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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 704931, 15 pages
http://dx.doi.org/10.1155/2012/704931
Research Article

Extension on Bifurcations of Traveling Wave Solutions for a Two-Component Fornberg-Whitham Equation

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received 20 September 2012; Accepted 16 November 2012

Academic Editor: Zhenya Yan

Copyright © 2012 Zhenshu Wen. 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

Fan et al. studied the bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. They gave a part of possible phase portraits and obtained some uncertain parametric conditions for solitons and kink (antikink) solutions. However, the exact explicit parametric conditions have not been given for the existence of solitons and kink (antikink) solutions. In this paper, we study the bifurcations for the two-component Fornberg-Whitham equation in detalis, present all possible phase portraits, and give the exact explicit parametric conditions for various solutions. In addition, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves are obtained. Our results extend the previous study.

1. Introduction

In 2011, Fan et al. [1] introduced the following two-component Fornberg-Whitham equation where denotes the height of the water surface above a horizontal bottom, and indicts the horizontal velocity field. They studied the bifurcations of traveling wave solutions for (1.1) through obtaining some uncertain parametric conditions for solitons, kink (antikink) solutions, and further gave some expressions of those solutions. However, they did not give the explicit parametric conditions for the existence of solitons and kink (antikink) solutions. In this paper, we further analyze the bifurcations for (1.1) systematically by exploiting the bifurcation method and qualitative theory of dynamical systems [27]. We present all possible phase portraits determinately and give all the exact explicit parametric conditions for various solutions. Additionally, we obtain explicit peakons and periodic cusp waves for (1.1), which were not included in [1].

2. Bifurcations of Phase Portraits

In this section, we will present the process of obtaining the bifurcations of phase portraits for (1.1).

For given constant , substituting , with into (1.1), it follows, where the prime stands for derivative with respect to the variable .

Integrating (2.1) once leads to where both and are integral constants.

From the second equation of system (2.2), we obtain

Substituting (2.3) into the first equation of system (2.2), it follows:

By setting , (2.4) becomes

Letting , we obtain a planar system with first integral

Note that when , system (2.6) and (2.7) become respectively.

Transformed by , system (2.6) becomes a Hamiltonian system

Since the first integral of system (2.6) is the same as that of the Hamiltonian system (2.10), system (2.6) should have the same topological phase portraits as system (2.10) except the straight line . Therefore, we should be able to obtain the topological phase portraits of system (2.6) from those of system (2.10).

Let

It is easy to obtain the two extreme points of as follows: from which we can obtain a critical curve for as follows:

Further, we obtain two bifurcation curves: from and , respectively. Note that when , obviously .

Additionally, we can obtain another two critical curves for , that is, from and , respectively.

Note that (2.16) can also be obtained by letting .

Let be one of the singular points of system (2.10), then the characteristic values of the linearized system of system (2.10) at the singular point are

From the qualitative theory of dynamical systems, we can determine the property of singular point by the sign of .

Based on the above analysis, we give the information of the singular points for system (2.10) and their relationship with , and when , as an illustration, in the following lemma.

Lemma 2.1. For , one has and the singular points of system (2.10) can be described as follows.(a)If , then there is only one singular point denoted as   . is a saddle point.(b)If , then there are two singular points denoted as and   , respectively. is a saddle point and is a degenerate saddle point.(c)If , then there are three singular points denoted as , , and   , respectively. and are saddle points and is a center.(d)If , then there are three singular points denoted as , , and   , respectively. and are saddle points and is a center.(e)If , then there are two singular points denoted as and   , respectively. is a degenerate saddle point and is a saddle point.(f)If , then there is only one singular point denoted as   . is a saddle point.

Proof. Lemma 2.1 follows easily from the graphics of the function , which can be obtained directly and shown in Figure 1 (note that ).

704931.fig.001
Figure 1: The graphics of when .

Remark 2.2. The case when follows easily from the similar analysis of system (2.8), and we just omit it here for simplicity.

For the other cases, the similar analysis can be taken to make the conclusions. We just omit these processes for simplicity. However, it is worth mentioning that, when and , there exist two saddle points and one center lying on the same side of the singular line . Hence, there may exist heteroclinic orbits for system (2.6). We will show the existence of heteroclinic orbits for system (2.6) under certain conditions in the following analysis.

When , we set the three solutions of to be , , and (), respectively. Through simple calculation, we can express and as the function of , that is,

It follows from that must satisfy condition

From , we obtain the expression of as the function of ,

Substituting (2.20) into , we obtain the expression of from as follows:

We can easily know that (2.22) does not satisfy (2.19), while (2.21) satisfies (2.19), if .

By substituting (2.21) into (2.20), we obtain the bifurcation curve (denoted by ) for as follows:

Note that if , then obviously we have . Additionally, we have indicated that when and , there exist two saddle points and one center lying on the same side of the singular line . Therefore, we obtain the fourth critical curve for from ,

Hence, we can express the existence of the heteroclinic orbits as follows.

Lemma 2.3. (1) For and , there exist heteroclinic orbits for system (2.6). (2) For or , there exists no heteroclinic orbit for system (2.6).

Proof. Note that when (or ), one saddle point and one center point lie on the left side of the singular line and the other saddle point on the right side of (or on) the singular line . Therefore, Lemma 2.3 follows easily from the above analysis.

