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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 483492, 18 pages
http://dx.doi.org/10.1155/2013/483492
Research Article

New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel-Korteweg-de Vries Equation

1Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
2Department of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, China

Received 26 April 2013; Accepted 3 July 2013

Academic Editor: Chuanzhi Bai

Copyright © 2013 Yun Wu and Zhengrong Liu. 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 nonlinear waves described by Schamel-Korteweg-de Vries equation . Two new types of nonlinear waves called compacton-like waves and kink-like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell-shape solitary waves, the blow-up waves, the valley-shape solitary waves, the kink-like waves, and the compacton-like waves. The second phenomenon is that the periodic-blow-up wave can be bifurcated from the smooth periodic wave.

1. Introduction and Preliminary

Consider the following Schamel-Korteweg-de Vries (S-KdV) equation [1, 2]: where , , and are constants.

Equation (1) arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space. Tagare and Chakraborti [1] showed that (1) has solitary wave solution by applying direct integral method. Lee and Sakthivel [3] gave some exact traveling wave solutions of (1) by using exp-function method.

When , (1) becomes the Schamel equation [4]:

When , (1) becomes a well-known KdV equation which has been studied successively by many authors (e.g., [58]).

The concept of compacton, soliton with compact support or strict localization of solitary waves, appeared in the work of Rosenau and Hyman [9] where a genuinely nonlinear dispersive equation is defined by They found certain solitary wave solutions which vanish identically outside a finite core region. These solutions are called compactons.

Several studies have been conducted in [1020]. The aim of these studies was to examine if other nonlinear dispersive equations may generate compactons structures.

In order to investigate the nonlinear waves of (1), letting be wave speed and substituting with into (1), it follows that Integrating (5), we get Setting yields the following planar system: Obviously, system (7) is a Hamiltonian system with Hamiltonian function If one puts then one can see the following facts.

When , has three zero points , , and which possess expressions When , has two zero points and which are denoted by When , has one zero point .

Letting be one of the singular points of system (7), then the characteristic values at are

From the qualitative theory of dynamical systems, we get the following conclusions:(1)if , then is a saddle point,(2)if , then is a center point,(3)if , then is a degenerate saddle point.

On parametric plane, let , , and , respectively, represent the following three curves: Let    represent the regions surrounded by , , , and the coordinate axes (see Figures 1 and 2).

483492.fig.001
Figure 1: The bifurcation phase portraits of system (7) when .
483492.fig.002
Figure 2: The bifurcation phase portraits of system (7) when .

According to the previous analysis, we obtain the bifurcation phase portraits of system (7) as in Figures 1 and 2.

In this paper, we study the nonlinear waves and their bifurcations in (1) by using the bifurcation method of dynamical systems [2123]. We point out that there are two new types of nonlinear waves, kink-like waves and compacton-like waves [2433]. Furthermore, we reveal two kinds of new bifurcation phenomena which are introduced in the abstract.

This paper is organized as follows. In Section 2, we display the two new types of nonlinear waves. We show the two kinds of new bifurcation phenomena in Sections 3 and 4. A brief conclusion is given in Section 5.

2. Two New Types of Nonlinear Waves

In this section, we display two new types of nonlinear waves defined by (1).

2.1. Kink-Like Waves

Proposition 1. (1) When the parameters satisfy , , and , (1) has a kink-like wave solution and an antikink-like wave solution , respectively, which are hidden in the following equations: where and .
(2)   When the parameters satisfy one of the following Cases.
Case  1.  , , and ,
Case  2.  , and ,
Case  3.  , , and ,
Case  4.  , , and .
Equation (1) has a kink-like wave solution and an antikink-like wave solution , respectively, which are hidden in the following equations: where and .

Proof. (1) Under the condition , , and , () is a saddle point and on its left side there are two orbits connecting with it (see Figure 3(a1)).
In (8), letting , it follows that
On suppose . Substituting (19) into (7) and integrating them along and , respectively, we get (14)–(16).
(2) Under one of Cases  1–4, () is a saddle point and on its left side there are two orbits connecting with it (see Figures 3(a2)–3(a4)).
In (8), letting , it follows that
Similar to the proof of (1), we get the results of (2).

483492.fig.003
Figure 3: The graphs of orbits and when (a1) and , (a2) and or or , (a3) and or or , and (a4) and .

