Further Results on the Traveling Wave Solutions for an Integrable Equation

which relates (1) with thewell-knownmodifiedKdV 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:


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  ∈ R,  ̸ = − 1, 0. They used the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems [8][9][10] to find all possible bounded traveling wave solutions and their parametric representations for the cases of || = 1/2, 2, respectively.In fact, when  = 1/2, (3) reads as a Harry Dym-type equation, which is actually the first member in the positive Camassa-Holm hierarchy [11][12][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  = 2.When  ≥ 2,  ∈ Z, 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  ̸ = 0.For the results of traveling wave solutions on (4), we refer the readers to [16][17][18][19][20][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 2 Journal of Applied Mathematics [21,22].Then, by using the bifurcation method of dynamical systems [21][22][23][24][25][26][27][28], we obtain some explicit traveling wave solutions of (3) for the case of  = −(/) ( ̸ =  and ,  ∈ Z + ).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.

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  =  −  ( ̸ = 0) 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  = 1/, we have Next, we also transform (4) into traveling wave system.Substituting (, ) = () with  =  1/3  −  into (4), we have Integrating (10) once and letting  = 2 5/3 leads to where  1 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  ∈ Z.When  ∈ Z, 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  ∉ Z. Next, we study the traveling wave solutions of (3) and their bifurcations for  ∉ Z.

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  = 0, from (9), we obtain the following planar system Letting we have Then, system (12) can be written as Under the transformation  = / −1 , we have System ( 16) has the first integral where Next, we discuss the phase portraits of system.( 16) for two different cases of  − .
At the singular point (, 0), 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 (  , 0) ( = 1, 2) as follow.
Therefore, we have the following results.
From the previous discussion, we get the phase portraits of system (13) as Figures 1(a Similar to the previous discussion, we obtain the following results. (1) When  >  and  > 0, ( 3 , 0) is a center point and ( 4 , 0) is a saddle point.
Through the discussion mentioned above, we obtain the phase portraits of Sy. ( 16) as Figures 1(c)-1(f).
(3) When  <  and − is odd, (3) has the solitary wave solution and the blow-up solution (4) When  < ,  −  is even and  > 0, (3) also has the solitary wave solution of the same expression as  2 (, ).
Next, we give the demonstration for the previous results of ( 3) by two cases.
Assume that ( 0 , 0) is an initial point of system ( 13), then we have the following results.
On the  −  plane, the orbits Γ 1 , Γ ± 2 , and Γ 3 have the same expression which can be written as Substituting the previous expression into  −1 / =  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  2 ().
When  <  and  < 0, we can obtain the solutions of the same expressions as  2 () and  3 ().
Case 2 ( > ).When  = −/,  > , − is odd and  > 0, 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.
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).

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  ̸ = , ,  ∈ Z + .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.
) and 1(b).(b)  −  is odd In this case, () has two zero points  3 and  4 , where