Journal of Applied Mathematics

Volume 2013 (2013), Article ID 490341, 17 pages

http://dx.doi.org/10.1155/2013/490341

Research Article

The Traveling Wave Solutions and Their Bifurcations for the BBM-Like Equations

Shaoyong Li1,2 and Zhengrong Liu²

¹College of Mathematics and Information Sciences, Shaoguan University, Shaoguan, Guangdong 512005, China

²Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

Received 3 June 2013; Accepted 3 July 2013

Academic Editor: Shiping Lu

Copyright © 2013 Shaoyong Li and Zhengrong Liu. 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.

Abstract.

We investigate the traveling wave solutions and their bifurcations for the BBM-like equations by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-like equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

1. Introduction

In recent years, the nonlinear phenomena exist in all fields including either the scientific work or engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, chemical kinematics, and chemical physics. Many nonlinear evolution equations are playing important roles in the analysis of the phenomena.

In order to find the traveling wave solutions of these nonlinear evolution equations, there have been many methods, such as Jacobi elliptic function method [1, 2], F-expansion and extended F-expansion method [3, 4], -expansion method [5, 6], and the bifurcation method of dynamical systems [7–11].

BBM equation or regularized long-wave equation (RLW equation)

was derived by Peregine [12, 13] and Benjamin et al. [14] as an alternative model to Korteweg-de Vries equation for small-amplitude, long wavelength surface water waves.

There are various generalized form related to (1). Shang [15] introduced a family of BBM-like equations with nonlinear dispersion

which were called BBM-like equations as alternative model to the nonlinear dispersive equations [16–18]. He presented a method called the extend sine-cosine method to seek exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of (2).

When , (2) reduces to the BBM-like equation

Jiang et al. [19] employed the bifurcation method of dynamical systems to investigate (3). Under different parametric conditions, they gave various sufficient conditions to guarantee the existence of smooth and nonsmooth traveling wave solutions. Furthermore, through some special phase orbits, they obtained some solitary wave solutions expressed by implicit functions, periodic cusp wave solution, compacton solution, and peakon solution.

Wazwaz [20] introduced a system of nonlinear variant RLW equations

and derived some compact and noncompact exact solutions by using the sine-cosine method and tanh method.

Feng et al. [21] studied the following generalized variant RLW equations:

By using four different ansatzs, they obtained some exact solutions such as compactons, solitary pattern solutions, solitons, and periodic solutions.

Kuru [22–24] considered the following BBM-like equations with a fully nonlinear dispersive term:

By means of the factorization technique, he obtained the traveling wave solutions of (6) in terms of the Weierstrass functions.

In the present paper, we use the bifurcation method and numerical simulation approach of dynamical systems to study the following BBM-like equations:

For BBM-like equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. We also reveal the relationships among these solutions theoretically. For BBM-like equation, we construct two elliptic periodic wave solutions and two hyperbolic blow-up solutions.

This paper is organized as follows. In Section 2, we state our main results which are included in two propositions. In Sections 3 and 4, we give the derivations for the two propositions, respectively. A brief conclusion is given in Section 5.

2. Main Results and Remarks

In this section, we list our main results and give some remarks. Firstly, let us recall some symbols. The symbols sn and cn denote the Jacobian elliptic functions sine amplitude and cosine amplitude . cosh , sinh , sech , and csch are the hyperbolic functions. Secondly, for the sake of simplification, we only consider the case , , and (the other cases can be considered similarly). To relate conveniently, for given constant wave speed , let

Proposition 1. Consider BBM-like equation

and its traveling wave equation

For given constants and , there are the following results.

When or , , (9) has two elliptic periodic blow-up solutions

where

For the graphs of   and , see Figures 1(a) and 1(b).

(a)

(b) .

Figure 1: The graphs of and when , , and .

When and , (9) has two trigonometric periodic blow-up solutions

The graphs of and are similar to Figure 1.

