Abstract
We use the bifurcation method of dynamical systems to study the bifurcations of traveling wave solutions for the Kundu equation. Various explicit traveling wave solutions and their bifurcations are obtained. Via some special phase orbits, we obtain some new explicit traveling wave solutions. Our work extends some previous results.
1. Introduction
In this paper, we consider the bifurcations of traveling wave solutions for the Kundu equation with the fifth-order nonlinear term: where , , , and are real constants. Equation (1a) was derived by Kundu [1] in the study of integrability, and it is an important special case of the generalized complex Ginzburg-Landau [2].
For convenience, we denote the Kundu equation by where , .
Meanwhile, (1a) and (1b) and their special cases arise in various physical and mechanical applications, such as plasma physics, nonlinear fluid mechanics, nonlinear optics, and quantum physics (see [2–7]). For example, (1b) contains the following particular cases.
(1) If , then (1a) and (1b) reduce to the derivative Schrödinger equation:
(2) If , , and , then (1a) and (1b) reduce to Chen-Lee-Liu equation [8]:
(3) If , , and , then (1a) and (1b) reduce to Gerdjikov-Ivanov equation [9]:
In [2], by using primary integral method, van Saarloos and Hohenberg obtained the exact solitary waves of the form for (2). Furthermore, Biswas [10] carried out the integration of the generalized Kundu equation to obtain the 1-soliton solution. Feng and Wang [11] gave the explicit solitary wave solutions by the algebraic curve method for (1a) and (1b). Bright/dark solitary wave solutions and triangular periodic wave solutions of (1a) and (1b) were obtained in [12]. Zhang et al. [13] obtained exact solitary waves of (1a) and (1b) by using the undetermined coefficient method, and they also investigated orbital stability of solitary waves. Zhang [14] obtained various exact traveling wave solutions of (1a) and (1b) by the general solutions of a kind of subequation. Besides, Guo and Wu proved that the kind of solitary waves (5) for (2) is stable in [15]. He and Meng [16] investigated the exact traveling waves for (4) by using the bifurcation theory and the method of phase portraits analysis.
In this paper, we use the bifurcation method of dynamical systems [17–23] to study the traveling wave solutions and their bifurcations for (1a) and (1b). Through bifurcation phase portraits, we obtain various traveling wave solutions of (1a) and (1b). Via some special phase orbits, we also obtain new explicit traveling wave solutions. Not only the existence of these solutions is proved, but also their concrete expressions are presented. Our work extends some previous results [10–16].
In order to derive traveling wave solutions, we assume that (1a) and (1b) have solutions of the form (5). Substituting (5) into (1a) and (1b) and making the real part and imaginary part of (1a) and (1b) equal to zero, we have
Let
Substituting (8) into (6) and equating the coefficients of these terms , , and to zero, we get , . Therefore, when (6) is identical to zero. Combining (9) and (7), we obtain the following equation: where
In fact, (10) is equivalent to the system which has the first integral
The rest of the paper is organized as follows. Our main results and remarks are arranged in Section 2. In Section 3, in order to derive the main results, we show various planar systems and their bifurcation phase portraits of (10). We state the theoretic derivation of our main results by the bifurcation method of dynamical systems in Section 4. Short conclusions are given in Section 5.
2. Main Results
In this section, we state our main results and give some corresponding remarks. Our main results are listed in the following two propositions.
Proposition 1. When the parameters and , (10) possesses the following solutions.
(1) If , the blowup solution is of expression
(2) If , the blowup solution is of expression
and the solitary wave solution possesses the expression
where
(3) If , the blowup solution is of expression
the kink wave solution is of expression
and the periodic blowup solution is of expression
where
(4) If , the solitary wave solution is of expression
and the kink wave solution is of expression
where
(5) If , the blowup solutions are of expressions
and the kink wave solution is of expression
where
Proposition 2. When the parameters and , (10) possesses the following solutions.
