Abstract and Applied Analysis

Volume 2013 (2013), Article ID 869438, 19 pages

http://dx.doi.org/10.1155/2013/869438

Research Article

Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations

Shaoyong Li1,2 and Rui Liu¹

¹Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

²College of Mathematics and Information Sciences, Shaoguan University, Shaoguan, Guangdong 512005, China

Received 22 December 2012; Revised 7 February 2013; Accepted 17 February 2013

Academic Editor: Chun-Lei Tang

Copyright © 2013 Shaoyong Li and Rui 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.

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations where , and are real parameters, is a complex function, and is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.

1. Introduction

Since the exact solutions to nonlinear wave equations help to understand the characteristics of nonlinear equations, seeking exact solutions of nonlinear equations is an important subject. For this purpose, there have been many methods such as the Jacobi elliptic function method [1, 2], F-expansion method [3, 4], and -expansion method [5, 6].

Recently, the bifurcation method of dynamical systems [7–9] has been introduced to study the nonlinear partial differential equations. Up to now, the method is widely used in literatures such as [10–16].

In this paper, we consider the generalized Zakharov equations [17], which read as

where , and are real parameters. is a complex function which represents the envelop of the electric field, and is a real function which represents the plasma density measured from its equilibrium value. Huang and Zhang [17] used Fan's direct algebraic method to obtain some exact travelling wave solutions of (1) as follows:

where are two constants and

When , (1) reduce to the equations

El-Wakil et al. [18] used the extended Jacobi elliptic function expansion method to obtain some Jacobi elliptic function expression solutions of (13).

When , (13) reduce to the equations

By multisymplectic numerical method, Wang [19] proved the preservation of discrete normal conservation law of (14) theoretically and investigated the propagation and collision behaviors of the solitary waves numerically. There are also many other researchers studying (1) or its special case; for more information, one can see [20–24].

In this paper, we investigate the nonlinear waves and the bifurcation phenomena of (1). Coincidentally, under some transformations, (1) reduce to a planar system (54) which is similar to the planar system obtained by Feng and Li [16]. Many exact explicit parametric representations of solitary waves, kink and antikink waves, and periodic waves were obtained in [16], and their work is very important for the model. In order to find the travelling wave solutions of (1), here we consider (1) by using the bifurcation method mentioned above; firstly, we obtain three types of explicit nonlinear wave solutions, this is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. Secondly, we point out that these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves under different parameter conditions. Thirdly, we reveal five kinds of interesting bifurcation phenomena mentioned in the abstract above.

The remainder of this paper is organized as follows. In Section 2, we give some notations and state our main results. In Section 3, we give derivations for our results. A brief conclusion is given in Section 4.

2. Main Results

In this section, we state our main results. To relate conveniently, let us give some notations which will be used in the latter statement and the derivations.

Let represent the following seven curves:

Let represent the regions surrounded by the curves and the coordinate axes (see Figure 1).

(a)

(b)

Figure 1: The locations of the regions and curves .

Let

where is the first integral which will be given later and is the integral constant.

In order to search for the solutions of (1) and studing the bifurcation phenomena, we only need to get the solution according to (22) and (23). For convenience, throughout the following work we only discuss the solution . Now let us state our main results in the following Propositions 1, 2, and 3.

2.1. When the Orbit Is Defined by

Proposition 1. (i) For , (1) have two fractional nonlinear wave solutions

where

If , then are 1-blow-up waves (refer to Figure 2(d)). If , then are 2-blow-up waves (refer to Figure 3(d)). If , then are symmetric solitary waves (refer to Figure 4(d)).

(ii) For , (1) have two nonlinear wave solutions

where

These solutions have the following properties and wave shapes.

(1) When or , then and are periodic-blow-up waves (refer to Figure 2(a) or Figure 3(a)). Specially, in region when , the periodic-blow-up waves become 1-blow-up waves , and for the varying process, see Figure 2, while the periodic-blow-up waves become a trivial wave . In region when , the periodic-blow-up waves become 2-blow-up waves , and for the varying process, see Figure 3.

 (2) When , then and are symmetric periodic waves (refer to Figure 5(a)). If , the symmetric periodic waves become symmetric solitary waves , and for the varying process, see Figure 4. If , the symmetric periodic waves and become two trivial waves , and for the varying process, see Figure 5.

(iii) For , (1) have four nonlinear wave solutions

where

is a nonzero arbitrary real constant and . Corresponding to or , these solutions have the following properties and wave shapes.

(1) For the case of , there are four properties as follows.