When and , (9) has two elliptic periodic blow-up solutions , and two symmetric elliptic periodic wave solutions ,

where

For the graphs of , see Figures 2–4.

(a)

(b)

(c)

(d)

Figure 2: The varying process for graphs of and when and where , , and (a) . (b) . (c) . (d) .

(a)

(b)

(c)

(d)

Figure 3: The varying process for graphs of when and where , , and (a) . (b) . (c) . (d) .

(a)

(b)

(c)

(d)

Figure 4: The varying process for graphs of when and where , , and (a) . (b) . (c) . (d) .

When and , (9) has a hyperbolic smooth solitary wave solution

a hyperbolic blow-up solution

a hyperbolic peakon wave solution

and a hyperbolic periodic peakon wave solution

where

For the graphs of and , see Figures 4(d) and 3(d). For the graphs of and , see Figures 7(a) and 7(b).

When and , (9) has two elliptic periodic blow-up solutions

where

For the graphs of and , see Figures 5(a) and 6(a).

(a)

(b)

(c)

(d)

Figure 5: The varying process for graphs of when and where , , and (a) . (b) . (c) . (d) .

(a)

(b)

(c)

(d)

Figure 6: The varying process for graphs of when and where , , and (a) . (b) . (c) . (d) .

(a)  

(b)

(c)

(d)

Figure 7: The graphs of , when , and , when , .

When , (9) has two elliptic periodic blow-up solutions

For the graphs of and , see Figures 7(c) and 7(d).

Remark 2. When and , the elliptic periodic blow-up solutions and become the trigonometric periodic blow-up solutions and , respectively.

Remark 3. When and , the elliptic periodic blow-up solutions and become the trigonometric periodic blow-up solutions and , respectively. The symmetric elliptic periodic wave solutions and become a trivial solution , and for the varying process, see Figure 2.

Remark 4. When and , the elliptic periodic blow-up solution becomes the hyperbolic blow-up solution , for the varying process, see Figure 3. The elliptic periodic wave solutions become the hyperbolic smooth solitary wave solution , and for the varying process, see Figure 4. The elliptic solutions and become a trivial solution .

Remark 5. When and , the elliptic periodic blow-up solution becomes the hyperbolic blow-up solution , and for the varying process, see Figure 5. The elliptic periodic blow-up solution becomes the hyperbolic smooth solitary wave solution , and for the varying process, see Figure 6.

Proposition 6. Consider BBM-like equation

and its traveling wave equation

For given constants and , there are the following results.

() When and , (32) has two elliptic periodic wave solutions

where

() When and , (32) has two hyperbolic blow-up solutions

where

For the graphs of , see Figure 8.

(a)  

(b)

(c)

(d)

Figure 8: The graphs of , when , and , when .

Remark 7. In order to confirm the correctness of these solutions, we have verified them by using the software Mathematica; for instance, about the commands are as follows:

3. The Derivations for Proposition 1

In this section, firstly, we derive the precise expressions of the traveling wave solutions for BBM-like equation. Secondly we show the relationships among these solutions theoretically. Substituting with into (9), it follows that

Integrating (39) once, we have

where is an integral constant.

Letting , we obtain the following planar system

Under the transformation

system (41) becomes

Clearly, system (41) and system (43) have the same first integral

where is an integral constant. Consequently, these two systems have the same topological phase portraits except for the straight line . Thus, we can understand the phase portraits of system (41) from that of system (43).

When the integral constant , (44) becomes

Solving equation , we get three roots , , and as (15), (16), and (17).

On the other hand, solving equation

we get three three singular points , where

According to the qualitative theory, we obtain the phase portraits of system (43) as Figures 9 and 10.

(a)  

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 9: The phase portraits of system (43) when and .

(a)  

(b)

(c)

Figure 10: The phase portraits of system (43) when .

