Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel ’ d-Sokolov System

and Applied Analysis 3 where

Deng et al. [4], by using the Weierstrass elliptic function method, presented many solutions of system (1) with  =  + 1.It also includes the above solutions (2) and (3).Zhang et al. [5] showed some solutions of system (1) under the special parameters via employing the bifurcation method.By means of the complete discrimination system for polynomial method, many solutions of system (1) were acquired in [6].
In [2], the system was introduced.Wang [7] gave recursion, Hamiltonian, symplectic and cosymplectic operator, roots of symmetries, and scaling symmetry for system (5).Wazwaz [8], by using the tanh method and the sine-cosine method, obtained many solutions with compact and noncompact structures of system (5), including Biazar and Ayati [9] obtained some solutions of system (5) through Exp-function method and modification of Expfunction method.Zhang et al. [10], by employing the complex method, gained all meromorphic exact solutions of system (5).Applying the auxiliary equation method, some exact solutions of system (5) were given in [11].El-Wakil and Abdou [12] got some new exact solutions of system ( 5) by means of modified extended tanh-function method.
In this paper, we are interested in system (1).We study the bifurcations of nonlinear waves for system (1).
In Section 2, we will consider (2, 1) system.Firstly, we will show that the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves.Secondly, we will demonstrate that the kink waves can be bifurcated from the smooth solitary waves and the singular waves.In Section 3, we will consider (1, 2) system.Firstly, we will confirm that the compactons can be bifurcated from the smooth solitary waves.Secondly, we will clarify that the peakons can be bifurcated from the solitary waves and the singular cusp waves.In Section 4, (2, 2) system will be considered.We will verify that the solitary waves can be bifurcated from the smooth periodic waves and the periodic singular waves.A short conclusion will be given in Section 5.

The Bifurcations of Solitary Waves and Kink Waves for 𝐷(2, 1) System
When  = 2 and  = 1, system (1) becomes (2, 1) system: We will reveal two kinds of interesting bifurcation phenomena to system (16).The first phenomenon is that fractional solitary waves can be bifurcated from two types of smooth periodic waves: trigonometric periodic waves and elliptic periodic waves.The second phenomenon is that the kink waves can be bifurcated from the smooth solitary waves and the singular waves.We state these results and give proof as follows.
The Derivations of Propositions 1-3.According to the qualitative theory, we obtain the bifurcation phase portraits of system (17) as in Figure 7. Employing some orbits in Figure 7, we derive the results of Propositions 1-3 as follows.

The Bifurcations of Compactons and Peakons for 𝐷(1, 2) System
When  = 1 and  = 2, system (1) becomes We will reveal two kinds of interesting bifurcation phenomena to system (51).The first phenomenon is that the smooth solitary waves can turn into the compactons.The second phenomenon is that the peakons can be bifurcated from the singular cusp waves and the solitary waves.The concrete results are stated as follows.
Note that (1, 2) system is read as where and ,  are given in (13).
Proposition 5.For given  1 < 0 and  ̸ = 0, if  3 () <  <  2 (), then system (51) has two types of nonlinear wave solutions which tend to the peakon solution   when  →  2 () − 0. For the varying process, see Figures 9  and 10.These two types of nonlinear wave solutions are singular cusp wave solutions and solitary wave solutions of the following expressions.

The Bifurcations of Solitary Waves for 𝐷(2, 2) System
When  = 2 and  = 2, system (1) becomes We will reveal the interesting bifurcation phenomenon to system (84).That is, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.The concrete results are stated as follows.
Note that (2, 2) system is read as where  2 is as (18) and ,  are given in (13).Let ) ) and let  1 () be as (20).Proposition 6.For fixed  2 and , system (84) has solitary wave solutions which can be bifurcated from the following two types of nonlinear wave solutions.
The Derivations of Proposition 6.According to the qualitative theory, we also obtain the bifurcation phase portraits of system (85) as in Figure 14.Employing some orbits in Figure 14, we derive the results of Proposition 6 as follows.
Note that when

Conclusions
In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we have revealed some interesting bifurcation phenomena of nonlinear waves for the (, ) system (1).Firstly, for (2, 1) system, we have pointed out that the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves (see Figures 1 and 2).In the meantime, the kink waves can be bifurcated from the solitary waves and the singular waves (see Figures 3-6).Secondly, for (1, 2) system, we have showed that the solitary waves can turn into the compactons (see Figure 8) and the peakons can be bifurcated from the singular cusp waves and the solitary waves (see Figures 9 and 10).Thirdly, for (2, 2) system, we have confirmed that the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves (see Figures 12 and 13).