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

Abstract

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.

1. Introduction

In 2008, Liu [1] introduced a class of nonlocal dispersive models, that is, 𝜃-equations, as follows: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢𝑢_𝑥=(1−𝜃)𝑢_𝑥𝑢_𝑥𝑥+𝜃𝑢𝑢_𝑥𝑥𝑥,𝑥∈ℝ,𝑡>0, (1.1) 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 two-component generalized 𝜃-equations: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢𝑢_𝑥−(1−𝜃)𝑢_𝑥𝑢_𝑥𝑥−𝜃𝑢𝑢_𝑥𝑥𝑥+𝜎𝜌𝜌_𝑥𝜌=0,𝑥∈ℝ,𝑡>0,_𝑡+𝜃𝜌_𝑥𝑢+(1−2𝜃)𝜌𝑢_𝑥=0,𝑥∈ℝ,𝑡>0, (1.2) 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.

In this paper, we study the bifurcation of traveling wave solutions for the following system: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢𝑢_𝑥−35𝑢_𝑥𝑢_𝑥𝑥−25𝑢𝑢_𝑥𝑥𝑥+𝜌𝜌_𝑥𝜌=0,𝑥∈ℝ,𝑡>0,_𝑡+25𝜌_𝑥1𝑢+5𝜌𝑢_𝑥=0,𝑥∈ℝ,𝑡>0, (1.3) 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.

2. 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 −𝑐𝜑^′+𝑐𝜑^′′′+𝜑𝜑^′−35𝜑^′𝜑^′′−25𝜑𝜑^′′′+𝜓𝜓^′=0,−𝑐𝜓𝜓^′+25𝜓𝜓^′1𝜑+5𝜓²𝜑^′=0. (2.1)

Integrating system (2.1) once leads to −𝑐𝜑+𝑐𝜑^′′+12𝜑²−1𝜑10^′2−25𝜑𝜑^′′+12𝜓²−𝑐=𝑔,2𝜓²+15𝜓²𝜑=𝐺, (2.2) where both 𝑔 and 𝐺 are integral constants, respectively.

From the second equation of system (2.2), we obtain 𝜓²=5𝐺()𝑐.𝜑−5/2 (2.3)

Substituting (2.3) into the first equation of system (2.2), it leads to 5𝜑−2𝑐²𝜑^′′1=−45𝜑−2𝑐𝜑^′2+545𝜑−2𝑐𝜑²−2𝑐𝜑−2𝑔+5𝐺. (2.4)

By setting 𝜑=𝜙+(3/2)𝑐, (2.4) becomes 𝜙−𝑐²𝜙^′′1=−4𝜙−𝑐𝜙^′2+543𝜙−𝑐𝜙−2𝑐²3−2𝑐𝜙−2𝑐1−2𝑔+5𝐺=−4𝜙−𝑐𝜙^′2+54𝜙³−74𝑐²3+2𝑔𝜙+4𝑐³+5𝐺+2𝑐𝑔. (2.5)

Letting 𝑦=𝜑^′, we obtain a planar system d𝜙d𝜉=𝑦,d𝑦d𝜉=−(1/4)𝜙−𝑐𝑦²+(5/4)𝜙³−(7/4)𝑐²+2𝑔𝜙+(3/4)𝑐³+5𝐺+2𝑐𝑔𝜙−𝑐², (2.6) with first integral 𝐻=1𝜙,𝑦2𝜙−𝑐𝑦²−4𝜙−𝑐³+20𝑐𝜙−𝑐²+25𝑐²−40𝑔𝜙−𝑐−100𝐺8,𝜙−𝑐for𝜙>𝑐, (2.7) or 𝐻=1𝜙,𝑦2𝑐−𝜙𝑦²−4𝑐−𝜙³−20𝑐𝑐−𝜙²+25𝑐²−40g𝑐−𝜙+100𝐺8,𝑐−𝜙for𝜙<𝑐. (2.8)

