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

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 𝑢_𝑡=𝑢_𝑥𝑥𝑡−𝑢_𝑥−𝑢𝑢_𝑥+3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝜌_𝑥,𝜌_𝑡=−𝜌𝑢_𝑥, (1.1) where 𝑢=𝑢(𝑥,𝑡) denotes the height of the water surface above a horizontal bottom, and 𝜌=𝜌(𝑥,𝑡) 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 [2–7]. 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 𝑐, substituting 𝑢(𝑥,𝑡)=𝜑(𝜉), 𝜌=𝜓(𝜉) with 𝜉=𝑥−𝑐𝑡 into (1.1), it follows, −𝑐𝜑^′+𝑐𝜑^′′′+𝜑^′+𝜑𝜑^′−3𝜑^′𝜑^′′−𝜑𝜑^′′′−𝜓^′=0,−𝑐𝜓^′+𝜑^′𝜓+𝜑𝜓^′=0, (2.1) where the prime stands for derivative with respect to the variable 𝜉.

Integrating (2.1) once leads to ()11−𝑐𝜑+2𝜑²−𝜑^′2+𝑐−𝜑𝜑^′′−𝜓=𝑔,𝜑−𝑐𝜓=𝐺, (2.2) where both 𝑔 and 𝐺 are integral constants.

From the second equation of system (2.2), we obtain 𝐺𝜓=.𝜑−𝑐 (2.3)

Substituting (2.3) into the first equation of system (2.2), it follows: ()11−𝑐𝜑+2𝜑²−𝜑^′2+𝑐−𝜑𝜑^′′−𝐺𝜑−𝑐=𝑔. (2.4)

By setting 𝜑=𝜙+𝑐−2/3, (2.4) becomes 23−𝜙²𝜙^′′=23−𝜙𝜙^′2+12𝜙³−163𝑐²1−6𝑐+4+6𝑔𝜙+279𝑐²−18𝑐+8+18𝑔−𝐺. (2.5)

Letting 𝑦=𝜙^′, we obtain a planar system 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=𝑦𝑑𝜉2/3−𝜙²+(1/2)𝜙³−(1/6)3𝑐²−6𝑐+4+6𝑔𝜙+(1/27)9𝑐²−18𝑐+8+18𝑔−𝐺2/3−𝜙², (2.6) with first integral 𝐻=2𝜙,𝑦3−𝜙²𝑦²−14𝜙⁴+163𝑐²−6𝑐+4+6𝑔𝜙²−2279𝑐²−18𝑐+8+18𝑔𝜙+2𝐺𝜙. (2.7)

Note that when 𝐺=0, system (2.6) and (2.7) become 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=𝑦𝑑𝜉²−(1/2)𝜙²−(1/3)𝜙+(1/6)3𝑐²−6𝑐+4+6𝑔−2/9,𝐻=22/3−𝜙𝜙,𝑦3−𝜙²𝑦²−14𝜙⁴+163𝑐²−6𝑐+4+6𝑔𝜙²−2279𝑐²−18𝑐+8+18𝑔𝜙, (2.8)(2.9) respectively.

Transformed by 𝑑𝜉=(2/3−𝜙)²𝑑𝜏, system (2.6) becomes a Hamiltonian system 𝑑𝜙=2𝑑𝜉3−𝜙²𝑦,𝑑𝑦=2𝑑𝜉3−𝜙𝑦²+12𝜙³−163𝑐²1−6𝑐+4+6𝑔𝜙+279𝑐²−18𝑐+8+18𝑔−𝐺. (2.10)

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 𝑙∶𝜙=2/3. Therefore, we should be able to obtain the topological phase portraits of system (2.6) from those of system (2.10).

Let 𝑓𝜙=12𝜙³−163𝑐²1−6𝑐+4+6𝑔𝜙+279𝑐²−18𝑐+8+18𝑔−𝐺. (2.11)

It is easy to obtain the two extreme points of 𝑓(𝜙) as follows: 𝜙^∗_±1=±33𝑐²1−6𝑐+4+6𝑔,for𝑔>−2𝑐²4−2𝑐+3, (2.12) from which we can obtain a critical curve for 𝑔 as follows: 𝑔₀(𝑐)1=−2𝑐²4−2𝑐+3. (2.13)

Further, we obtain two bifurcation curves: 𝐺₁=1273𝑐²−6𝑐+4+6𝑔3𝑐²−6𝑐+4+6𝑔+9𝑐²,𝐺−18𝑐+8+18𝑔₂=1−273𝑐²−6𝑐+4+6𝑔3𝑐²−6𝑐+4+6𝑔+9𝑐²−18𝑐+8+18𝑔, (2.14) from 𝑓(𝜙^∗₋)=0 and 𝑓(𝜙^∗₊)=0, respectively. Note that when 𝑔>𝑔₀(𝑐), obviously 𝐺₁>𝐺₂.

