Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 571672, 8 pages
http://dx.doi.org/10.1155/2015/571672

Research Article

Bifurcation and Solitary-Like Solutions for Compound KdV-Burgers-Type Equation

Yin Li, Ruiying Wei, and Guoming Jian

School of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China

Received 22 October 2014; Accepted 22 January 2015

Academic Editor: Morteza Rafei

Copyright © 2015 Yin Li et al. 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.

Firstly, based on the improved sub-ODE method and the bifurcation method of dynamical systems, we investigate the bifurcation of solitary waves in the compound KdV-Burgers-type equation. Secondly, numbers of solitary patterns solutions are given for each parameter condition and numerical simulations are used to display the dynamical characteristics. Finally, we obtain twelve solitary patterns solutions under some parameter conditions, such as the trigonometric function solutions and the hyperbolic function solutions.

1. Introduction and Main Results

Consider the following compound KdV-Burgers-type equation with nonlinear terms of any order:

and the compound KdV-type equation with nonlinear terms of any order:

These equations include a number of equations which have been studied by many authors [1–3]. Song et al. [4, 5] gave some solitary wave solutions and bifurcation phase portraits of (1) with being odd or even. It is necessary to point out that when we take the different parameters values, the following equations can be derived from (1).

If , , and , (1) becomes the equation

which was studied in [1].

If , , , and , (1) becomes the mKdV and combined KdV equation [6]:

If , , , and , (1) becomes Gardner’s equation [7]:

Biswas and Zerrad [7] also studied the soliton perturbation theory of (5).

If , and , and , (1) becomes the equation [8]

Dey and Coffey [9, 10] considered the kink-profile solitary wave solutions for (1) with and . The compound KdV-Burgers-type equation (2) with is a model for long-wave propagation in nonlinear media with dispersion and dissipation [11]; Coffey [3] considered soliton solutions, conservation laws, Backlund transformation, and other properties for (1) with and so on.

In this paper, we apply the bifurcation method of dynamical systems [12–23] and the improved sub-ODE method [24–30] with the help of symbolic computation Mathematica and Maple to study solitary wave solutions of (1).

Based on the bifurcation method and the improved sub-ODE method with the help of symbolic computation Mathematica, the bifurcation for (1) is derived by use of a proper transformation. To our knowledge, this type of transformation obtained has not been ever seen before in the literature. Then based on this transformation, some new solutions for (1) and (2) are found.

This paper is organized as follows. In Section 2, we derive the bifurcations of phase portraits of (1). In Section 3, we give the main results via the sub-ODE method. In Section 4, the main results and some exact travelling wave solutions for (1) are obtained. Conclusions are given in the last section.

2. Bifurcations of Phase Portraits of (1)

In this section, we discuss some bifurcation phase portraits of (1). To study dynamical behavior of (1), we first need to give the lemmas [21, 31] in this section.

Lemma 1. Let be a nilpotent singular point of the vector field , where and are analytic functions in a neighborhood of the origin at least with quadratic terms in the variables and . Let be the solution of the equation in a neighborhood of . Assume that the development of the function is of the form (higher order terms) and with , , and . For or , one gives the property of singular point as in Tables 1 and 2.

Table 1: The property of singular point .

Conditions
Property of

The origin is the saddle node.

, or
The origin is a cusp.

Table 2: The property of singular point .

Conditions
Property of

even, or even,
Stable node

odd, or odd,
Elliptic saddle

or
Focus or center

Case 1. For this case, is even.

Case 2. For this case, is odd and .

To this end, substituting with into (1), we have

If tends to , we obtain the solitary wave solutions and kink wave solutions tend to some constants. This implies that , , and tend to zero, when tends to . Integrating (7) once we get

where is the constant of integration. Letting , we obtain a planar system:

where .

System (9) has the following first integral:

where is the constant of integration. To study the distribution of singular points of system (9), we need to investigate the zero points of . If , is odd, , and , then we get three zero points of as

where . Suppose is a singular point of system (9). Let

be the coefficient matrix of the linearized system of (9). Then the eigenvalue of at is

According to Lemma 1 and the theory of dynamical systems [4, 5, 12–16], we obtain the following conclusions.

(i)If , is a saddle point.(ii)If , is a degenerate saddle point.(iii)If , is a center point. Solving equations and , respectively, we get two bifurcation curves and as follows:

From the above analysis, we obtain the bifurcation phase portraits of system (9) as in Table 3.

Table 3: The local bifurcation phase portraits of system (9).

=
,

, , , is odd.

, , , is odd.

, , . 

, , .

, , , is even.

, , , is even.

3. The Improved Method and Statement of the Main Results

In this section, we will give some exact parametric representations of travelling wave solutions of (1). To study solitary wave solutions and the qualitative behavior of system (1), we first need to introduce the improved method based on [25] as follows. Suppose a nonlinear PDE,

where is a polynomial in its arguments.