Note that when 𝐺=0, systems (2.6), (2.7), and (2.8) become, respectively, d𝜙d𝜉=𝑦,d𝑦d𝜉=−(1/4)𝑦²+(5/4)𝜙²+𝑐𝜙−(7/4)𝑐²−2𝑔,𝜙−𝑐𝐻=1𝜙,𝑦2𝜙−𝑐𝑦²−184𝜙−𝑐^5/2+20𝑐𝜙−𝑐^3/2+25𝑐²−40𝑔𝜙−𝑐^1/2,for𝜙>𝑐,𝐻=1𝜙,𝑦2𝑐−𝜙𝑦²−184𝑐−𝜙^5/2−20𝑐𝑐−𝜙^3/2+25𝑐²−40𝑔𝑐−𝜙^1/2,for𝜙<𝑐. (2.9)(2.10)(2.11)

Transformed by d𝜉=(𝜙−𝑐)²d𝜏, system (2.6) becomes a Hamiltonion system d𝜙d𝜏=𝜙−𝑐²𝑦,d𝑦d𝜏1=−4𝜙−𝑐𝑦²+54𝜙³−74𝑐²3+2𝑔𝜙+4𝑐³+5𝐺+2𝑐𝑔. (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 𝑓𝜙=𝜙³−74𝑐²3+2𝑔𝜙+4𝑐³+5𝐺+2𝑐𝑔. (2.13)

It is easy to obtain the two extreme points of 𝑓(𝜙) as follows: 𝜙^∗_±=±7𝑐²+8𝑔,12for7𝑔>−8𝑐², (2.14) from which we can obtain a critical curve for 𝑔 as follows: 𝑔₀(𝑐)7=−8𝑐². (2.15)

We obtain two bifurcation curves: 𝐺₁1=−18072𝑐𝑔+27𝑐³+8𝑔+7𝑐²21𝑐²,𝐺+24𝑔₂1=−18072𝑐𝑔+27𝑐³−8𝑔+7𝑐²21𝑐²+24𝑔, (2.16) from 𝑓(𝜙^∗₋)=0 and 𝑓(𝜙^∗₊)=0, respectively. Note that when 𝑔<𝑔₀(𝑐), obviously 𝐺₁<𝐺₂. For convenience, we assume that 𝑔∝𝑐² in this paper, then we have 𝐺₁∝𝑐³ and 𝐺₂∝𝑐³.

Further, from 𝐺₁=0 or 𝐺₂=0, we can obtain another two critical curves for 𝑔, that is, 𝑔₁(𝑐)1=−2𝑐²,𝑔₂(𝑐)=58𝑐². (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 𝜆_±1=±25𝜙^∗−𝑐²𝑓^′𝜙^∗. (2.19)

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>𝑔₂(𝑐), one has 𝐺₁<0<𝐺₂ and the singular points of system (2.12) can be described as follows.(a)If 𝐺<𝐺₁, then there is only one singular point denoted as 𝑆₁(𝜙₁,0)(𝜙^∗₋<𝑐<𝜙^∗₊<𝜙₁). 𝑆₁ is a saddle point.(b)If 𝐺=𝐺₁, then there are two singular points denoted as 𝑆₁(𝜙₁,0) and 𝑆₂(𝜙₂,0)(𝜙₁=𝜙^∗₋<𝑐<𝜙^∗₊<𝜙₂), respectively. 𝑆₁ is a degenerate saddle point and 𝑆₂ is a saddle point.(c)If 𝐺₁<𝐺<0, then there are three singular points denoted as 𝑆₁(𝜙₁,0), 𝑆₂(𝜙₂,0), and 𝑆₃(𝜙₃,0)(𝜙₁<𝜙^∗₋<𝜙₂<𝑐<𝜙^∗₊<𝜙₃), respectively. 𝑆₁ and 𝑆₃ are saddle points and 𝑆₂ is a center.(d)If 0<𝐺<𝐺₂, then there are three singular points denoted as 𝑆₁(𝜙₁,0), 𝑆₂(𝜙₂,0), and 𝑆₃(𝜙₃,0)(𝜙₁<𝜙^∗₋<𝑐<𝜙₂<𝜙^∗₊<𝜙₃), respectively. 𝑆₁ and 𝑆₃ are saddle points and 𝑆₂ is a center.(e)If 𝐺=𝐺₂, then there are two singular points denoted as 𝑆₁(𝜙₁,0) and 𝑆₂(𝜙₂,0)(𝜙₁<𝜙^∗₋<𝑐<𝜙^∗₊=𝜙₂), respectively. 𝑆₁ is a saddle point and 𝑆₂ is a degenerate saddle point.(f)If 𝐺>𝐺₂, then there is only one singular point denoted as 𝑆₁(𝜙₁,0)(𝜙₁<𝜙^∗₋<𝑐<𝜙^∗₊). 𝑆₁ is a saddle point.

