Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

Abstract

By using the bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation: 𝑚_𝑡=−3𝑚𝑢_𝑥−𝑚_𝑥𝑢+𝑘𝜌𝜌_𝑥, 𝜌_𝑡=−𝜌_𝑥𝑢+2𝜌𝑢_𝑥, the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.

1. Introduction

Based on the deformation of bi-Hamiltonian structure of the hydrodynamic type, Chen et al. [1] obtained a two-component Camassa-Holm equation: 𝑚_𝑡=−2𝑚𝑢_𝑥−𝑚_𝑥𝑢+𝑒𝜌𝜌_𝑥,𝜌_𝑡=−𝜌_𝑥𝑢−𝜌𝑢_𝑥, (1) where 𝑚=𝑢−𝑢_𝑥𝑥 and 𝑒=±1. This system is integrable in the sense that it has Lax pair and it is a generalization form of the Camassa-Holm equation [2]. A good fact is that it has the peakon and multikink solutions [3].

A two-component generalization of the Degasperis-Procesi equation [4, 5] is read as follows: 𝑚_𝑡=−3𝑚𝑢_𝑥−𝑚_𝑥𝑢+𝑘𝜌𝜌_𝑥,𝜌_𝑡=−𝜌_𝑥𝑢+2𝜌𝑢_𝑥, (2) where 𝑚=𝑢−𝑢_𝑥𝑥, 𝑘 is a real parameter. When 𝜌=0, (2) is reduced to the Degasperis-Procesi equation [6]. Jin and Guo [4] analyze some aspects of blowup mechanism, traveling wave solutions and the persistence properties for (2). The self-similar solutions of (2) have been obtained by yuen [5].

In the present paper, we will investigate the bifurcation set of (2) using the bifurcation theory and the method of phase portraits analysis [7–9] and obtain some exact peakon, solitary wave and smooth periodic wave solutions.

Let 𝜉=𝑥−𝑐𝑡, where 𝑐≠0 is the wave speed. By using the travelling wave transformation 𝑢()(𝜉)+4𝑥,𝑡=𝜙3𝜌()(𝜉),𝑐,𝑥,𝑡=𝜌 (3) we reduce (2) to the following ordinary differential equations: 1𝜙+3𝑐𝜙^′′′+3𝜙^′𝜙^′′−4𝜙𝜙^′−133𝑐𝜙^′+𝑘𝜌𝜌^′=0,2𝜌𝜙^′−1𝜙+3𝑐𝜌^′=0, (4) where "′" is the derivative with respect to 𝜉.

Suppose that 4𝜌=𝜙+3𝑐²+𝑎₁4𝜙+3𝑐+𝑎₀, (5) where 𝑎₁, 𝑎₀ are constants to be determined later.

Substituting (5) into the second equation of (4), we have 2𝑐+𝑎₁𝜙𝜙^′+83𝑐²+73𝑎₁𝑐+2𝑎₀𝜙^′=0. (6)

Integrating (6) once with respect to 𝜉 and setting the integral constant to zero yield 122𝑐+𝑎₁𝜙²+83𝑐²+73𝑎₁𝑐+2𝑎₀𝜙=0. (7)

From (7), we obtain 2𝑐+𝑎₁8=0,3𝑐²+73𝑎₁𝑐+2𝑎₀=0. (8)

Solving (8), we find the following set of solutions: 𝑎₁=−2𝑐,𝑎₀=𝑐². (9)

Substituting (9) into (5), we have 1𝜌=𝜙+3𝑐². (10)

Substituting (10) into the first equation of (4), integrating once with respect to 𝜉, and setting the integral constant to (1/162)𝑘𝑐⁴−(32/9)𝑐² yield 1𝜙+3𝑐𝜙^′′−4𝜙+3𝑐×−12𝑘𝜙³+12−3𝑘𝑐²𝜙+1027𝑘𝑐³+53𝑐+𝜙^′2=0. (11)

Letting 𝑦=𝑑𝜙/𝑑𝜉, we get the following planar system: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=4𝑑𝜉𝜙+3𝑐×−12𝑘𝜙³+12−3𝑘𝑐²𝜙+1027𝑘𝑐³+53𝑐−𝑦²×1𝜙+3𝑐⁻¹. (12)

System (12) is a two-parameter planar dynamical system depending on the parameter set (𝑐,𝑘). Since the phase orbits defined by the vector field of system (12) determine all travelling wave solutions of (11), we should investigate the bifurcations of phase portraits of system (12) in (𝜙,𝑦)-phase plane as the parameters 𝑐,𝑘 are changed.