When , that is, , and , then and become

    which represent four low-kink waves (refer to Figure 6(d) or Figure 7(d)). Specially, let , then and become and . If belongs to one of , and , then , and they represent four symmetric solitary waves (refer to Figure 6(a)). Specially, when and , then the four symmetric solitary waves and become the four low-kink waves and , and for the varying process, see Figure 6. If belongs to one of , , and , then , and they represent four 1-blow-up waves (refer to Figure 7(a)). Specially, when and , then the four 1-blow-up waves and become the four low-kink waves and , and for the varying process, see Figure 7. If belongs to one of , and , then . Specially, when and , then tend to two trivial solutions .

(a)

(b)

(c)

(d)

Figure 2: (The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 3: (The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when and , where ,   and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 4: (The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of when and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 5: (The symmetric periodic waves become two trivial waves.) The varying process for the figures of and when and , where ,   and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 6: (The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of and when and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 7: (The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of and when , and , where , and (a) , (b) , (c) , and (d) .

(2) For the case of , there are three properties as follows.

If , that is, , and , then and become and which represent four 1-blow-up waves (refer to Figure 8(d)). Specially, let , then and become the hyperbolic 1-blow-up wave solutions and

If and , then , and they represent four 2-blow-up waves (refer to Figure 8(a)). Specially, when , then the four 2-blow-up waves become four 1-blow-up waves and , and for the varying process, see Figure 8. If and , then of forms

    which represent hyperbolic blow-up waves. When , then tend to two trivial solutions .

(a)

(b)

(c)

(d)

Figure 8: (The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of and when , and , where , and (a) , (b) , (c) , and (d) .

2.2. When the Orbit Is Defined by

Proposition 2. If belongs to one of the regions or curves , and , then (1) have four nonlinear wave solutions

where

is a nonzero arbitrary real constant, is given in (24), and

Let

About , one has the following fact:

Corresponding to and , these solutions have the following properties and wave shapes.

   For the case of , there are six properties as follows.

If , that is, , and , then and become

    which represent four low-kink waves (refer to Figure 9(d)). Specially, let , then and .  If , and , then , and they represent four tall-kink waves (refer to Figure 9(a)). Specially, when and , then and become and , and for the varying process, see Figure 9.  If , and , then . When , they represent four antisymmetric solitary waves (refer to Figure 10(a)). When , they represent four symmetric solitary waves (refer to Figure 11(a)). Specially, when , if , then and become and , and for the varying process, see Figure 10. When and , then and tend to two trivial solutions , and for the varying process, see Figure 11. If and , then of forms

 which represent two tall-kink waves and tend to a trivial wave when . If and , then of forms

 which represent two antisymmetric solitary waves, tend to a trivial wave when , and tend to two trivial waves when . If and , then which represent two symmetric solitary waves. Specially, when and , then tend to two trivial waves .

   For the case of , there are five properties as follows.

If , then and represent four 1-blow-up waves (refer to Figure 12(d)). Specially, let , then and . If and , then , and they represent four pairs of 1-blow-up waves (refer to Figure 12(a)). In particular, when , if , then the four pairs of 1-blow-up waves become two pairs of 1-blow-up waves, and the varying process is displayed in Figure 12. If , then the four pairs of 1-blow-up waves become two trivial waves . If and , then , and they represent four tall-kink waves. If , when , then which represent two pairs of 1-blow-up waves. When , then which represent two pairs of 1-blow-up waves. If and , then which represent two tall-kink waves.

(a)

(b)

(c)

(d)

Figure 9: (The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of and when , and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 10: (The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of and when , and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 11: (The four symmetric solitary waves become two trivial waves.) The varying process for the figures of and when , and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 12: (The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of and when , and , where , and (a) , (b) , (c) , and (d) .

2.3. When the Orbit Is Defined by

Proposition 3. (i) If belongs to one of , and , then (1) have four nonlinear wave solutions

where

When , or , then and represent periodic blow-up wave solutions (refer to Figure 13(a)). When , then and represent periodic wave solutions (refer to Figure 14(a)).

(ii) If or , then (1) have two fractional wave solutions

where

When ,   represent two 1-blow-up waves (refer to Figure 13(d)). When ,   represent two tall-kink waves (refer to Figure 14(d)).

(iii) If and , then the periodic blow-up waves tend to two trivial solutions , while tend to the 1-blow-up waves , and the varying process is displayed in Figure 13. If and , then the periodic waves tend to two trivial solutions , while tend to the tall-kink waves , and for the varying process, see Figure 14.

(a)

(b)

(c)

(d)

Figure 13: (The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of when and , where , and (a) , (b) , (c) , and (d) .