Next, we simulate the planar graphs of the kink-like waves for those data given in Example 2.

Example 2 (Corresponding to Proposition 1  (1)). Giving , , , and , we get and . Note that orbits have expressions (19). From (19) we get and . These imply that passes point and passes point . Thus letting and as the initial conditions of (6), we get the simulation of the integral curve which corresponds to as Figure 4(a1). Meanwhile, choosing and as the initial conditions of (6), we get the simulation of the integral curve which corresponds to as in Figure 4(a2).

483492.fig.004
Figure 4: (Corresponding to Example 2). The simulation of integral curve of (6) when , , , and . (a1) Initial conditions and . (a2) Initial conditions and .
2.2. Compacton-Like Waves

Proposition 3. Let be an initial value when parameters and initial value satisfy one of the following Cases.
Case  1. , , and ,
Case  2. , , , and ,
Case  3. , , , and ,
Case  4. , , and ,
Case  5. , , , and ,
Case  6. , , , and ,
Case  7. , , , and ,
Case  8. , , , and ,
Case  9. , , , and or .
Equation (1) has compacton-like wave solutions and , respectively, which are hidden in the following equations: where and .

Proof. Under one of Cases  1–9, there is an orbit passing point () (see Figure 5 (a1)–5(a6)).
In (8), letting , it follows that
On suppose . Substituting (24) into (7) and integrating it along , respectively, we obtain (21)–(23).

483492.fig.005
Figure 5: The graphs of orbit when (a1) and or or , (a2) and , (a3) and or or , (a4) and or or , (a5) and , and (a6) and or or .

Next, we simulate the planar graphs of the compacton-like waves for those data given in Example 4.

Example 4 (Corresponding to Proposition 3 Case  (3)). Giving , , , and , we get and . Note that orbit has expression (24). Letting , it follows that . From (24) we get . Thus letting and as the initial conditions of (6), we get the simulation of the integral curve as in Figure 6(a1). Meanwhile, choosing and as the initial conditions of (6), we get the simulation of the integral curve as Figure 6(a2).

483492.fig.006
Figure 6: (Corresponding to Example 4). The simulation of integral curve of (6) when , , , and . (a1) Initial conditions and . (a2) Initial conditions and .

3. Bifurcation of the Kink Waves

In this section, we show that the kink waves can be bifurcated from five other waves.

3.1. Bifurcation from Bell-Shape Solitary Waves

Proposition 5. For and , (1) has two nonlinear wave solutions where and is an arbitrary real number. These solutions possess the following properties.(1)If , , and , then and becomewhich represent a kink wave and an antikink wave.
In particular, when , , , , and , and become which was given by Lee and Sakthivel [3]. This implies that is the special case of or .(2)If , then and become
When belongs to one of the regions , , represents a hyperbolic solitary wave.
In particular, when , and , becomes which was obtained by Tagare and Chakraborti [3]. This implies that is the special case of or .(3)Under one of the following Cases.
Case  1. , , and belongs to one of the regions , ,
Case  2. , , and , and they represent two bell-shape solitary waves.
In particular, when in Case  1 and , and become which are the solutions of the Schamel equation.
When in Case  1 and , the two bell-shape solitary waves and become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 7 and 8.

fig7
Figure 7: (The kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph of when , , , and . Where , , , and (a) . (b) . (c) . (d) .
fig8
Figure 8: (The anti-kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph of when , , , and . Where , , , and (a) . (b) . (c) . (d) .

Proof. In (8), letting , it follows that Substituting (35) into , we have Let , (36) becomes Integrating (37), we have where is an arbitrary constant.
Completing the previous integral and solving the equation for , it follows that where is an arbitrary real number. From (39) we obtain the solutions and as (25).
In (25) letting , we get (29). From (25) and (29), we get the result (1) of Proposition 5.
When , via (25) it follows that (see (31)).
Thus, we get the result (2) of Proposition 5.
In (38), letting (see (27)), it follows that
Letting , then We have (see (33)).
Similarly, we have (see (34)).
From (41)–(46), we get result of Proposition 5.