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

Figure 1 

The graphics of 𝑓(𝜙) when 𝑔>𝑔2(𝑐).

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 𝑔₀(𝑐)<𝑔<𝑔₂(𝑐) and 𝐺₁<𝐺<𝐺₂(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 𝐺₁<𝐺<𝐺₂, 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, 𝜙_𝑠=−𝜙_𝑚−8𝑔+7𝑐²−3𝜙²_𝑚2,𝜙_𝑏=−𝜙_𝑚+8𝑔+7𝑐²−3𝜙²_𝑚2. (2.20)

It follows from 𝜙_𝑠<𝜙_𝑚<𝜙_𝑏 that 𝜙_𝑚 must satisfy condition 𝜙²_𝑚<8𝑔+7𝑐².12 (2.21) From 𝐻(𝜙_𝑠,0)=𝐻(𝜙_𝑏,0), we obtain the expression of 𝐺 as the function of 𝜙_𝑚, 1𝐺=1009𝑐³+24𝑐𝑔−8𝑔+15𝑐²𝜙_𝑚−8𝑐𝜙²_𝑚+4𝜙³_𝑚+2𝑐²−16𝑔−4𝑐𝜙_𝑚4𝜙²_𝑚−8𝑔+4𝑐𝜙_𝑚−3𝑐³. (2.22)

Substituting (2.22) into 𝑓(𝜙_𝑚)=0, we obtain the expression of 𝜙_𝑚 from 𝑓(𝜙_𝑚)=0 as follows: 𝜙_𝑚1=165𝑐−2𝑐²−6𝑔,𝜙_𝑚2=165𝑐+2𝑐²−6𝑔,𝜙_𝑚3=−7𝑐²+8𝑔3,𝜙_𝑚4=7𝑐²+8𝑔3,𝜙_𝑚5=12−𝑐−2𝑐²+2𝑔,𝜙_𝑚6=12−𝑐+2𝑐²+2𝑔, (2.23)(2.24)(2.25)(2.26)(2.27)(2.28)

Note that from (2.23)–(2.28), we obtain three critical curves for 𝑔, that is, 𝑔₀(𝑐), in (2.12), 𝑔₁(𝑐) in (2.15), and 𝑔₃(𝑐)=16𝑐². (2.29)

We then check the condition 𝜙²_𝑚<(8𝑔+7𝑐²)/12 (i.e., (2.21)) for the above 𝜙_𝑚s one by one and give the results in the following lemma.