(a)

(b)

(c)

(d)

Figure 14: (The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of when and , where , and (a) , (b) , (c) , and (d) .

3. The Derivations of Main Results

To derive our results, substituting (23) and with into (1), it follows that

Integrating the first equation of (52) twice with respect to and taking the integral constants to zero, we get (22). Substituting (22) into the second equation of (52) and letting , it follows that

Form (53) we obtain the planar system

with the first integral

According to the qualitative theory, we obtain the bifurcation phase portraits of system (54) as Figures 15 and 16. Using the information given by Figures 15 and 16, we give derivations to Propositions 1, 2, and 3, respectively.

Figure 15: The bifurcation phase portraits of system (54) for .

Figure 16: The bifurcation phase portraits of system (54) for .

3.1. The Derivations to Proposition 1

When the orbit is defined by , from (55) we obtain

Substituting (56) into the first equation of (54) and integrating it, we have

where is an arbitrary constant or .

For , letting or and completing the above integral, it follows that

which yields as (29).

For , completing (57) and solving for , it follows that

where is an arbitrary real constant. Let , respectively, and we obtain the solutions and as (3) and (31).

When , that is, , let

Thus, we have

When , that is, , similarly we get

When , that is, , if , then and .

Thus,

For , completing (57) and solving for , it follows that

where is an arbitrary real constant.

Note that if is a solution of (53), so is . Thus, from (64) we obtain the solutions and as in (33).

For the case of . When , that is, and . In (33) letting , we get (34). Furthermore, in (34) letting , it follows that

When , by (33) it follows that

For the case of , similarly we can obtain (7), (35), and (36), and here we omit the process. Hereto, we have completed the derivations for Proposition 1.

3.2. The Derivations to Proposition 2

When the orbit is defined by , from (55) we obtain

where and are given in (25) and (40). Substituting (67) into the first equation of (54) and integrating it, we have

where is an arbitrary constant or .

Completing the above integral and solving for , it follows that

where is given in (39), is an arbitrary constant, and

Similar to the derivations for and , we get and (see (38)) from (69).

For the case of , when , that is, , then and . From (38) and (41), it is easy to check that and become and (see (42)), respectively. Furthermore, in (42) letting , it follows that

If and , that is, , and , we have

If and , that is, , and , we have

If and , that is, , and , we have

For the case of , similarly we can obtain the relations of the solutions , and , and here we omit the process. Hereto, we have completed the derivations for Proposition 2.

3.3. The Derivations to Proposition 3

When the orbit is defined by , firstly, if belongs to one of , and , then from (55) we obtain

where is given in (16) and . Substituting (75) into the first equation of (54) and integrating it, we have

where is an arbitrary constant or .

Completing the above integral and solving for , it follows that

where is given in (49),   is an arbitrary constant, and

In (77) letting , respectively, we obtain the solutions and as (47) and (48).

Secondly, if or , that is, and , thus from (75), we obtain

Similarly, we have

which yields as in (51).

When , it follows that

Thus, we have

Hereto, we have completed the derivations for our main results.

4. Conclusions

In this paper, we have investigated the explicit expressions of the nonlinear waves and bifurcation phenomena in (1). Firstly, we obtained three types of explicit nonlinear wave solutions. The first type is the fractional expressions and (see (29) and (51)). The second type is the trigonometric expressions and (see (47) and (48)). The third type is the exp-function expressions , , , and (see (33) and (38)).

Secondly, we revealed five kinds of interesting bifurcation phenomena in (1). The first phenomena is that the 1-blow-up waves can be bifurcated from the periodic-blow-up (see Figure 2 or Figure 13) and 2-blow-up waves (see Figure 8). The second phenomena is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves (see Figure 3). The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves (see Figure 4). The fourth kind is that the low-kink waves can be bifurcated from the symmetric solitary waves (see Figure 6), the 1-blow-up waves (see Figure 7), the tall-kink waves (see Figure 9), and the antisymmetric solitary waves (see Figure 10). The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves (see Figure 14).

Thirdly, we showed that some previous results are our special cases. For instants, , , and are included in , , , and (see (4), (5), (7), (33), and (38)). are included in and (see (6) and (33)).

Finally, we employed the software Mathematica to check the correctness of these solutions. For example, the commands for and are as follows:

  ,  ,  ,  ,  ,  ,  ,  ,  ,     0 0

Acknowledgment

This paper is supported by the National Natural Science Foundation (no. 11171115), Natural Science Foundation of Guangdong Province (no. S2012040007959) and the Fundamental Research Funds for the Central Universities (no. 2012ZM0057).

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