From Figures 9 and 10, one can see that there are six kinds of orbits , where passes , passes and , and pass and , and passes . Now, we will derive the explicit expressions of solutions for the BBM-like equation, respectively.

When or , , has the expression

where and are complex numbers.

Substituting (48) into and integrating it, we have

Completing the integrals in the above two equations and noting that , we obtain and as (11).

When and , has the expression

where , .

Substituting (50) into and integrating it, we have

Completing the integrals in the two above equations and noting that , we obtain and as (19) and (20).

When and , and have the expressions

Substituting (52) into and integrating them, we have

Completing the integrals in the above four equations and noting that , we obtain as (21).

When and , and have the expressions

where , .

Substituting (54) into and integrating them, we have

Completing the integrals in the two equations above and noting that , we obtain and as (23) and (24).

When and , has the expression

where and are complex numbers.

Substituting (56) into and integrating it, we have

Completing the integrals in the two above equations and noting that , we obtain and as (28).

When , has the expression

where and are complex numbers.

Substituting (58) into and integrating it, we have

Completing the integrals in the two above equations and noting that , we obtain and as (30).

When and , there are two special kinds of orbits surrounding the center point (see Figure 11(d)) and surrounding the center point (see Figure 12(a)), which are the boundaries of two families of closed orbits. Note that the periodic waves of (9) correspond to the periodic integral curves of (40), and the periodic integral curves correspond to the closed orbits of system (41). For given constants , and the corresponding initial value , we simulate the integral curves of (40) as shown in Figures 11 and 12.

(a)

(b)

(c)

(d)

Figure 11: The simulation of integral curves of (40) when , , , (a) initial value , (b) initial value , and (c) initial value .

(a)

(b)

(c)

(d)

Figure 12: The simulation of integral curves of (40) when , , , (b) initial value , (c) initial value , and (d) initial value .

From Figure 11, we see that when the initial value tends to , the periodic integral curve tends to peakon. This implies that the orbit corresponds to peakon. On plane, has the expression

where , .

Substituting (60) into and integrating it, we have

Completing the integral in the above equation and noting that , we obtain as (25).

Similarly, from Figure 12, we see that when the initial value tends to , the periodic integral curve tends to a periodic peakon. This implies that the orbit corresponds to a periodic peakon. On plane, has the expression

Substituting (62) into and integrating it, we have

Completing the integral in the above equation and noting that , we obtain as (26), where

Hereto, we have finished the derivations for the solutions . In what follows, we shall derive the relationships among these solutions.

() When and , it follows that

Thus, we have

() When and , it follows that

Thus, we have

Hereto, we have completed the derivations for Proposition 1.

4. The Derivations for Proposition 6

In this section, we derive the precise expressions of the traveling wave solutions for BBM-like equation. Similar to the derivations in Section 3, substituting with into (32) and integrating it, we have the following planar system:

with the first integral

where and are the integral constants.

When , , and , (70) becomes

Substituting (71) into the first equation of (69) and integrating it from to or to , respectively, it follows that

Completing the integrals in the two above equations and noting that , we obtain the two elliptic periodic wave solutions and as (34).

When , , and , (70) becomes

where

Substituting (73) into the first equation of (69) and integrating it from to or to , respectively, it follows that

Completing the integrals in the two above equations and noting that , we obtain the two hyperbolic blow-up solutions and as (36).

Hereto, we have completed the derivations for Proposition 6.

5. Conclusion

In this paper, we have investigated the nonlinear wave solutions and their bifurcations for BBM-like equations. For BBM-like equation, we obtain some precise expressions of traveling wave solutions (see ), which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. We also reveal the relationships among these solutions theoretically (see Remarks 2–5 and the corresponding derivations). For BBM-like equation, we construct two elliptic periodic wave solutions and two hyperbolic blow-up solutions (see ). We would like to study the BBM-like equations further.

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.

Acknowledgment

This work is supported by the National Natural Science Foundation (no. 11171115).

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