Lemma 2.2. Starting from interval (−(7/8)𝑐²,(5/8)𝑐²), one has the following.(1)For 𝑔∈(−(7/8)𝑐²,(1/6)𝑐²)⊂(−(7/8)𝑐²,(5/8)𝑐²) and 𝑐>0, 𝜙_𝑚=(1/6)(5𝑐−2𝑐²−6𝑔) (i.e., (2.23)) satisfies (2.21).(2)For 𝑔∈(−(7/8)𝑐²,(1/6)𝑐²)⊂(−(7/8)𝑐²,(5/8)𝑐²) and 𝑐<0, 𝜙_𝑚=(1/6)(5𝑐+2𝑐²−6𝑔) (i.e., (2.24)) satisfies (2.21).(3)For any 𝑔∈(−(7/8)𝑐²,(5/8)𝑐²), (2.25) does not satisfy (2.21). (4)For any 𝑔∈(−(7/8)𝑐²,(5/8)𝑐²), (2.26) does not satisfy (2.21). (5)For 𝑔∈(−(1/2)𝑐²,(5/8)𝑐²)⊂(−(7/8)𝑐²,(5/8)𝑐²) and 𝑐<0, 𝜙_𝑚=(1/2)(−𝑐−2𝑐²+2𝑔) (i.e., (2.27)) satisfies (2.21). (6)For 𝑔∈(−(1/2)𝑐²,(5/8)𝑐²)⊂(−(7/8)𝑐²,(5/8)𝑐²) and 𝑐>0, 𝜙_𝑚=(1/2)(−𝑐+2𝑐²+2𝑔) (i.e., (2.28)) satisfies (2.21).

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

From Lemma 2.2, substituting (2.23) and (2.24) into 𝑓(𝜙_𝑚)=0, respectively, we obtain another two bifurcation curves (denoted by 𝐺^∗₁ and 𝐺^∗₂) for 𝐺 as follows: 𝐺^∗₁=4135−𝑐³+9𝑐𝑔+𝑐²−6𝑔𝑐²−6𝑔,for𝑔₀(𝑐)<𝑔<𝑔₃(𝑐)𝐺,𝑐>0,_∗2=4135−𝑐³+9𝑐𝑔−𝑐²−6𝑔𝑐²−6𝑔,for𝑔₀(𝑐)<𝑔<𝑔₃(𝑐),𝑐<0. (2.30)

Similarly, substituting (2.27) and (2.28) into 𝑓(𝜙_𝑚)=0, we have 𝐺_∗=0,for𝑔₁(𝑐)<𝑔<𝑔₃(𝑐),𝑐<0or𝑔₁(𝑐)<𝑔<𝑔₃(𝑐),𝑐>0. (2.31)

Note that we have indicated that when 𝑔₀(𝑐)<𝑔<𝑔₂(𝑐) and 𝐺₁<𝐺<𝐺₂(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^∗₁=0(𝑐>0) or 𝐺^∗₂=0(𝑐<0), 𝑔₄(𝑐)=0. (2.32)

Lemma 2.3. (1) For 𝑔∈(𝑔₀(𝑐),𝑔₄(𝑐))∪(𝑔₄(𝑐),𝑔₃(𝑐)), and 𝐺=𝐺^∗₁,𝑐>0 (or 𝐺=𝐺^∗₂,𝑐<0), there exist heteroclinic orbits for system (2.6).
(2) For any 𝑔∉(𝑔₀(𝑐),𝑔₄(𝑐))∪(𝑔₄(𝑐),𝑔₃(𝑐)) or 𝐺≠𝐺^∗₁,𝑐>0 and 𝐺≠𝐺^∗₂,𝑐<0, 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.

Figure 2 

The phase portraits of system (2.6) when 𝑔>𝑔2(𝑐).

Figure 3 

The phase portraits of system (2.6) when 𝑔=𝑔2(𝑐).

Figure 4 

The phase portraits of system (2.6) when 𝑔3(𝑐)≤𝑔<𝑔2(𝑐).

Figure 5 

The phase portraits of system (2.6) when 𝑔4(𝑐)<𝑔<𝑔3(𝑐).

Figure 6 

The phase portraits of system (2.6) when 𝑔=𝑔4(𝑐).

Figure 7 

The phase portraits of system (2.6) when 𝑔1(𝑐)<𝑔<𝑔4(𝑐).

Figure 8 

The phase portraits of system (2.6) when 𝑔=𝑔1(𝑐).

Figure 9 

The phase portraits of system (2.6) when 𝑔0(𝑐)<𝑔<𝑔1(𝑐).