Additionally, we can obtain another two critical curves for 𝑔, that is, 𝑔₁(𝑐)1=−2𝑐²,𝑔−2𝑐+1₂(𝑐)1=−2𝑐²−2𝑐, (2.15)(2.16) from 𝐺₁=0 and 𝐺₂=0, respectively.

Note that (2.16) can also be obtained by letting 𝜙^∗₊=2/3.

Let (𝜙^∗,0) 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 (𝜙^∗,0) are 𝜆_±=±23−𝜙^∗2𝑓^′𝜙^∗. (2.17)

From the qualitative theory of dynamical systems, we can determine the property of singular point (𝜙^∗,0) by the sign of 𝑓^′(𝜙^∗).

Based on the above analysis, we give the information of the singular points for system (2.10) and their relationship with 𝜙^∗₋, 𝜙^∗₊ and 2/3 when 𝑔>𝑔₂(𝑐), as an illustration, in the following lemma.

Lemma 2.1. For 𝑔>𝑔₂(𝑐), one has 𝐺₁<0<𝐺₂ and the singular points of system (2.10) can be described as follows.(a)If 𝐺<𝐺₂, then there is only one singular point denoted as 𝑆₁(𝜙₁,0)  (𝜙₁<𝜙^∗₋<2/3<𝜙^∗₊). 𝑆₁ is a saddle point.(b)If 𝐺=𝐺₂, then there are two singular points denoted as 𝑆₁(𝜙₁,0) and 𝑆₂(𝜙₂,0)  (𝜙₁<𝜙^∗₋<2/3<𝜙^∗₊=𝜙₂), respectively. 𝑆₁ is a saddle point and 𝑆₂ is a degenerate saddle point.(c)If 𝐺₂<𝐺<0, then there are three singular points denoted as 𝑆₁(𝜙₁,0), 𝑆₂(𝜙₂,0), and 𝑆₃(𝜙₃,0)  (𝜙₁<𝜙^∗₋<2/3<𝜙₂<𝜙^∗₊<𝜙₃), respectively. 𝑆₁ and 𝑆₃ are saddle points and 𝑆₂ is a center.(d)If 0<𝐺<𝐺₁, then there are three singular points denoted as 𝑆₁(𝜙₁,0), 𝑆₂(𝜙₂,0), and 𝑆₃(𝜙₃,0)  (𝜙₁<𝜙^∗₋<𝜙₂<2/3<𝜙^∗₊<𝜙₃), respectively. 𝑆₁ and 𝑆₃ are saddle points and 𝑆₂ is a center.(e)If 𝐺=𝐺₁, then there are two singular points denoted as 𝑆₁(𝜙₁,0) and 𝑆₂(𝜙₂,0)  (𝜙₁=𝜙^∗₋<2/3<𝜙^∗₊<𝜙₂), respectively. 𝑆₁ is a degenerate saddle point and 𝑆₂ is a saddle point.(f)If 𝐺>𝐺₁, then there is only one singular point denoted as 𝑆₁(𝜙₁,0)  (𝜙^∗₋<2/3<𝜙^∗₊<𝜙₁). 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 (note that 𝑓(2/3)=𝐺).

Figure 1 

The graphics of 𝑓(𝜙) when 𝑔>𝑔2(𝑐).

Remark 2.2. The case when 𝐺=0 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 𝑔₀(𝑐)<𝑔<𝑔₂(𝑐) and 𝐺₂<𝐺<min(0,𝐺₁), there exist two saddle points and one center lying on the same side of the singular line 𝑙∶𝜙=2/3. 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 𝐺₁<𝐺<𝐺₂, we set the three solutions of 𝑓(𝜙)=0 to 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−(4/3)3𝑐²−6𝑐+4+6𝑔−3𝜙²_𝑚2,𝜙_𝑏=−𝜙_𝑚+(4/3)3𝑐²−6𝑐+4+6𝑔−3𝜙²_𝑚2. (2.18)

