Bifurcation of Traveling Wave Solutions for a Two-Component Generalized θ-Equation

We study the bifurcation of traveling wave solutions for a two-component generalized θ-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.


Introduction
In 2008, Liu 1 introduced a class of nonlocal dispersive models, that is, θ-equations, as follows: where u x, t denotes the velocity field at time t in the spatial x direction.Recently, Ni 2 further investigated the cauchy problem for the following twocomponent generalized θ-equations: x ∈ R, t > 0, where σ takes 1 or −1.This system includes two components u x, t and ρ x, t .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.

Mathematical Problems in Engineering
In this paper, we study the bifurcation of traveling wave solutions for the following system: uu xxx ρρ x 0, x ∈ R, t > 0, which is a special form of system 1.2 through taking θ 2/5 and σ 1, 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.

Bifurcation of Phase Portraits
For given constant c, multiplying both sides of the second equation of system 1.3 by ρ x, t and substituting u x, t ϕ ξ , ρ ψ ξ with ξ x − ct into system 1.3 , it follows that ψψ ϕ 1 5 ψ 2 ϕ 0.

2.1
Integrating system 2.1 once leads to where both g and G 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

2.12
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 l : φ c.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 f φ as follows: 14 from which we can obtain a critical curve for g as follows:

2.15
We obtain two bifurcation curves: from f φ * − 0 and f φ * 0, respectively.Note that when g < g 0 c , obviously G 1 < G 2 .For convenience, we assume that g ∝ c 2 in this paper, then we have Further, from G 1 0 or G 2 0, we can obtain another two critical curves for g, that is, 2.17

2.18
Note that 2.18 can also be obtained by letting φ * c, c > 0 or φ * − c, c < 0. Let φ * , 0 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 φ * , 0 are From the qualitative theory of dynamical systems, we can determine the property of singular point φ * , 0 by the sign of f φ * and whether φ * equals to c or not.However, we also know that H c, y ∞ from 2.7 and 2.8 .Therefore, φ c is an isolated orbit, dividing φ, y -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 φ * − , φ * and c in the following lemma.
Lemma 2.1.For g > g 2 c , one has G 1 < 0 < G 2 and the singular points of system 2.12 can be described as follows.
a If G < G 1 , then there is only one singular point denoted as , then there are two singular points denoted as S 1 φ 1 , 0 and S 2 φ 2 , 0 φ 1 φ * − < c < φ * < φ 2 , respectively.S 1 is a degenerate saddle point and S 2 is a saddle point.c If G 1 < G < 0, then there are three singular points denoted as S 1 φ 1 , 0 , S 2 φ 2 , 0 , and , respectively.S 1 and S 3 are saddle points and S 2 is a center.
then there are three singular points denoted as S 1 φ 1 , 0 , S 2 φ 2 , 0 , and Proof.Lemma 2.1 follows easily from the graphics of the function f φ 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 g 0 c < g < g 2 c and G 1 < G < G 2 G / 0 , there exist two saddle points and one center lie on the same side of singular line φ c.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 G 1 < G < G 2 , we set three solutions of f φ 0 be φ s , φ m , and φ b φ s < φ m < φ b , respectively.Through simple calculation, we can express φ s and φ b as the function of φ m , that is,

2.20
It follows from φ s < φ m < φ b that φ m must satisfy condition

2.21
From H φ s , 0 H φ b , 0 , we obtain the expression of G as the function of φ m ,

2.22
Substituting 2.22 into f φ m 0, we obtain the expression of φ m from f φ m 0 as follows: Note that from 2.23 -2.28 , we obtain three critical curves for g, that is, g 0 c , in 2.12 , g 1 c in 2.15 , and

2.29
We then check the condition φ 2 m < 8g 7c

2.30
Similarly, substituting 2.27 and 2.28 into f φ m 0, we have Note that we have indicated that when g 0 c < g < g 2 c and G 1 < G < G 2 G / 0 , there exist two saddle points and one center lying on the same side of singular line φ c.Therefore, we obtain the fifth critical curve for g from G * The phase portraits of system 2.6 when g > g 2 c .
The phase portraits of system 2.6 when g g 2 c .
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.

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 ϕ φ 2/3 c and the expression of the second component of system 1.3 in the following theorems.
Theorem 3.1.For constant wave speed c, integral constants g and G, one has the following.
1 If c, g, G satisfy one of the following conditions: 1 and c > 0; then there exist smooth solitons for system 1.3 , which can be implicitly expressed as where 2 If c, g, G satisfy one of the following conditions: 2 < G < G 2 and c < 0; then there exist smooth solitons for system 1.3 , which can be implicitly expressed as where

3.4
3 If c, g, G satisfy one of the following conditions: Mathematical Problems in Engineering then there exist smooth solitons for system 1.3 , which can be implicitly expressed as where

3.6
4 If c, g, G satisfy one of the following conditions: and c > 0; then there exist smooth solitons for system 1.3 , which can be implicitly expressed as: where snu 2 sin φ.

3.8
Proof. 1 From the phase portraits in Figures 2-9, we see that when c, g, G 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 φ 1 and φ * 1 can be obtained from 2.8 .
The phase portraits of system 2.6 when The phase portraits of system 2.6 when g 4 c < g < g 3 c .Substituting 3.9 into the first equation of system 2.6 , and integrating along the homoclinic orbits, it follows that

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

Figure 6:
The phase portraits of system 2.6 when g g 4 c . 2 When c, g, G 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 φ 2 and φ * 2 can be obtained from 2.7 .

Figure 8:
The phase portraits of system 2.6 when g g 1 c .3.12 From 3.12 , we obtain the solitons 3.3 along with 3.4 .
3 When c, g, G 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 φ 3 , φ * 31 and φ * 32 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 3.14 From 3.14 8 , we obtain the solitons 3.5 along with 3.6 .4 When c, g, G 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 φ 4 , φ * 41 , and φ * 42 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 Proof.We have showed that, when g 0 c < g < g 4 c or g 4 c < g < g 3 c , and G G * 1 c > 0 or G G * 2 c < 0 , there exist heteroclinic orbits for system 2.6 .The heteroclinic orbits can be expressed as where

3.18
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

3.20
The case when c < 0, can be analyzed similarly.We omit it here for the ease of simplicity.

3.30
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 c > 0, in fact, we can analyze the case when c < 0 following the same procedure.We just omit it here.

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 θ-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 .

Figure 1 :
Figure 1: The graphics of f φ when g > g 2 c .

Figure 7 :
Figure 7:The phase portraits of system 2.6 when g 1 c < g < g 4 c .