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 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:(i)𝑔>𝑔₂(𝑐), 𝐺₁<𝐺<0 and 𝑐≠0;(ii)𝑔₁(𝑐)<𝑔≤𝑔₂(𝑐), 𝐺₁<𝐺<0 and 𝑐>0;(iii)𝑔₁(𝑐)≤𝑔<𝑔₄(𝑐), 𝑔₄(𝑐)<𝑔<𝑔₃(𝑐), 0<𝐺<𝐺^∗₁ and 𝑐>0;(iv)𝑔₀(𝑐)<𝑔<𝑔₁(𝑐), 𝐺₁<𝐺<𝐺^∗₁ and 𝑐>0; then there exist smooth solitons for system (1.3), which can be implicitly expressed as 𝑐−𝜙/𝜙^∗₁−𝜙−1𝑐−𝜙/𝜙^∗₁⋅−𝜙+1(𝑐−𝜙)/(𝜙^∗₁−𝜙)+√𝛼^𝛼(𝑐−𝜙)/(𝜙^∗₁−𝜙)−√𝛼^𝛼=𝑒^|𝜉|, (3.1) where 𝛼=𝑐−𝜙₁𝜙^∗₁−𝜙₁. (3.2)(2) If 𝑐, 𝑔, 𝐺 satisfy one of the following conditions:(v)𝑔>𝑔₂(𝑐), 0<𝐺<𝐺₂ and 𝑐≠0;(vi)𝑔₁(𝑐)<𝑔≤𝑔₄(𝑐), 0<𝐺<𝐺₂ and 𝑐<0;(vii)𝑔₁(𝑐)≤𝑔<𝑔₄(𝑐), 𝑔₄(𝑐)<𝑔<𝑔₃(𝑐), 𝐺^∗₂<𝐺<0 and 𝑐<0;(viii)𝑔₀(𝑐)<𝑔<𝑔₁(𝑐), 𝐺^∗₂<𝐺<𝐺₂ and 𝑐<0; then there exist smooth solitons for system (1.3), which can be implicitly expressed as 𝜙−𝑐/𝜙−𝜙^∗₂−1𝜙−𝑐/𝜙−𝜙^∗₂⋅+1(𝜙−𝑐)/(𝜙−𝜙^∗₂)+𝛽^𝛽(𝜙−𝑐)/(𝜙−𝜙^∗₂)−𝛽^𝛽=𝑒^|𝜉|, (3.3) where 𝜙𝛽=²−𝑐𝜙₂−𝜙^∗₂. (3.4)(3) If 𝑐, 𝑔, 𝐺 satisfy one of the following conditions:(ix)𝑔₃(𝑐)≤𝑔<𝑔₂(𝑐), 𝐺₁<𝐺<0 and 𝑐<0;(x)𝑔₀(𝑐)<𝑔<𝑔₃(𝑐), 𝐺₁<𝐺<𝐺^∗₂ and 𝑐<0;  then there exist smooth solitons for system (1.3), which can be implicitly expressed as 𝑔₁^𝑢1₀d𝜙𝑢−³−𝑐𝜙₃−𝜙^∗₃₁^𝑢1₀dn²𝑢1−𝛾²₁sn²𝑢d𝑢=𝑒^|𝜉|, (3.5) where 𝑔₁=2𝜙^∗₃₂,𝛾−𝑐₂₁=𝑘²₁𝜙^∗₃₂−𝜙₃𝜙^∗₃₁−𝜙₃,𝑘²₁=𝜙^∗₃₁−𝑐𝜙^∗₃₂,−𝑐sn𝑢₁=sin𝜙. (3.6)(4) If 𝑐, 𝑔, 𝐺 satisfy one of the following conditions:(xi)𝑔₃(𝑐)≤𝑔<𝑔₂(𝑐), 