#### Abstract

The objective of this paper is to extend some results of pioneers for the nonlinear equation introduced by Qiao. The equivalent relationship of the traveling wave solutions between the integrable equation and the generalized KdV equation is revealed. Moreover, when and , we obtain some explicit traveling wave solutions by the bifurcation method of dynamical systems.

#### 1. Introduction

Qiao [1] introduced the following equation: which is the second positive member in a new completely integrable hierarchy. Equation (1) possesses a Lax representation and bi-Hamiltonian structure [1, 2]. In [1, 2] the traveling wave solutions of (1) were studied. Sakovich [3] found the transformation which relates (1) with the well-known modified KdV equation and obtained three types of smooth solitons of (1). Moreover, the equivalence of (1) and the modified CBS equation is proved in [4]. Yang and Chen [5] obtained two potentials and two pseudopotentials of (1). Equation (1) is derived from the two-dimensional Euler equation and proven to have Lax pair and bi-Hamiltonian structures [6].

To study the bifurcations of traveling wave solutions, Li and Qiao [7] considered the following nonlinear equation: where , , . They used the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems [8–10] to find all possible bounded traveling wave solutions and their parametric representations for the cases of , , respectively. In fact, when , (3) reads as a Harry Dym-type equation, which is actually the first member in the positive Camassa-Holm hierarchy [11–13]. The Harry Dym equation is an important integrable model in soliton theory [13]. It is also related to the classical string problem and has many applications in theoretical and experimental physics [14].

Sakovich [3] established the equivalent relationship between (1) and the modified KdV equation by the transformation for . When , , one objective of this paper is to find some transformations which relate the traveling wave solutions of (3) with that of the generalized KdV equation [15]: where is a positive integer and . For the results of traveling wave solutions on (4), we refer the readers to [16–21]. The other objective is to extend the results of Li and Qiao [7]. We continue to consider the problems on explicit traveling wave solutions of (3) and their bifurcations, but we do not use the theory of the singular traveling wave systems. Instead, we apply the transformations which transform (3) into a traveling wave system without singular straight line [21, 22]. Then, by using the bifurcation method of dynamical systems [21–28], we obtain some explicit traveling wave solutions of (3) for the case of . Not only the existence of these solutions are proved, but also their concrete expressions are presented.

The rest of the paper is organized as follows. In Section 2, we reveal the equivalent relationship of the traveling wave solutions between (3) and (4). In Section 3, various planar systems and their bifurcation phase portraits of (3) are given. We state the explicit traveling wave solutions of (3) and present their theoretical derivation in Section 4. Some conclusions are given in Section 5.

#### 2. Equivalent Relationship of (3) and (4)

In order to study the equivalent relationship of the traveling wave solutions between (3) and (4), we transform both (3) and (4) into traveling wave systems.

First of all, we substitute with into (3). Then, we get where is a constant wave speed. Letting and integrating (5) once, we have where is an integral constant. Letting then (6) can be rewritten as

Letting , we have

Next, we also transform (4) into traveling wave system. Substituting with into (4), we have Integrating (10) once and letting leads to where is an integral constant.

Finally, according to (9) and (11), we know that, from the traveling wave solution (3), we can drive the solutions of (4) for . When , one can notice that the traveling wave system of (4) is more general than (9) because of the arbitrary coefficient . However, the traveling wave solutions of (3) cannot be derived from the solutions of (4) for . Next, we study the traveling wave solutions of (3) and their bifurcations for .

#### 3. Planar Systems and Their Bifurcation Phase Portraits

In this section, we derive the traveling wave systems of (3) for the different cases of and draw their bifurcation phase portraits which are the basis for constructing nonlinear wave solutions.

When , that is, , putting and , from (9), we obtain the following planar system Letting we have

Then, system (12) can be written as Under the transformation , we have

System (16) has the first integral where

Next, we discuss the phase portraits of system. (16) for two different cases of .

(a) is even and .

Letting

we have

Solving , we get

At the singular point , it is easy to obtain that the linearized system of system (16) has the eigenvalues

From (12) and (18), we get the properties of the singular points as follow.(i) If , then is a saddle point of system (16).(ii) If , then is a center point of system (16).(iii) If , then is a degenerate singular point of system (16).

Therefore, we have the following results.(1) When , is a center point, and and are saddle points. Due to , the orbits connecting with and are heteroclinic orbits.(2) When , is a saddle point, and and are center points.