Thereby, based on the above analysis, we obtain the bifurcations of phase portraits for system (2.6) in Figures 2, 3, 4, 5, 6, 7, and 8 under the corresponding conditions.

704931.fig.002
Figure 2: The phase portraits of system (2.6) when .
704931.fig.003
Figure 3: The phase portraits of system (2.6) when .
704931.fig.004
Figure 4: The phase portraits of system (2.6) when .
704931.fig.005
Figure 5: The phase portraits of system (2.6) when .
704931.fig.006
Figure 6: The phase portraits of system (2.6) when .
704931.fig.007
Figure 7: The phase portraits of system (2.6) when .
704931.fig.008
Figure 8: The phase portraits of system (2.6) when .

3. Main Results and the Theoretic Derivations of Main Results

In this section, we state our results about solitons, kink (antikink) solutions, peakons, and periodic cusp waves for the first component of system (1.1). To relate conveniently, we omit and the expression of the second component of system (1.1) in the following theorems.

Theorem 3.1. For constant wave speed , integral constants and , one has the following.(1) If , , and satisfy one of the following conditions: (i) and ; (ii) and ; (iii) and ; (iv) and ,then there exist soliton solutions for (1.1), which can be implicitly expressed as where (2) If , , and satisfy condition: (v) and ,then there exist soliton solutions for (1.1), which can be implicitly expressed as where If , , and satisfy one of the following conditions: (vi) and ; (vii) and ,then there exist solitons solution for (1.1), which can be implicitly expressed as where , , and are given in (3.4), (3.5), and (3.6) respectively.

Remark 3.2. (3.1) and (3.7) are the same as those given in [1]; however, (3.3) is not shown in [1].

Remark 3.3. We give all possible homoclinic orbits in Figure 9, while it seems that Figures 9(b), 9(d), and 9(e) are not given in [1].

fig9
Figure 9: The different kinds of homoclinic orbits for system (2.6).

Proof. (1) From the phase portraits in Figures 2, 3, 4, 5, 6, 7, and 8, we see that when , , and satisfy one of the conditions, that is, (i), (ii), (iii), or (iv), there exist homoclinic orbits as showed individually in Figures 9(a), 9(b), or 9(d). The expressions of the homoclinic orbits can be given as follows:
Substituting (3.8) into the first equation of system (2.6) and integrating along the homoclinic orbits, it follows that
From (3.9), we obtain the soliton solutions (3.1).
(2) When , , and satisfy one of the conditions, that is, (v), (vi), or (vii), there exist homoclinic orbits as showed individually in Figures 9(c) or 9(e). The expressions of the homoclinic orbits can be given as follows: or
Substituting (3.10) and (3.11) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that
From (3.12), we obtain the soliton solutions (3.3) and (3.7).

Theorem 3.4. If integral constants and satisfy and , then there exist kink and antikink solutions.

Proof. We have showed that when and , there exist heteroclinic orbits for system (2.6). The heteroclinic can be expressed as where which can be obtained by substituting (2.23) into (2.11).
Substituting (3.13) into the first equation of system (2.6) and integrating along the heteroclinic orbits, it follows that where is the initial value.
From (3.15), we have
If we take , (3.16) becomes (3.16) or (3.17) are kink (antikink) solutions.

Theorem 3.5. (1) If and , then there exist peakons for (1.1), which can be explicitly expressed as
(2) If and , then (1.1) has periodic cusp waves where ,  , and with

Remark 3.6. When , system (2.8) is the exact planar system of the Fornberg-Whitham equation [8, 9], and peakons (3.18) and periodic cusp waves (3.19) are the same as those in [9].

Proof. (1) When and , from Figure 5, we see that there is a triangle orbit, which can be expressed as
Substituting (3.22) into the first equation of system (2.6) and integrating along the triangle orbits, it follows that
From (3.24), we obtain peakons (3.18).
(2) When and , from Figure 4, we see that there is an semiellipse orbit, which can be expressed as
Substituting (3.25) into the first equation of system (2.6) and integrating along the semiellipse orbit, it follows that
From (3.27), we obtain periodic cusp waves (3.19) along with (3.20) and (3.21).

Remark 3.7. Our results are more general than those in [10] from the perspective of parameters, because the results in [10] are obtained by setting integral constant to a special value ( in [10]), while our results are obtained under all possible parametric conditions.

Remark 3.8. The system investigated in this paper is relatively complicated for all parameters. That is why Fan et al. [1] did not show the exact parametric conditions for kink (antikink) solutions. However, we figure out a new way to obtain the exact explicit parametric conditions for kink (antikink) solutions and obtain solitons under more general conditions.

4. Conclusions

Based on a previous paper [1], we further study the bifurcations of traveling wave solutions for the two-component Fornberg-Whitham equation, present all possible phase portraits determinately, and show all the exact explicit parametric conditions under which there exist solitons and kink (or antikink) solutions for (1.1). In addition, we obtain peakons and periodic cusp waves with explicit expressions for (1.1). Our results extend the previous study [1], which only gives some possible phase portraits, some undetermined parametric conditions, and some implicit expressions of solitons and kink (or antikink) solutions.

Acknowledgment

This research is supported by Huaqiao University (no. 12BS223).

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