#### Abstract

We study the bifurcation of traveling wave solutions for a two-component generalized -equation. We show all the explicit bifurcation parametric conditions and all possible phase portraits of the system. Especially, the explicit conditions, under which there exist kink (or antikink) solutions, are given. Additionally, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves with explicit expressions, are obtained.

#### 1. Introduction

In 2008, Liu [1] introduced a class of nonlocal dispersive models, that is, -equations, as follows: where denotes the velocity field at time in the spatial direction.

Recently, Ni [2] further investigated the cauchy problem for the following two-component generalized -equations: where takes or . This system includes two components and . The first one describes the horizontal velocity of the fluid, while the other one describes the horizontal deviation of the surface from equilibrium, both are measured in dimensionless units.

In this paper, we study the bifurcation of traveling wave solutions for the following system: which is a special form of system (1.2) through taking and , by employing the bifurcation method and qualitative theory of dynamical systems [3–7]. We give all the explicit bifurcation parametric conditions for various solutions and all possible phase portraints of the system, from which not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves are obtained.

#### 2. Bifurcation of Phase Portraits

For given constant , multiplying both sides of the second equation of system (1.3) by and substituting with into system (1.3), it follows that

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

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

Substituting (2.3) into the first equation of system (2.2), it leads to

By setting , (2.4) becomes

Letting , we obtain a planar system with first integral or

Note that when , systems (2.6), (2.7), and (2.8) become, respectively,

Transformed by , system (2.6) becomes a Hamiltonion system

Since the first integral of system (2.6) is the same as that of the Hamiltonian system (2.12), system (2.6) should have the same topological phase portraits as system (2.12) except the straight line . Therefore, we should be able to obtain the topological phase portraits of system (2.6) from those of system (2.12).

Let

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

We obtain two bifurcation curves: from and , respectively. Note that when , obviously . For convenience, we assume that in this paper, then we have and .

Further, from or , we can obtain another two critical curves for , that is,

Note that (2.18) can also be obtained by letting , or , .

Let be one of the singular points of system (2.12), then the characteristic values of the linearized system of system (2.12) at the singular point are

From the qualitative theory of dynamical systems, we can determine the property of singular point by the sign of and whether equals to or not. However, we also know that from (2.7) and (2.8). Therefore, is an isolated orbit, dividing -plane into two parts.

Based on the above analysis, we give the property of the singular points for system (2.12) and their relationship with , and in the following lemma.

Lemma 2.1. *For , one has and the singular points of system (2.12) 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 degenerate saddle point and is a 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 saddle point and is a degenerate 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.

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

If , we set three solutions of 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 (2.22) into , we obtain the expression of from as follows:

Note that from (2.23)–(2.28), we obtain three critical curves for , that is, , in (2.12), in (2.15), and

We then check the condition (i.e., (2.21)) for the above s one by one and give the results in the following lemma.

Lemma 2.2. *Starting from interval , one has the following.*(1)*For and , (i.e., (2.23)) satisfies (2.21).*(2)*For and , (i.e., (2.24)) satisfies (2.21).*(3)*For any , (2.25) does not satisfy (2.21). *(4)*For any , (2.26) does not satisfy (2.21). *(5)*For and , (i.e., (2.27)) satisfies (2.21). *(6)*For and , (i.e., (2.28)) satisfies (2.21).*

*Proof. *Lemma 2.2 follows easily from the definitional domain of the s and general logical reasoning.

From Lemma 2.2, substituting (2.23) and (2.24) into , respectively, we obtain another two bifurcation curves (denoted by and ) for as follows:

Similarly, substituting (2.27) and (2.28) into , we have

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

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

*Proof. *Lemma 2.3 follows easily from the above analysis.

Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system (2.6) in Figures 2, 3, 4, 5, 6, 7, 8, and 9 under corresponding conditions.

#### 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 (1.3). To relate conveniently, we omit and the expression of the second component of system (1.3) in the following theorems.