0<𝐺<𝐺₂ and 𝑐>0;(xii)𝑔₀(𝑐)<𝑔<𝑔₃(𝑐), 𝐺^∗₁<𝐺<𝐺₂ and 𝑐>0; then there exist smooth solitons for system (1.3), which can be implicitly expressed as: 𝑔₂^𝑢2₀d𝑢−𝑐−𝜙₄𝜙^∗₄₁−𝜙₄^𝑢2₀dn²𝑢1−𝛾²₂sn²𝑢d𝑢=𝑒^|𝜉|, (3.7) where 𝑔₂=2𝑐−𝜙^∗₄₂,𝛾²₂=𝑘²₂𝜙₄−𝜙^∗₄₂𝜙₄−𝜙^∗₄₁,𝑘²₂=𝑐−𝜙^∗₄₁𝑐−𝜙^∗₄₂,sn𝑢₂=sin𝜙. (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: 𝑦=±𝜙−𝜙₁𝜙^∗₁−𝜙𝑐−𝜙,𝜙₁≤𝜙≤𝜙^∗₁<𝑐, (3.9) where 𝜙₁ and 𝜙^∗₁ 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 ^𝜙∗1_𝜙√𝑐−𝑠d𝑠𝑠−𝜙₁𝜙^∗₁=𝜉.−𝑠 (3.10)
From (3.10), we obtain the solitons (3.1) along with (3.2).
(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: 𝑦=±𝜙₂−𝜙𝜙−𝜙^∗₂𝜙−𝑐,𝑐<𝜙^∗₂≤𝜙≤𝜙₂, (3.11) where 𝜙₂ and 𝜙^∗₂ 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 ^𝜙_𝜙∗2√𝑠−𝑐d𝑠𝜙₂−𝑠𝑠−𝜙^∗₂=𝜉. (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: 𝑦=±𝜙−𝜙₃𝜙^∗₃₁−𝜙𝜙^∗₃₂−𝜙𝜙−𝑐,𝑐<𝜙₃≤𝜙≤𝜙^∗₃₁<𝜙^∗₃₂, (3.13) where 𝜙₃, 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 ^𝜙∗31_𝜙√𝑠−𝑐d𝑠𝑠−𝜙₃𝜙^∗₃₁−𝑠𝜙^∗₃₂=𝜉.−𝑠 (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: 𝑦=±𝜙₄−𝜙𝜙−𝜙^∗₄₁𝜙−𝜙^∗₄₂𝑐−𝜙,𝜙^∗₄₂<𝜙^∗₄₁≤𝜙≤𝜙₄<𝑐, (3.15) where 𝜙₄, 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 ^𝜙_𝜙∗41√𝑐−𝑠d𝑠𝜙₄−𝑠𝑠−𝜙^∗₄₁𝑠−𝜙^∗₄₂=𝜉. (3.16)
From (3.16) [8], we obtain the solitons (3.7) along with (3.8).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10 