3.2. Bifurcation from Blow-Up Waves

Proposition 6. For and , (1) has two nonlinear wave solutions as and . These solutions possess the following properties.
(1) Under one of the following Cases.
Case  1. , , and belongs to one of the regions , ,
Case  2. , , and belongs to any one of the regions , , and , and they represent two blow-up waves.
In particular, when in Case  1 and , the two blow-up waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 9 and 10.

fig9
Figure 9: (The kink wave is bifurcated from the blow-up wave). The varying process for the graph of when , , , , and , where , , , , and (a) . (b) . (c) . (d) .
fig10
Figure 10: (The antikink wave is bifurcated from the blow-up wave). The varying process for the graph of when , , , , and , where , , , , and (a) . (b) . (c) . (d) .

Similar to the proof of Proposition 5, we get the results of Proposition 6.

3.3. Bifurcation from Valley-Shape Solitary Waves

Proposition 7. When the parameters satisfy and , (1) has two valley-shape solitary wave solutions and , respectively, which are hidden in the following equations: In particular, when , the two valley-shape solitary waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 11 and 12.

fig11
Figure 11: (The kink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph of when , , , and , where , , , and (a) . (b) . (c) . (d) .
fig12
Figure 12: (The antikink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph of when , , , and , where , , , and (a) . (b) . (c) . (d) .

Proof. When and , () is a saddle point and on its left side there is an orbit connecting with it (see Figure 13).
In (8), letting , it follows that Substituting (49) into (7) and integrating it along the orbit , we get (47) and (48).
Letting , it follows that
When , completing the integrals in (47) and (48), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition 7.

483492.fig.0013
Figure 13: The graph of orbit when and .
3.4. Bifurcation from Kink-Like Waves

Proposition 8. When the parameters satisfy , , and , the kink-like wave and the antikink-like wave, respectively, become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures 14 and 15.

fig14
Figure 14: (The kink wave is bifurcated from the kink-like wave). The varying process for the graph of when , , , and , where , , , and (a) . (b) . (c) . (d) .
fig15
Figure 15: (The antikink wave is bifurcated from the antikink-like wave). The varying process for the graph of when , , , and , where , , , and (a) . (b) . (c) . (d) .

Proof. Letting , it follows that (see (50)) and
When , and , completing the integrals in (17), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition 8.

3.5. Bifurcation from Compacton-Like Waves

Proposition 9. When the parameters satisfy , , and , the two compacton-like waves become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures 16 and 17.

fig16
Figure 16: (The kink wave is bifurcated from the compacton-like wave). The varying process for the graph of when , , , and , where , , , and (a) . (b) . (c) . (d) .
fig17
Figure 17: (The antikink wave is bifurcated from the compacton-like wave). The varying process for the graph of when , , , and , where , , , and (a) . (b) . (c) . (d) .

Proof. Letting , it follows that
When , , and , completing the integrals in (21), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition 9.

4. Bifurcation of Smooth Periodic Wave

Proposition 10. For and , (1) has a nonlinear wave solution where and is an arbitrary real number. The solution possesses the following properties.(1)if belongs to any one of the regions , , then represents periodic blow-up wave solution,(2)if belongs to , then represents periodic wave solution.
In particular, when , the periodic wave becomes a periodic blow-up wave. For the varying process, see Figure 18. When , the periodic wave tends to a trivial wave . For the varying process, see Figure 19.

fig18
Figure 18: (The periodic blow-up wave is bifurcated from the periodic wave). The varying process for the graph of when , , , , and , where , , , , and (a) . (b) . (c) . (d) .
fig19
Figure 19: (The periodic wave become the trivial wave). The varying process for the graph of when , , , , and , where , , , , and (a) . (b) . (c) . (d) .

Proof. Completing the integral in (38), we get as (55). is an arbitrary real number.
When , in (38) letting (see (28)), we have Letting , then We have
Obviously, will blow up when .
Hereto, we have completed the proofs for all propositions.

5. Conclusion

In this paper, we have studied the bifurcation behavior of S-KdV equation. Two new types of nonlinear waves called kink-like waves and compacton-like waves have been displayed in Propositions 13. Furthermore, two kinds of new bifurcation phenomena have been revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which have been stated in Propositions 59. The second phenomenon is that the periodic blow-up wave can be bifurcated from the periodic wave which has been explained in Proposition 10. At the same time, we have got three new explicit expressions for traveling waves which were given in (25) and (55). Two previous results are our some special cases (see (30) and (32)).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11171115) and the Science and Technology Foundation of Guizhou (no. LKS[2012]14).

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