The Bifurcations of Traveling Wave Solutions of the Kundu Equation

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.


Introduction
In this paper, we consider the bifurcations of traveling wave solutions for the Kundu equation with the fifth-order nonlinear term: where  3 ,  5 ,  2 , 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].
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  = /2,  = ( + )/4.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.

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  4 > 0 and  2 < 0, (10) possesses the following solutions.
(1) If  1 = 2 2 /3 4 , the blowup solution is of expression and the solitary wave solution possesses the expression where the kink wave solution is of expression and the periodic blowup solution is of expression where

the solitary wave solution is of expression
and the kink wave solution is of expression where (5) If  1 = 0, the blowup solutions are of expressions and the kink wave solution is of expression where Proposition 2. When the parameters  4 < 0 and  2 > 0, (10) possesses the following solutions.

the solitary wave solution is of expression
the kink wave solution is of expression and the periodic wave solution is of expression where (32)

the solitary wave solution is of expression
and the kink wave solution is of expression where where (5) If  1 = 0, the solitary wave solution is of expression Remark 1.The traveling wave solutions   () ( = 3, 5, ) are the same as the ones in [10][11][12][13][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.

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 () = 0, we get five roots as follows: Second, let (, 0) be one of the singular points of system (12).The characteristic values of the linearized system of system ( 12) at (, 0) are From the qualitative theory of dynamical systems, we get the properties of the singular point (, 0) as follows.
By using the property of equilibrium points and bifurcation theory, we obtain four bifurcation curves in the ( 2 ,  1 )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.

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.

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).