(1) If , the solitary wave solution is of expression
(2) If , the solitary wave solution is of expression
the kink wave solution is of expression
and the periodic wave solution is of expression
where
(3) If , the solitary wave solution is of expression
and the kink wave solution is of expression
where
(4) If , the periodic wave solution is of expression
where
(5) If , the solitary wave solution is of expression
Remark 1. The traveling wave solutions are the same as the ones in [10–14], while the rest of the solutions are new, which cannot be found in the literature.
Remark 2. Choosing we can obtain all the exact solutions reported in [16].
Remark 3. The correctness of the above solutions is tested by using the software Mathematica.
3. The Bifurcation Phase Portraits
In order to derive the main results mentioned in Section 2, we draw the bifurcation phase portraits of system (12) by the qualitative theory of dynamical systems.
First, letting we have Solving , we get five roots as follows:
Second, let be one of the singular points of system (12). The characteristic values of the linearized system of system (12) at are From the qualitative theory of dynamical systems, we get the properties of the singular point as follows.(i)If , then is a saddle point of system (12).(ii)If , then is a center point of system (12).(iii)If , then is a degenerate singular point of system (12).
By using the property of equilibrium points and bifurcation theory, we obtain four bifurcation curves in the -parameter plane as follows:
Then, according to the qualitative theory, we obtain the bifurcation phase portraits of system (12) as shown in Figures 1 and 2.


4. The Derivations for Proposition 1
In this section, by using the above bifurcation phase portraits (Figure 1) and (12) and (13), we will give the exact traveling wave solutions of (10) under the given parameter conditions shown in Section 2.
(1) When , the curve possesses the following expression: where . Substituting (45) into and integrating it along , we have
In (46) completing the integral and solving the equation for , we get the solution .
(2) When , the curves and have the following expressions:
Substituting (47) into and integrating them along and , respectively, it follows that
Completing the integrals above and solving them for , we obtain the solutions and .
(3) When , the curves , , and have the following expressions:
Substituting (49) into and integrating them along , , and , respectively, it follows that
Completing the integrals above, we obtain the solutions .
(4) When , the curves and have the following expressions:
Substituting (51) into and integrating them along and , respectively, it follows that
Completing the integrals above, we obtain the solutions and .
(5) If , the curves , , and have the following expressions:
Substituting (53) into and integrating them along , , and , respectively, it follows that
Completing the integrals above, we obtain the solutions and complete the derivations for Proposition 1.
5. The Derivations for Proposition 2
Similarly, we will complete the derivations for Proposition 2 by using the bifurcation phase portraits (Figure 2) and (12) and (13).
(1) When , the curve possesses the following expression: where and . Substituting (55) into and integrating it along , we have
In (56) completing the integral and solving the equation for , we get the solution .
(2) When , the curves , , and have the following expressions:
Substituting (57) into and integrating them along , , and , respectively, it follows that
Completing the integrals above, we obtain the solutions .
(3) When , the curves , and have the following expressions: where , and ,
Substituting (59) and (60) into and integrating them along and , respectively, it follows that
Completing the integrals above, we obtain the solutions and .
(4) When , the curve possesses the following expression:
Substituting (62) into and integrating it along , we have
In (63) completing the integral above, we get the solution .
(5) If , the curve possesses the following expression:
Substituting (64) into and integrating it along , we get
Similar to the process above, we obtain and complete the derivations for Proposition 2. Hereto, we have completed the derivations for our main results mentioned in Section 2.
6. Conclusions
In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the bifurcations of traveling wave solutions for the Kundu equations (1a) and (1b). In order to derive the traveling wave solutions, we show the bifurcation phase portraits of system (12). Through the phase portraits, we obtain various explicit traveling wave solutions. Also, our work includes some previous results. On the other hand, from some special orbits, we also obtain some new explicit traveling wave solutions.
Note that there are two problems waiting to solve. The first one is that we do not give the elliptic periodic wave solutions for the Kundu equations (1a) and (1b). The relationship between the elliptic periodic wave solutions and the solutions given in this paper is not revealed. The second one is that Zhang et al. [13] showed that the solitary wave solutions are of orbital stability. However, we do not know whether the other solitary wave solutions obtained in this paper are of orbital stability.
Acknowledgment
This research is supported by the National Natural Science Foundation of China (no. 11171115).