Clearly, on such straight line 𝜙=−(1/3)𝑐 in the phase plane (𝜙,𝑦), system (12) is discontinuous. Such system is called a singular travelling wave system by one of authors in [10].

Definition 1 (see [7, 10]). Suppose that 𝜙(𝜉) is a solution of system (12) for 𝜉∈(−∞,∞). (i) 𝜙(𝜉) is called peakon solution if 𝜙(𝜉) is smooth locally on either side of 𝜉₀ and lim_{𝜉→𝜉0−0}𝜙(𝜉)=lim_{𝜉→𝜉0+0}𝜙(𝜉)=𝑎 and 𝑎≠0, 𝑎≠±∞. (ii) 𝜙(𝜉) is called a solitary wave solution if lim_𝜉→+∞𝜙(𝜉)=lim_{𝜉→−∞}𝜙(𝜉). Usually, a peakon solution of (2) corresponds to two heteroclinic orbits of system (12) and a solitary wave solution of (2) corresponds to a homoclinic orbit of system (12). Similarly, a periodic orbit of system (12) corresponds to a smooth periodic wave solution of (2).

Thus, to investigate peakons, solitary waves and smooth periodic waves of (2), we should find all heteroclinic, homoclinic, and periodic orbits of system (12) depending on the parameter space of this system.

The rest of this paper is organized as follows. In Section 2, we discuss the bifurcations of phase portraits of system (12), where explicit parametric conditions will be derived. In Section 3, we give some exact parametric representations of peakon, solitary wave and smooth periodic wave solutions of (2). A short conclusion will be given in Section 4.

2. Bifurcation Sets and Phase Portraits of System (12)

Using the transformation 𝑑𝜉=(𝜙+(1/3)𝑐)𝑑𝜏, it carries (12) into the Hamiltonian system 𝑑𝜙=1𝑑𝜏𝜙+3𝑐𝑦,𝑑𝑦=4𝑑𝜏𝜙+3𝑐×−12𝑘𝜙³+12−3𝑘𝑐²𝜙+1027𝑘𝑐³+53𝑐−𝑦². (13)

Since both systems (12) and (13) have the same first integral 3𝜙+𝑐²𝑦²+1𝜙5481𝑘𝜙⁵+162𝑘𝑐𝜙⁴+915𝑘𝑐²−52𝜙³+60𝑐𝑘𝑐²−27𝜙²−2𝑐²114𝑘𝑐²+891𝜙−80𝑐³2𝑘𝑐²+9=ℎ, (14) then two systems above have the same topological phase portraits except for the line 𝜙=−(1/3)𝑐. Therefore, we can obtain the bifurcation phase portraits of system (12) from that of system (13). We consider the equilibrium points and their properties for system (13) as follows.

When 𝑘=0, it is easy to see that system (13) has two equilibrium points (−(4/3)𝑐,0),(−(5/6)𝑐,0) on 𝜙-axis and two equilibrium points (−(1/3)𝑐,−𝑐),(−(1/3)𝑐,𝑐) on the line 𝜙=−(1/3)𝑐.

When 𝑘≠0, let 𝑓𝜙=𝜙³−2𝑘12−3𝑘𝑐²𝜙−1023𝑘9𝑘𝑐²+1. (15) Obviously, (−(4/3)𝑐,0) is an equilibrium point of system (13) and (𝜙,0) is a equilibrium point of system (13) if and only if 𝑓(𝜙)=0 when 𝜙≠−(4/3)𝑐. Denoting that Ω=(5725+1575√14)^1/3, 𝑘_∗=(Ω²−14Ω−125)/6Ω𝑐², we draw the graph of 𝑓=𝑓(𝜙) for given 𝑐≠0 as Figure 1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1 

The graphics of the function 𝑓(𝜙) when 𝑐≠0. (a) 𝑐<0,  𝑘<0. (b) 𝑐<0, 0<𝑘<𝑘∗. (c) 𝑐<0, 𝑘=𝑘∗. (d): 𝑐<0, 𝑘>𝑘∗. (e) 𝑐>0, 𝑘<0. (f) 𝑐>0, 0<𝑘<𝑘∗. (g) 𝑐>0, 𝑘=𝑘∗. (h) 𝑐>0, 𝑘>𝑘∗.