The different kinds of homoclinic orbits for system (2.6).

Theorem 3.2. If constant wave speed 𝑐, integral constants 𝑔 and 𝐺 satisfy 𝑔₀(𝑐)<𝑔<𝑔₄(𝑐) or 𝑔₄(𝑐)<𝑔<𝑔₃(𝑐), and 𝐺=𝐺^∗₁ (𝑐>0) or 𝐺=𝐺^∗₂ (𝑐<0), then there exist kink (antikink) solutions for system (1.3).

Proof. We have showed that, when 𝑔₀(𝑐)<𝑔<𝑔₄(𝑐) or 𝑔₄(𝑐)<𝑔<𝑔₃(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 𝑦=±𝜙−𝜙_𝑠𝜙_𝑏−𝜙,𝑐−𝜙for𝑐>0, (3.17) where 𝜙_𝑠=112−5𝑐+2𝑐²−6𝑔−1511𝑐²+4𝑐𝑐²,𝜙−6𝑔+24𝑔_𝑏=112−5𝑐+2𝑐²−6𝑔+1511𝑐²+4𝑐𝑐²−6𝑔+24𝑔, (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 ^𝜙_𝜙0√𝑐−𝑠d𝑠𝑠−𝜙_𝑠𝜙_𝑏−𝑠=±𝜉, (3.19) where 𝜙₀∈(𝜙_𝑠,𝜙_𝑏) is the initial value.
From (3.19), we have 𝑐−𝜙_𝑠−𝑐−𝜙^{𝑐−𝜙𝑠/(𝜙𝑏−𝜙𝑠)}𝑐−𝜙_𝑠+𝑐−𝜙^{𝑐−𝜙𝑠/(𝜙𝑏−𝜙𝑠)}⋅𝑐−𝜙+𝑐−𝜙_𝑏^{𝑐−𝜙𝑏/(𝜙𝑏−𝜙𝑠)}𝑐−𝜙−𝑐−𝜙_𝑏^{𝑐−𝜙𝑏/(𝜙𝑏−𝜙𝑠)}=𝑐−𝜙_𝑠−𝑐−𝜙₀^{𝑐−𝜙𝑠/(𝜙𝑏−𝜙𝑠)}𝑐−𝜙_𝑠+𝑐−𝜙₀^{𝑐−𝜙𝑠/(𝜙𝑏−𝜙𝑠)}⋅𝑐−𝜙₀+𝑐−𝜙_𝑏^{𝑐−𝜙𝑏/(𝜙𝑏−𝜙𝑠)}𝑐−𝜙₀−𝑐−𝜙_𝑏^{𝑐−𝜙𝑏/(𝜙𝑏−𝜙𝑠)}𝑒^±𝜉. (3.20) The case when 𝑐<0, can be analyzed similarly. We omit it here for the ease of simplicity.

Theorem 3.3. (1) If 𝑔=𝑔₄(𝑐), 𝐺=0 and 𝑐≠0, then there exist peakons for system (1.3), which can be explicitly expressed as 5𝜙=2𝑐𝑒^{−|𝑥−𝑐𝑡|}−32𝑐. (3.21)
(2) If 𝑔₄(𝑐)≤𝑔<𝑔₂(𝑐), 𝐺=0 and 𝑐≠0, then system (1.3) has periodic cusp waves 𝑢()()+3𝑥,𝑡=𝜙𝜉−2𝑖𝑇2𝑐, (3.22) where 𝑖=0,±1,±2,…,𝜉=𝑥−𝑐𝑡∈[(2𝑖−1)𝑇,(2𝑖+1)𝑇], and 𝜙(𝜉)=145𝑐−25𝑐²−40𝑔𝑒^{|𝑥−𝑐𝑡|}+145𝑐+25c²−40𝑔𝑒^{−|𝑥−𝑐𝑡|}−32𝑐, (3.23) with 𝑇=ln5𝑐+25𝑐²−40𝑔2.10𝑔 (3.24)

Proof. (1) When 𝑔=𝑔₄(𝑐), 𝐺=0 and 𝑐≠0, from Figure 6, we see that there is a triangle orbit, which can be expressed as 3𝑦=±𝜙+2𝑐,for−32𝑐≤𝜙≤𝑐(𝑐>0),𝜙=𝑐,for𝑦≤√52𝑐(𝑐>0). (3.25)(3.26)
Substituting (3.25) into the first equation of system (2.6), and integrating along the triangle orbits, it follows that ^𝑐_𝜙d𝑡()𝑐=𝜉.𝑡+3/2 (3.27)
From (3.27), we obtain peakons (3.21).
(2) When 𝑔₄(𝑐)≤𝑔<𝑔₂(𝑐), 𝐺=0 and 𝑐≠0, from Figures 4 and 5, we see that there is a semiellipse orbit, which can be expressed as 𝑦=±𝜙²9+3𝑐𝜙+4𝑐²−10𝑔,for12−3𝑐+210𝑔≤𝜙≤𝑐(𝑐>0),𝜙=𝑐,for𝑦≤5𝑐²−8𝑔2(𝑐>0). (3.28)(3.29)
Substituting (3.28) into the first equation of system (2.6), and integrating along the semiellipse orbits, it follows that ^𝑐_𝜙d𝑡𝑡²+3𝑐𝑡+(9/4)𝑐²=𝜉.−10𝑔 (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.

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 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).

Acknowledgment

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