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.

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.

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