From Figure 1 and paying attention to that (−(4/3)𝑐,0) is always an equilibrium point of system (13), we have the following.(i)If 𝑘<0, then system (13) has two equilibrium points on 𝜙-axis.(ii)If 0<𝑘<𝑘_∗, then system (13) has four equilibrium points on 𝜙-axis.(iii)If 𝑘=𝑘_∗, then system (13) has three equilibrium points on 𝜙-axis.(iv)If 𝑘>𝑘_∗, then system (13) has two equilibrium points on 𝜙-axis.

On the other hand, it is also easy to see that if 𝑘>−2/𝑐², then system (13) has two equilibrium points (−(1/3)𝑐,±(√2𝑐/2)√𝑘𝑐²+2) on the line 𝜙=−(1/3)𝑐 and if   𝑘=−2/𝑐², then system (13) has one equilibrium point (−(1/3)𝑐,0) on the line 𝜙=−(1/3)𝑐.

Let 𝑀(𝜙_𝑒,𝑦_𝑒) be the coefficient matrix of the linearized system of the system (13) at an equilibrium point (𝜙_𝑒,𝑦_𝑒). Then, we have 𝑀𝜙_𝑒,𝑦_𝑒=𝑦_𝑒𝜙_𝑒+13𝑐−2𝑘𝜙³_𝑒−2𝑘𝑐𝜙²_𝑒1+22−3𝑘𝑐²𝜙_𝑒+13𝑐213−9𝑘𝑐²−2𝑦_𝑒, (16) and at this equilibrium point, we have Trace𝑀𝜙_𝑒,𝑦_𝑒=−𝑦_𝑒,𝐽𝜙_𝑒,𝑦_𝑒=det𝑀𝜙_𝑒,𝑦_𝑒=−2𝑦²_𝑒+2𝑘𝜙⁴_𝑒+𝑘𝑐𝜙³_𝑒1+43𝑘𝑐²−1𝜙²_𝑒+13𝑐89𝑘𝑐²−17𝜙_𝑒+19𝑐²29𝑘𝑐²−13. (17)

By the theory of planar dynamical systems, we know that for an equilibrium point (𝜙_𝑒,𝑦_𝑒) of a planar integrable system if 𝐽(𝜙_𝑒,𝑦_𝑒)<0, then the equilibrium point is a saddle point. If 𝐽(𝜙_𝑒,𝑦_𝑒)>0 and Trace(𝑀(𝜙_𝑒,𝑦_𝑒))=0, then it is a center point. If 𝐽(𝜙_𝑒,𝑦_𝑒)=0 and the Poincaré index of the equilibrium point is zero, then it is a cusp.

For a fixed ℎ, the level curve 𝐻(𝜙,𝑦)=ℎ defined by (14) determines a set of invariant curves of system (13) which contains different branches of curves. As ℎ is varied, it defines different families of orbits of (13) with different dynamical behaviors.

