Liu, “Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa-Holm equation,” Nonlinear Analysis. Real World Applications, vol. 12, no. 3, pp. 1698–1707, 2011. [14] B. He, Q. Meng, J. Zhang, and Y. Long, “Periodic loop solutions and their limit forms for the Kudryashov-Sinelshchikov equation,” Mathematical Problems in Engineering, vol. 2012, Article ID 320163, 10 pages, 2012. [15] Z. Wen, “Extension on bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation,” Abstract and Applied Analysis, vol. 2012, Article ID 704931, 15 pages, 2012. [16] Z. Wen, “Bifurcation of traveling wave solutions for a twocomponent generalized 𝜃-equation,” Mathematical Problems in Engineering, vol. 2012, Article ID 597431, 17 pages, 2012. [17] S. Li and Z. Liu, “The traveling wave solutions and their bifurcations for the bbm-like B(m, n) equations,” Journal of Applied Mathematics, vol. 2013, Article ID 490341, 17 pages, 2013. [18] Y. Long and C. Chen, “Existence analysis of traveling wave solutions for a generalization of KdV equation,” Mathematical Problems in Engineering, vol. 2013, Article ID 462957, 7 pages, 2013. [19] M. Song, B. S. Ahmed, and A. Biswas, “Topological soliton solution and bifurcation analysis of the Klein-Gordon-Zakharov equation in (1 + 1)-dimensions with power law nonlinearity,” Journal of Applied Mathematics, vol. 2013, Article ID 972416, 7 pages, 2013. [20] Y. Wu and Z. Liu, “New types of nonlinear waves and bifurcation phenomena in Schamel-Korteweg-de Vries equation,” Abstract and Applied Analysis, vol. 2013, Article ID 483492, 18 pages, 2013. [21] Z. Wen, “Bifurcation of solitons, peakons and periodic cusp waves for 𝜃-equation,” Nonlinear Dynamics, 2014.

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