Step 1 (reduce NPDE to nonlinear ODE). By taking the transformation , , where is arbitrary nonzero constants, and (15) transform it to the ordinary differential equation reduces to be

Step 2 (determine the parameters). Determine the highest order nonlinear term and the linear term of the highest order term in (15) or (16). Then, in the resulting terms, balance the highest order nonlinear term and the linear term of the highest order term; we get a balance constant ( is usually a positive integer). If is a negative integer or a fraction, we make the following transformation:

We obtain the solutions of (15) to be as the following forms.

Case 1. If in (19)-(20),

where and satisfy the following equations:

which admits the first integral with ,

where . When , ,   and , (20) becomes a projective Riccati equation [25].

Case 2. If in (19)-(20),

where .

Step 3. (I) If , substitute (18) along with conditions (19)-(20) into (15).
(II) If , substituting (21) along with into (15) yields a set of algebraic equations for , . Setting the coefficients of these terms (or ) to zero yields a set of overdetermined algebraic equations in , ,  , , and .

Step 4. According to the theory of dynamical systems with the aid of Mathematica, solving the above set of equations yields the values of , , , and .

Step 5. We know that (15) admits the following solutions.

Case 1. If , ,

Case 2. If , ,

Case 3. If ,

where is a constant. According to (18), (21), (22)–(24), and the conclusions in Step 4, we can obtain many solutions for (15).

Under the above method and some parameter conditions, exact solitary wave solutions are obtained. Our results are as follows.

Proposition 2. (1) If , (1) has two solitary wave trigonometric function solutions:

where and .
(2) If , (1) has two solitary wave solutions:

where and is arbitrary constant.

Proposition 3. (1) If , (1) has two solitary wave solutions:

where ,    , and .
(2) If , (1) has two solitary wave solutions:

where , , and is arbitrary constant.

Proposition 4. (1) If , , (1) has two solitary wave hyperbolic function solutions:

where and and are arbitrary constants.
(2) If , (1) has two solitary wave solutions:

where , and is arbitrary constant.

Proposition 5. If , , (1) has two solitary wave solutions:

where

4. The Derivations of Theorem

4.1. The Derivations of Proposition Using the Improved Sub-ODE Method

In this section, we make the travelling transformation to (1) , , where is a constant to be determined later, and thus (1) becomes (8). Integrating (8) once with regard to , we obtain

with the integration constants taken to be zero.

According to Step 1 in Section 3, if , , , , and , by balancing and in (34), we get . Therefore, we make the following transformation:

then substituting (34) into (35) yields

where

According to Step 1 in Section 3, by balancing (or ) and in (36), we get . Therefore, we suppose that (37) has the following formal solutions:

where satisfy (19)-(20), where are constants to be determined later.

With the aid of Mathematica and Maple, substituting (38) along with (18)-(19) into (36) yields a set of algebraic equations for   . Setting the coefficients of these terms to zero yields a set of overdetermined algebraic equations with respect to , , , , and .

Case 1. Consider

Case 2. Consider

Case 3. Consider

Case 4. Consider

where is an arbitrary constant.

Case 5. Consider

Case 6. Consider

Case 7. Consider

Case 8. Consider

where and are arbitrary constants and and are the same as (37).

Solving the overdetermined algebraic equations, we get Propositions 2–5.

4.2. The Derivations of Proposition Using the Integral Method

In the following, we study the derivations of theorem using the integral method [4, 5, 13–16].

If , (9) reduces to the following system:

From Table 3, we obtain two homoclinic orbits () of (9).

In plane, is described as

where

We obtain the results by substituting (48) into :

If , then, integrating (50) along homoclinic orbit , we have

We obtain by completing the above two integrals

where

We obtain as formula (31) via (53).

If , integrating (50) along homoclinic orbit , we have

We obtain by completing the above two integrals

where

Our results (52) and (55) can be compared with the result of Song et al. [5] by assigning special values to the arbitrary parameters involved.

From (52), (55), and traveling wave transformation (34), we get solitary wave solutions , just as expressed in (31) and (32).

Similarly, we can get other solutions. Here we complete the derivation of Propositions 2–4.

5. Conclusion

In this paper, we were devoted to investigate the dynamical survey and the phase orbits of the compound KdV-Burgers-type equation via the improved sub-ODE method and the integral method. Some solitary wave solutions are given for each parameter condition. At the same time, the method can be widely applied to other nonlinear equations. Not only do we obtain the explicit solitary wave solutions but also we can get other kinds of solutions such as the trigonometric function solutions, the hyperbolic function solutions, and the rational solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor Yulin Zhao and Professor Zheng-an Yao of Sun Yat-sen University and Professors Y. Chen of East China Normal University, B. Li of NingBo University, and C. L. Zheng and Shaoyong Li of Shaoguan University for their fruitful help and discussions. The authors are also in debt to express thanks to the referees for valuable suggestions and for their positive comments. This work is supported by the NSF of China (no. 11171355), Guangdong Provincial Culture of Seedling of China (no. 2013LYM0081), Guangdong Provincial NSF of China (no. S2012010010069), Guangdong Provincial Department of Education research platform project (Guangdong Young Creative Talents (no. 2014208)). The Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University.

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