Using the property of the equilibrium points and bifurcation theory, we obtain the following results.(i)When 𝑘=0, there is one bifurcation line 𝑐=0 which divides the region (−∞,∞) into 2 subregions: 𝐴₁={𝑐∣𝑐<0}, 𝐴₂={𝑐∣𝑐>0}.(ii)When 𝑘≠0, there are five bifurcation curves as follows: 𝐿₁∶𝑐=0,𝐿₂∶𝑘=0,𝐿₃𝜇∶𝑘=𝑐²,𝐿₄1∶𝑘=2𝑐²,𝐿₅2∶𝑘=−𝑐², (18)which divide the (𝑐,𝑘)-parameter plane into 16 subregions as follows: 𝐵₁=(𝑐,𝑘)∣𝑐<0,𝑘<−2/𝑐²,𝐵₂=(𝑐,𝑘)∣𝑐<0,𝑘=−2/𝑐²,𝐵₃=(𝑐,𝑘)∣𝑐<0,−2/𝑐²,𝐵<𝑘<0₄=(𝑐,𝑘)∣𝑐<0,0<𝑘<1/2𝑐²,𝐵₅=(𝑐,𝑘)∣𝑐<0,𝑘=1/2𝑐²,𝐵₆=(𝑐,𝑘)∣𝑐<0,1/2𝑐²<𝑘<𝜇/𝑐²,𝐵₇=(𝑐,𝑘)∣𝑐<0,𝑘=𝜇/𝑐²,𝐵₈=(𝑐,𝑘)∣𝑐<0,𝑘>𝜇/𝑐²,𝐵₉=(𝑐,𝑘)∣𝑐>0,𝑘<−2/𝑐²,𝐵₁₀=(𝑐,𝑘)∣𝑐>0,𝑘=−2/𝑐²,𝐵₁₁=(𝑐,𝑘)∣𝑐>0,−2/𝑐²,𝐵<𝑘<0₁₂=(𝑐,𝑘)∣𝑐>0,0<𝑘<1/2𝑐²,𝐵₁₃=(𝑐,𝑘)∣𝑐>0,𝑘=1/2𝑐²,𝐵₁₄=(𝑐,𝑘)∣𝑐>0,1/2𝑐²<𝑘<𝜇/𝑐²,𝐵₁₅=(𝑐,𝑘)∣𝑐>0,𝑘=𝜇/𝑐²,𝐵₁₆=(𝑐,𝑘)∣𝑐>0,𝑘>𝜇/𝑐², (19) where 𝜇=(Ω²−14Ω−125)/6Ω, Ω=(5725+1575√14)^1/3.

The bifurcation sets and phase portraits of system (13) are shown in Figures 2 and 3.

(a)

(b)

Figure 2 

The bifurcation sets and phase portraits of system (13) when 𝑘=0. (a) 𝑐∈𝐴1. (b) 𝑐∈𝐴2.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Figure 3 

The bifurcation sets and phase portraits of system (13) when 𝑘≠0. (a) (𝑐,𝑘)∈𝐵1. (b) (𝑐,𝑘)∈𝐵2. (c) (𝑐,𝑘)∈𝐵3. (d) (𝑐,𝑘)∈𝐵4. (e) (𝑐,𝑘)∈𝐵5 (f) (𝑐,𝑘)∈𝐵6. (g) (𝑐,𝑘)∈𝐵7. (h) (𝑐,𝑘)∈𝐵8. (i) (𝑐,𝑘)∈𝐵9. (j) (𝑐,𝑘)∈𝐵10. (k) (𝑐,𝑘)∈𝐵11. (l) (𝑐,𝑘)∈𝐵12. (m) (𝑐,𝑘)∈𝐵13. (n) (𝑐,𝑘)∈𝐵14. (o) (𝑐,𝑘)∈𝐵15. (p) (𝑐,𝑘)∈𝐵16.

3. Exact Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions of (2)

In this section, we present all possible exact parametric representations of peakon, solitary wave and smooth periodic wave solutions through some special phase orbits. Next, we always suppose that ℎ₁=((16/9)−(608/243)𝑘𝑐²)𝑐⁴, ℎ₂=(𝑐⁴/23328000)((895537−238014√14)Ω² + (970075+382725√14)Ω−45865625), 𝜙_𝑠=−(1/3)𝑐, 𝜙₁=−(4/3)𝑐, 𝜙₂=((1/1500)𝑐)((6√14−23)Ω²−5(3√14+1)Ω−625), and Ω=(5725+1575√14)^1/3.