Theorem 3.1. *For constant wave speed , integral constants and , one has the following.*(1)* If , , satisfy one of the following conditions:(i) , and ;(ii), and ;(iii), , and ;(iv), and ; then there exist smooth solitons for system (1.3), which can be implicitly expressed as
*

*where*(2)

*If , , satisfy one of the following conditions:(v)*

*, and ;*(vi)*, and ;*(vii)*, , and ;*(viii)*, and ;**then there exist smooth solitons for system (1.3), which can be implicitly expressed as**where*(3)

*If , , satisfy one of the following conditions:(ix)*

*, and ;*(x)*, and ;**then there exist smooth solitons for system (1.3), which can be implicitly expressed as**where*(4)

*If , , satisfy one of the following conditions:(xi)*

*, and ;*(xii)*, and ;**then there exist smooth solitons for system (1.3), which can be implicitly expressed as:**where*

*Proof. *(1) From the phase portraits in Figures 2–9, we see that when , , satisfy one of the conditions, that is, (i), (ii), (iii), or (iv), there exist homoclinic orbits as showed individually in Figures 10(a) and 10(b). The expressions of the homoclinic orbits can be given as follows:
where and can be obtained from (2.8).

Substituting (3.9) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that

From (3.10), we obtain the solitons (3.1) along with (3.2).

(2) When , , satisfy one of the conditions, that is, (v), (vi), (vii), or (viii), there exist homoclinic orbits as showed individually in Figures 8(c) and 8(d). The expressions of the homoclinic orbits can be given as follows:
where and can be obtained from (2.7).

Substituting (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 solitons (3.3) along with (3.4).

(3) When , , satisfy one of the conditions, that is, (ix) or (x), there exist homoclinic orbits as showed individually in Figure 8(e). The expressions of the homoclinic orbits can be given as follows:
where , and can be obtained from (2.7).

Substituting (3.13) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that

From (3.14) [8], we obtain the solitons (3.5) along with (3.6).

(4) When , , satisfy one of the conditions, that is, (xi) or (xii), there exist homoclinic orbits as showed individually in Figure 8(f). The expressions of the homoclinic orbits can be given as follows:
where , , and can be obtained from (2.8).

Substituting (3.15) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that

From (3.16) [8], we obtain the solitons (3.7) along with (3.8).

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

Theorem 3.2. *If constant wave speed , integral constants and satisfy or , and () or (), then there exist kink (antikink) solutions for system (1.3).*

*Proof. *We have showed that, when or , and () or (), there exist heteroclinic orbits for system (2.6). The heteroclinic orbits can be expressed as
where
which can be obtained by substituting (2.23) into (2.20).

Substituting (3.17) into the first equation of system (2.6), and integrating along the heteroclinic orbits, it follows that
where is the initial value.

From (3.19), we have
The case when , can be analyzed similarly. We omit it here for the ease of simplicity.

Theorem 3.3. *(1) If , and , then there exist peakons for system (1.3), which can be explicitly expressed as
**(2) If , and , then system (1.3) has periodic cusp waves
**
where , and
**
with
*

*Proof. *(1) When , and , from Figure 6, we see that there is a triangle orbit, which can be expressed as

Substituting (3.25) into the first equation of system (2.6), and integrating along the triangle orbits, it follows that

From (3.27), we obtain peakons (3.21).

(2) When , and , from Figures 4 and 5, we see that there is a semiellipse orbit, which can be expressed as

Substituting (3.28) into the first equation of system (2.6), and integrating along the semiellipse orbits, it follows that

From (3.30), we obtain periodic cusp waves (3.22) along with (3.23) and (3.24).

Note that we only show the case when , in fact, we can analyze the case when following the same procedure. We just omit it here.

#### 4. Conclusions

In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the bifurcation of traveling wave solutions for a two-component generalized -equation (1.3), show all the explicit parametric conditions and all the phase portraits of system (1.3) determinately. Through the phase portraits, we can investigate various kinds of solutions. Specifically, the implicit expressions of the solitons, kink (antikink) solutions for system (1.3) are given. Besides, we also obtain peakons and periodic cusp waves with explicit expressions for system (1.3).

#### Acknowledgment

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