From the previous discussion, we get the phase portraits of system (13) as Figures 1(a) and 1(b).

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

(b) is odd.

In this case, has two zero points and , where

Similar to the previous discussion, we obtain the following results.(1)When and , is a center point and is a saddle point.(2)When and , is a saddle point and is a center point.(3)When and , is a saddle point and is a center point.(4)When and , is a saddle point and is a center point.

Through the discussion mentioned above, we obtain the phase portraits of Sy. (16) as Figures 1(c)–1(f).

#### 4. New Exact Solutions and Theoretical Derivation

When , , , , we have the following results.(1)When and is odd, (3) has the following exact solution: where .(2)When , is even and , (3) also has the solution of the same expression as .(3)When and is odd, (3) has the solitary wave solution and the blow-up solution (4)When , is even and , (3) also has the solitary wave solution of the same expression as .

Next, we give the demonstration for the previous results of (3) by two cases.

*Case 1 (). * When , is odd and , system (16) has two singular points and , where . The singular point is a center point, and is a saddle point.

When , is even and , system (16) has three singular points , and , where . The singular points and are center points, and the singular point is a saddle point.

Assume that is an initial point of system (13), then we have the following results.(1)When , the boundary of the closed orbit denoted by is a homoclinic orbit which passes and connects with (see Figure 2(a)).(2)When , there are two special orbits denoted by and which pass through (see Figure 2(a)). (3)When , we use to sign the orbit which passes through and connects with , where (see Figure 2(b)).

**(a)**

**(b)**

On the plane, the orbits , , and have the same expression which can be written as

Substituting the previous expression into and integrating it along different orbits, we have

In (28), completing the integration and solving the equation for , it follows that

Similarly, via (29) we have

Via (30), we get the solution of the same expression as .

When and , we can obtain the solutions of the same expressions as and .

Note that the transformation , we get the smooth solitary wave solution and the blow-up solution of (3).

*Case 2 (). * When , , is odd and , there is an interesting phenomenon concerning the traveling wave solutions of (3). Generally speaking, homoclinic orbit is corresponding to solitary wave solution. However, after applying the transformation to (3), we do not get solitary wave solution from homoclinic orbit. Therefore, we have the reason to believe that the transformation influences the corresponding relations between bifurcation orbits and traveling wave solutions.(1) When , is odd and , system (16) has two singular points and , where . The singular point is a center point and the singular point is a saddle point. The boundary of the closed orbit denoted by is a homoclinic orbit which passes through and connects with , where and (see Figure 3(a)).(2) When , is even and , Sy. (16) has three singular points , , and , where . The singular points and are saddle points. The singular point is a center point. Due to , the orbits connecting with and are two heteroclinic orbits. Here we only consider two special orbits which pass through , where . We denote the orbits by (see Figure 3(b)).

**(a)**

**(b)**

On plane, the orbit and the orbits have the same expression

We take , that is, or as an initial point for system (16). Substituting the previous expression into and integrating it along and , respectively, it follows that

In (34), completing the integration and solving the equation for , it follows that where .

From the transformation , we obtain the traveling wave solution of (3).

The theoretical derivation of the other cases can be finished similarly. We omit it for convenience. Hereto, we have completed the demonstrations to the previous results of (3).

#### 5. Conclusions

In this paper, we find some transformations which relate (3) with (4). Applying these transformations, we reveal the relationship of traveling wave solutions between (3) and (4). By using the bifurcation method of dynamical systems, we consider the further results on the explicit traveling wave solutions of (3) for the special case of , where , , . The correctness of these solutions is tested as well by using the software Mathematica.

Note that in this paper, there are two problems waiting to solve. The first one is that we only discuss the equivalent relationship of the traveling wave solutions between (3) and (4). We do not know whether the relationship of the other solutions of (3) and (4) is equivalent. The second one is that we have investigated the traveling wave solutions of (3) for the special cases of . But the traveling wave solutions of (3) for the other cases of await further study.

#### Conflicts of Interests

The authors have declared that there is no conflict of interests.

#### Acknowledgment

This research is supported by the National Natural Science Foundation of China (no. 11171115), Natural Science Foundation of Guangdong Province (no. S2012040007959), and the Fundamental Research Funds for the Central Universities (no. 2012ZM0057).