3.1. Exact Parametric Representations of Peakon Solutions

(i)   For the given ℎ=(16/9)𝑐⁴ in Figure 2(a), the level curve is shown in Figure 4(a). From Figure 4(a), we see that there are two heteroclinic orbits connecting with saddle points (𝜙_𝑠,±𝑐) and (𝜙₁,0) of system (13) when (𝑐,𝑘)∈𝐴₁. Their expressions are 𝑦=±𝜙−𝜙₁,𝜙_𝑠≤𝜙<𝜙₁. (20)

(a) 𝑐<0, ℎ∗=(16/9)𝑐4

(b) 𝑐>0, ℎ∗=(16/9)𝑐4

(c) 𝑐<0, ℎ∗=ℎ1

(d) 𝑐<0, ℎ∗=(128/243)𝑐4

(e) 𝑐<0, ℎ∗=ℎ2

(f) 𝑐>0, ℎ∗=ℎ1

(g) 𝑐>0, ℎ∗=(128/243)𝑐4

(h) 𝑐>0, ℎ∗=ℎ2

Figure 4 

The level curves of system (13) defined by ℎ=ℎ∗.

Substituting (20) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the heteroclinic orbits yield ^𝜙_𝜙𝑠𝑑𝑠𝑠−𝜙₁=−𝜉. (21)

Completing the above integral and solving the equation for 𝜙, it follows that 𝜙(𝜉)=𝑐e^−|𝜉|−43𝑐. (22)

Noting that 𝑢(𝑥,𝑡)=𝜙(𝑥−𝑐𝑡)+(4/3)𝑐,𝜌(𝑥,𝑡)=(𝜙(𝜉)+(1/3)𝑐)²,and𝜉=𝑥−𝑐𝑡, we get the parametric representation of peakon solution as follows: 𝑢(𝑥,𝑡)=𝑐e^{−|𝑥−𝑐𝑡|},𝜌(𝑥,𝑡)=𝑐²e^{−|𝑥−𝑐𝑡|}−1². (23)

(ii)   For the given ℎ=(16/9)𝑐⁴ in Figure 2(b), the level curve is shown in Figure 4(b). From Figure 4(b), we see that there are two heteroclinic orbits connecting with saddle points (𝜙_𝑠,±𝑐) and (𝜙₁,0) of system (13) when (𝑐,𝑘)∈𝐴₂. Their expressions are 𝑦=±𝜙−𝜙₁,𝜙₁<𝜙≤𝜙_𝑠. (24)

Substituting (24) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the heteroclinic orbits yield ^𝜙𝑠_𝜙𝑑𝑠𝑠−𝜙₁=𝜉. (25)

Completing the above integral, we can get the parametric representation of peakon solution which is the same as (23).

3.2. Exact Parametric Representations of Solitary Wave Solutions

(i) For the given ℎ=ℎ₁ in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is a homoclinic orbit connecting with a saddle point (𝜙₁,0) of system (13) and passing point (𝛽₂,0), and at the same time, there is a homoclinic orbit connecting with the saddle point (𝜙₁,0) and passing point (𝛽₁,0) when (𝑐,𝑘)∈𝐵₄, and their expressions are 𝑦=±𝑘6𝜙−𝜙₁𝛽₁−𝜙𝜙−𝛽₂𝜙−𝛽₃𝜙−𝛽₄𝜙−𝜙_𝑠,𝛽₂≤𝜙<𝜙₁,𝑦=±𝑘6𝜙−𝜙₁𝛽₁−𝜙𝜙−𝛽₂𝜙−𝛽₃𝜙−𝛽₄𝜙−𝜙_𝑠,𝜙₁<𝜙≤𝛽₁, (26)(27) respectively, where 𝛽₁, 𝛽₂, 𝛽₃, and 𝛽₄(𝛽₄<𝛽₃<𝛽₂<𝛽₁) are four real roots of 𝜓⁴−23𝑐𝜓³+13𝑘5𝑘𝑐²−18𝜓²−4𝑐27𝑘17𝑘𝑐²+27𝜓+2𝑐²81𝑘38𝑘𝑐²−27=0, (28) which are obtained by the Cardan formula. For example, 𝛽₁≈3.392753049, 𝛽₂≈0.8389789971, 𝛽₃≈0.09041647394, 𝛽₄≈−4.988815187 when 𝑐=−1, 𝑘=0.3; 𝛽₁≈1.942359082, 𝛽₂≈0.3828621123, 𝛽₃≈0.05356030588, 𝛽₄≈−2.712114834 when 𝑐=−0.5, 𝑘=1; and 𝛽₁≈2.712114834, 𝛽₂≈−0.05356030588, 𝛽₃≈−0.3828621123, 𝛽₄≈−1.942359082 when 𝑐=0.5, 𝑘=1.