It follows from 𝜙_𝑠<𝜙_𝑚<𝜙_𝑏 that 𝜙_𝑚 must satisfy condition 𝜙²_𝑚<3𝑐²−6𝑐+4+6𝑔9. (2.19)

From 𝐻(𝜙_𝑠,0)=𝐻(𝜙_𝑏,0), we obtain the expression of 𝐺 as the function of 𝜙_𝑚, 1𝐺=8𝜙³_𝑚+1279𝑐²−18𝑐+8+18𝑔. (2.20)

Substituting (2.20) into 𝑓(𝜙_𝑚)=0, we obtain the expression of 𝜙_𝑚 from 𝑓(𝜙_𝑚)=0 as follows: 𝜙_𝑚1𝜙=0,_𝑚22=−33𝑐²𝜙−6𝑐+4+6𝑔,_𝑚3=233𝑐²−6𝑐+4+6𝑔. (2.21)(2.22)

We can easily know that (2.22) does not satisfy (2.19), while (2.21) satisfies (2.19), if 𝑔>𝑔₀(𝑐).

By substituting (2.21) into (2.20), we obtain the bifurcation curve (denoted by 𝐺^∗) for 𝐺 as follows: 𝐺^∗=1279𝑐²−18𝑐+8+18𝑔. (2.23)

Note that if 𝑔>𝑔₀(𝑐), then obviously we have 𝐺₂<𝐺^∗<𝐺₁. Additionally, we have indicated that when 𝑔₀(𝑐)<𝑔<𝑔₂(𝑐) and 𝐺₂<𝐺<min(0,𝐺₁), there exist two saddle points and one center lying on the same side of the singular line 𝜙=2/3. Therefore, we obtain the fourth critical curve for 𝑔 from 𝐺^∗=0, 𝑔₃(𝑐)1=−2𝑐²8−2𝑐+9. (2.24)

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

Lemma 2.3. (1) For 𝑔∈(𝑔₀(𝑐),𝑔₃(𝑐)) and 𝐺=𝐺^∗, there exist heteroclinic orbits for system (2.6). (2) For 𝑔∉(𝑔₀(𝑐),𝑔₃(𝑐)) or 𝐺≠𝐺^∗, there exists no heteroclinic orbit for system (2.6).

Proof. Note that when 𝑔₃(𝑐)<𝑔<𝑔₂(𝑐) (or 𝑔=𝑔₃(𝑐)), one saddle point and one center point lie on the left side of the singular line 𝑙∶𝜙=2/3 and the other saddle point on the right side of (or on) the singular line 𝑙∶𝜙=2/3. 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.

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 𝑔=𝑔3(𝑐).

Figure 6 

The phase portraits of system (2.6) when 𝑔1(𝑐)<𝑔<𝑔3(𝑐).

