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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 189486, 20 pages

http://dx.doi.org/10.1155/2014/189486
Research Article

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

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 29 March 2014; Accepted 28 May 2014; Published 7 July 2014

Academic Editor: Chun-Lei Tang

Copyright © 2014 Huixian Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov system called system. We reveal some interesting bifurcation phenomena as follows. (1) For system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) For system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) For system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

1. Introduction

The system [1, 2] is read as where , , , , and are constants. Through some transformations, Xie and Yan [3] got some compacton and solitary pattern solutions of system (1) with , including where , are constants, Deng et al. [4], by using the Weierstrass elliptic function method, presented many solutions of system (1) with . 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 Exp-function 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.

The other generalized Drinfel'd-Sokolov system [13] is considered in [1422]. In [2331], many exact solutions of system (7) were obtained. Clearly, system (1) and system (7) are two different systems.

In this paper, we are interested in system (1). We study the bifurcations of nonlinear waves for system (1).

Under the transformations system (1) is reduced to Integrating the first equation of system (9) once, we have where is an integral constant. Substituting (10) into the second equation of system (9) and integrating it once yield the following equation: where is another integral constant. Letting , we obtain the planar system where is given in (4) and Letting , system (12) becomes which is called system. Clearly, systems (12) and (14) possess the same first integral

Employing (14) and (15), we reveal some interesting bifurcation phenomena listed in the above abstract.

In Section 2, we will consider 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 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,   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.

2. The Bifurcations of Solitary Waves and Kink Waves for System

When and , system (1) becomes 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.

Note that system is read as where and , are given in (13). When , let

Proposition 1. For given and , if   , then , , and are real and system (16) has two types of special periodic wave solutions which become the fractional solitary wave solution when . These two types of special periodic wave solutions are as follows.(1)Trigonometric periodic wave solution: where (2)Elliptic periodic wave solution: where For the varying process, see Figure 1.

fig1
Figure 1: The solitary wave is bifurcated from two types of periodic waves. The varying process for the figures of and when , , and where and .

Proposition 2. For given and , if , then , , and are real and system (16) has two types of special periodic wave solutions which become a fractional solitary wave solution: when . These two types of special periodic wave solutions are as follows.(1)Trigonometric periodic wave solution: where (2)Elliptic periodic wave solution: where For the varying process, see Figure 2.

fig2
Figure 2: The solitary wave is bifurcated from two types of periodic waves. The varying process for the figures of and when , , and where and .

Proposition 3. For given and , if , then system (16) has four nonlinear wave solutions which become two kink wave solutions: when . These four nonlinear wave solutions are as follows: where These four nonlinear wave solutions possess the following properties. (a)If , then and represent two solitary waves which tend to two kink waves (see Figure 3) when . (b)If , then and represent two singular waves which tend to two kink waves (see Figure 4) when . (c)If , then and represent two singular waves which tend to two kink waves (see Figure 5) when . (d)If , then and represent two solitary waves which tend to two kink waves (see Figure 6) when .

fig3
Figure 3: Two kink waves are bifurcated from two peak-solitary waves. The varying process for the figures of and when , , and where and .
fig4
Figure 4: Two kink waves are bifurcated from two singular waves. The varying process for the figures of and when , , and where and .
fig5
Figure 5: Two kink waves are bifurcated from two singular waves. The varying process for the figures of and when , , and where and .
fig6
Figure 6: Two kink waves are bifurcated from two valley-solitary waves. The varying process for the figures of and when , , and where and .

The Derivations of Propositions 13. 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 13 as follows.

fig7
Figure 7: The bifurcation phase portraits of system (17).

(1) When , , and (as in Figure 7(a)), the closed curves and possess the following expressions: Substituting (34) into and integrating them along and , respectively, it follows that Completing the integrals above and solving the equations for , we obtain the solutions , (see (22)) and , (see (24)).

(2) When , , and (as in Figure 7(a)), the closed curves and possess the following expressions: Substituting (36) into and integrating them along and , respectively, it follows that Completing the integrals above and solving the equations for , we obtain the solutions , (see (27)) and , (see (29)).

When , it follows that , , and tend to and , tend to . Further, we have Thus, we have

Similarly, we can prove the limit property of and when .

(3) When , , and (as in Figure 7(b)), the curve connecting with embraces the expression Substituting (40) into and integrating it, we have where Completing the integral above and solving the equation for , we obtain From , we get

(4) When , , and (as in Figure 7(b)), the curve connecting with embraces the expression Substituting (45) into and integrating it, we have where Completing the integral above and solving the equation for , we obtain From , we get

Note that when , it follows that , , , and . Thus, we have

Similarly, we can also get , , and when . These complete the derivations of Propositions 13.

3. The Bifurcations of Compactons and Peakons for System

When and , 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 system is read as where and , are given in (13).

For fixed , put

Proposition 4. For given and , if , then system (51) has a family of solitary wave solutions which become the compacton solution when . The solitary wave solutions are as follows.

(1) When , the solitary wave solutions possess the expressions and , and is given by the implicit function where

(2) When , the solitary wave solutions possess the expressions and , and is given by the implicit function where For the varying process, see Figure 8.

fig8
Figure 8: The solitary waves turn into the compactons. The varying process for the figures of and when , , and where .

Proposition 5. For given and , if , then system (51) has two types of nonlinear wave solutions which tend to the peakon solution when . 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.

(1) When , the singular cusp wave solutions possess the expression and the solitary wave solutions possess the expressions and , and is given by the implicit function where

(2) When , the singular cusp wave solutions possess the expression and the solitary wave solutions possess the expressions and , and is given by the implicit function where

fig9
Figure 9: The peakon is bifurcated from the singular cusp wave and the solitary wave. The varying process for the figures of and when , , and where and .
fig10
Figure 10: The peakon is bifurcated from the singular cusp wave and the solitary wave. The varying process for the figures of and when , , and where and .

The Derivations of Propositions 4 and 5. According to the qualitative theory, we obtain the bifurcation phase portraits of system (52) as in Figure 11. Through some orbits in Figure 11, we derive the results of Propositions 4 and 5 as follows.

fig11
Figure 11: The bifurcation phase portraits of system (52).

(1) When , , and (as in Figure 11(a)), the homoclinic orbit owns the expression Substituting (67) into and integrating it, we have Completing the integral above, we get

(2) When , , and (as in Figure 11(a)), the homoclinic orbit is expressed by Substituting (70) into and integrating it, we have Completing the integral above, we get

Note that when and , it follows that , , and . Further, we have Thus, from we have Solving (75) for , we can get (see (55)), which is the same as (2).

Similarly, from (72), we can also get solution when and .

(3) When , , and (as in Figure 11(b)), the curves and own the expression Substituting (76) into and integrating it, we have Completing the integrals above and solving the equations for , we get (see (61)) and (see (64)).

(4) When , , and (as in Figure 11(b)), two homoclinic orbits can be expressed as Substituting (78) and (79) into and integrating it, we have Completing the integrals above, we get (62) and (65).

Note that when and , it follows that , , , , and