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

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.


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:   +  2     +  3  2   +  4   = 0,  > 0, ( 2 ,  3 ,  4 ) ∈  3 . (2) These equations include a number of equations which have been studied by many authors [1][2][3].Song et al. [4,5] gave some solitary wave solutions and bifurcation phase portraits of (1) with  4 = 1,  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).
Dey and Coffey [9,10] considered the kink-profile solitary wave solutions for (1) with  = 1 and  = 2.The compound KdV-Burgers-type equation (2) with  > 1 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  = 1 and so on.

Conditions
Property of (0, 0)  even,  > 2 + 1 or  even,  = 2 + 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.

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 (0, 0) 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  + (, ) = 0 in a neighborhood of (0, 0).Assume that the development of the function (, ()) is of the form     + (higher order terms) and () = (/+ /)(, ()) =     +  with   ̸ = 0,  ≥ 2, and  ≥ 1.For  = 2 or  = 2 + 1, one gives the property of singular point (0, 0) as in Tables 1 and 2.
Case 1.For this case,  is even.
Case 2. For this case,  is odd and   < 0.
equations (  ) = 0 and (  , 0) = 0, respectively, we get two bifurcation curves Γ 1 and Γ 2 as follows: From the above analysis, we obtain the bifurcation phase portraits of system (9) as in Table 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.
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.
Under the above method and some parameter conditions, exact solitary wave solutions are obtained.Our results are as follows.

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.

Case 1. Consider
where  1 is an arbitrary constant.
Case 5. Consider Case 8. Consider where  ̸ = 1 and  1 are arbitrary constants and  0 and  1 are the same as (37).
Similarly, we can get other solutions.Here we complete the derivation of Propositions 2-4.

Conclusion
In this paper, we were devoted to investigate the dynamical survey and the phase orbits of the compound KdV-Burgerstype 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.