Figure 7 

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

Figure 8 

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 (1.1). To relate conveniently, we omit 𝜑=𝜙+𝑐−2/3 and the expression of the second component of system (1.1) 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, and 𝐺 satisfy one of the following conditions: (i)𝑔≥𝑔₂(𝑐) and 0<𝐺<𝐺₁; (ii)𝑔₃(𝑐)≤𝑔<𝑔₂(𝑐) and 𝐺^∗<𝐺<𝐺₁; (iii)𝑔₁(𝑐)<𝑔<𝑔₃(𝑐) and 0≤𝐺<𝐺₁; (iv)𝑔₀(𝑐)<𝑔<𝑔₃(𝑐) and 𝐺^∗<𝐺<0, then there exist soliton solutions for (1.1), which can be implicitly expressed as 2𝜙₁₂−𝜙₁𝜙₁₃−𝜙₁𝜙₁₂−𝜙𝜙₁₃−𝜙+4𝜙₁𝜙+2𝜙₁₂𝜙₁₃+2𝜙²₁^𝛼1×2𝜙₁₂−𝜙𝜙₁₃−𝜙+2𝜙+2𝜙₁×(𝜙−𝜙₁)^𝛼1𝜙₁₃−𝜙₁₂^𝛼1+1−1=𝑒^|𝜉|/2, (3.1) where 𝜙₁₂=−𝜙₁−233𝑐²−6𝑐+4+6𝑔−2𝜙²₁,𝜙₁₃=−𝜙₁+233𝑐²−6𝑐+4+6𝑔−2𝜙²₁,𝛼₁=2/3−𝜙₁𝜙₁₂−𝜙₁𝜙₁₃−𝜙₁. (3.2)(2) If 𝑐, 𝑔, and 𝐺 satisfy condition: (v)𝑔>𝑔₂(𝑐) and 𝐺₂<𝐺<0, then there exist soliton solutions for (1.1), which can be implicitly expressed as 2𝜙₃−𝜙₃₁𝜙₃−𝜙₃₂𝜙−𝜙₃₁𝜙−𝜙₃₂+4𝜙₃𝜙+2𝜙₃₁𝜙₃₂+2𝜙²₃^𝛼3×2𝜙−𝜙₃₁𝜙−𝜙₃₂+2𝜙+2𝜙₃×𝜙₃−𝜙^𝛼3𝜙₃₁−𝜙₃₂^𝛼3+1−1=𝑒^−|𝜉|/2, (3.3) where 𝜙₃₁=−𝜙₃−233𝑐²−6𝑐+4+6𝑔−2𝜙²₃,𝜙₃₂=−𝜙₃+233𝑐²−6𝑐+4+6𝑔−2𝜙²₃,𝛼₃=(2/3)−𝜙₃𝜙₃−𝜙₃₁𝜙₃−𝜙₃₂. (3.4)(3.5)(3.6) If 𝑐, 𝑔, and 𝐺 satisfy one of the following conditions: (vi)𝑔₃(𝑐)≤𝑔<𝑔₂(𝑐) and 𝐺₂<𝐺<0; (vii)𝑔₀(𝑐)<𝑔<𝑔₃(𝑐) and 𝐺₂<𝐺<𝐺^∗, then there exist solitons solution for (1.1), which can be implicitly expressed as 2𝜙₃−𝜙₃₁𝜙₃−𝜙₃₂𝜙−𝜙₃₁𝜙−𝜙₃₂+4𝜙₃𝜙+2𝜙₃₁𝜙₃₂+2𝜙²₃^𝛼3×2𝜙−𝜙₃₁𝜙−𝜙₃₂+2𝜙+2𝜙₃×𝜙₃−𝜙^𝛼3𝜙₃₁−𝜙₃₂^𝛼3+1−1=𝑒^|𝜉|/2, (3.7) where 𝜙₃₁, 𝜙₃₂, and 𝛼₃ 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].

(a)

(b)

(c)

(d)

(e)

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 𝑐, 𝑔, and 𝐺 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: 𝑦=±𝜙−𝜙₁𝜙₁₂−𝜙𝜙₁₃−𝜙22/3−𝜙,𝜙₁≤𝜙≤𝜙₁₂<23<𝜙₁₃. (3.8)
Substituting (3.8) into the first equation of system (2.6) and integrating along the homoclinic orbits, it follows that ^𝜙12_𝜙(2/3−𝑠)𝑑𝑠𝑠−𝜙₁𝜙₁₂−𝑠𝜙₁₃=1−𝑠2𝜉. (3.9)
From (3.9), we obtain the soliton solutions (3.1).
(2) When 𝑐, 𝑔, and 𝐺 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: 𝜙𝑦=±³−𝜙𝜙−𝜙₃₁𝜙−𝜙₃₂2𝜙−2/3,𝜙₃₂<23<𝜙₃₁≤𝜙≤𝜙₃, (3.10) or 𝜙𝑦=±³−𝜙𝜙−𝜙₃₁𝜙−𝜙₃₂22/3−𝜙,𝜙₃₂<𝜙₃₁≤𝜙≤𝜙₃<23. (3.11)
Substituting (3.10) and (3.11) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that ^𝜙_𝜙31(𝑠−2/3)𝑑𝑠𝜙₃−𝑠𝑠−𝜙₃₁𝑠−𝜙₃₂=12𝜉,_𝜙𝜙31(2/3−𝑠)𝑑𝑠𝜙₃−𝑠𝑠−𝜙₃₁𝑠−𝜙₃₂=12𝜉. (3.12)
From (3.12), we obtain the soliton solutions (3.3) and (3.7).

Theorem 3.4. If integral constants 𝑔 and 𝐺 satisfy 𝑔₀(𝑐)<𝑔<𝑔₃(𝑐) and 𝐺=𝐺^∗, then there exist kink and antikink solutions.

Proof. We have showed that when 𝑔₀(𝑐)<𝑔<𝑔₃(𝑐) and 𝐺=𝐺^∗, there exist heteroclinic orbits for system (2.6). The heteroclinic can be expressed as 𝑦=±𝜙−𝜙_𝑠𝜙_𝑏−𝜙2/3−𝜙,𝜙_𝑠≤𝜙≤𝜙_𝑏, (3.13) where 𝜙_𝑠=−133𝑐²,𝜙−6𝑐+4+6𝑔_𝑏=133𝑐²−6𝑐+4+6𝑔, (3.14) 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 ^𝜙_𝜙0(2/3−𝑡)𝑑𝑡𝑡−𝜙_𝑠𝜙_𝑏1−𝑡=±2𝜉, (3.15) where 𝜙₀∈(𝜙_𝑠,𝜙_𝑏) is the initial value.
From (3.15), we have 𝜙−𝜙_𝑠^{(2/3−𝜙𝑠)/(𝜙𝑏−𝜙𝑠)}⋅𝜙_𝑏−𝜙^{(2/3−𝜙𝑏)/(𝜙𝑏−𝜙𝑠)}=𝜙₀−𝜙_𝑠^{(2/3−𝜙𝑠)/(𝜙𝑏−𝜙𝑠)}⋅𝜙_𝑏−𝜙₀^{(2/3−𝜙𝑏)/(𝜙𝑏−𝜙𝑠)}𝑒^±𝜉/2. (3.16)
If we take 𝜙₀=(𝜙_𝑠+𝜙_𝑏)/2, (3.16) becomes 𝜙−𝜙_𝑠^{(2/3−𝜙𝑠)/(𝜙𝑏−𝜙𝑠)}⋅𝜙_𝑏−𝜙^{(2/3−𝜙𝑏)/(𝜙𝑏−𝜙𝑠)}=𝜙_𝑏−𝜙_𝑠2^{4/3/(𝜙𝑏−𝜙𝑠)}𝑒^±𝜉/2. (3.17) (3.16) or (3.17) are kink (antikink) solutions.

Theorem 3.5. (1) If 𝑔=𝑔₃(𝑐) and 𝐺=0, then there exist peakons for (1.1), which can be explicitly expressed as 2𝜙=32𝑒^{−(1/2)|𝑥−𝑐𝑡|}−1. (3.18)
(2) If 𝑔₃(𝑐)<𝑔<𝑔₂(𝑐) and 𝐺=0, then (1.1) has periodic cusp waves 𝑢()()2𝑥,𝑡=𝜙𝜉−2𝑖𝑇+𝑐−3, (3.19) where 𝑖=0,±1,±2,…,  𝜉=𝑥−𝑐𝑡∈[(2𝑖−1)𝑇,(2𝑖+1)𝑇], and 𝜙(𝜉)=1243−1366𝑐−3𝑐²−6𝑔𝑒^{(1/2)|𝑥−𝑐𝑡|}+1243+1366𝑐−3𝑐²−6𝑔𝑒^{−(1/2)|𝑥−𝑐𝑡|}−23, (3.20) with 𝑇=2ln4+66𝑐−3𝑐²−6𝑔29𝑐²−18𝑐+8+18𝑔. (3.21)

Remark 3.6. When 𝐺=0, 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 𝑔=𝑔₃(𝑐) and 𝐺=0, from Figure 5, we see that there is a triangle orbit, which can be expressed as 1𝑦=±22𝜙+32,for−32≤𝜙≤3,2𝜙=32,for−32≤𝑦≤3. (3.22)(3.23)
Substituting (3.22) into the first equation of system (2.6) and integrating along the triangle orbits, it follows that _𝜙^2/3𝑑𝑡=1𝑠+2/32𝜉. (3.24)
From (3.24), we obtain peakons (3.18).
(2) When 𝑔₃(𝑐)<𝑔<𝑔₂(𝑐) and 𝐺=0, from Figure 4, we see that there is an semiellipse orbit, which can be expressed as 1𝑦=±2𝜙²+434𝜙+3−233𝑐²1−6𝑐+4+6𝑔,for329𝑐²2−18𝑐+8+18𝑔−2≤𝜙≤3,2𝜙=31,for−666𝑐−3𝑐²1−6𝑔≤𝑦≤666𝑐−3𝑐²−6𝑔. (3.25)(3.26)
Substituting (3.25) into the first equation of system (2.6) and integrating along the semiellipse orbit, it follows that _𝜙^2/3𝑑𝑡𝜙²+(4/3)𝜙+(4/3)−(2/3)3𝑐²=1−6𝑐+4+6𝑔2𝜉. (3.27)
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 (ℎ=(3𝑐/32)(𝑐²−8𝑔) 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).