Substituting (26) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝜙_𝛽2𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₁𝛽₁−𝑠𝑠−𝛽₂𝑠−𝛽₃𝑠−𝛽₄=−𝑘6𝜉. (29)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (30)𝜙 is confirmed by 𝜙𝜒=𝛽₂𝛽₃−𝛽₁+𝛽₃𝛽₁−𝛽₂sn²𝜔𝜒,𝑚𝛽₃−𝛽₁+𝛽₁−𝛽₂sn²,𝜉𝜒=Ω𝛼𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝛼²𝛼,𝑚+𝜔²₁−𝛼²𝛽₂−𝜙₁𝜙_𝑠−𝜙₁𝜒, (31) where Ω=((𝜙_𝑠−𝜙₁)/𝜔𝛼²(𝛽₂−𝜙₁))(6/𝑘), 𝛼²=(𝜙₁−𝛽₃)(𝛽₁−𝛽₂)/(𝜙₁−𝛽₂)(𝛽₁−𝛽₃), 𝛼²₁=(𝛽₁−𝛽₂)/(𝛽₁−𝛽₃), 𝜔=(1/2)(𝛽₁−𝛽₃)(𝛽₂−𝛽₄), 𝑚=(𝛽₁−𝛽₂)(𝛽₃−𝛽₄)/(𝛽₁−𝛽₃)(𝛽₂−𝛽₄), Π(⋅,⋅,⋅) is Legendre's incomplete elliptic integral of the third kind, sn(⋅,⋅) is the Jacobian elliptic function (see [11]), and 𝜒 is a new variable.

Substituting (27) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝛽1_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₁𝛽₁−𝑠𝑠−𝛽₂𝑠−𝛽₃𝑠−𝛽₄=𝑘6𝜉. (32)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (33)𝜙 is confirmed by 𝜙𝜒=𝛽₁𝛽₂−𝛽₄+𝛽₄𝛽₁−𝛽₂sn²𝜔𝜒,𝑚𝛽₂−𝛽₄+𝛽₁−𝛽₂sn²,𝜉𝜒=Ω𝛼𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝛼²𝛼,𝑚+𝜔²₁−𝛼²𝛽₁−𝜙₁𝜙_𝑠−𝜙₁𝜒, (34) where Ω=((𝜙_𝑠−𝜙₁)/𝜔𝛼²(𝜙₁−𝛽₁))(6/𝑘), 𝛼²=(𝜙₁−𝛽₄)(𝛽₁−𝛽₂)/(𝛽₁−𝜙₁)(𝛽₂−𝛽₄), 𝛼²₁=(𝛽₂−𝛽₁)/(𝛽₂−𝛽₄), 𝜔=(1/2)(𝛽₁−𝛽₃)(𝛽₂−𝛽₄), and 𝑚=(𝛽₁−𝛽₂)(𝛽₃−𝛽₄)/(𝛽₁−𝛽₃)(𝛽₂−𝛽₄).

(ii) For the given ℎ=(128/243)𝑐⁴ in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is a homoclinic orbit connecting with a cusp (𝜙₁,0) of system (13) and passing point (𝛽₁,0) when (𝑐,𝑘)∈𝐵₅, and its expression is 1𝑦=±2√×3𝑐𝜙−𝜙₁𝛽₁−𝜙𝜙−𝜙₁𝜙−𝛽₂𝜙−𝛽₃𝜙−𝜙_𝑠,𝜙₁<𝜙≤𝛽₁, (35) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of 𝜓³−2𝑐𝜓²−233𝑐²8𝜓−𝑐27³=0, (36) which are obtained by the Cardan formula. For example, 𝛽₁≈1.917558777, 𝛽₂≈0.03905297305, 𝛽₃≈−3.956611750 when 𝑐=−1; 𝛽₁≈3.956611750, 𝛽₂−0.03905297305, 𝛽₃≈−1.917558777 when 𝑐=1; and 𝛽₁≈0.9587793884, 𝛽₂≈0.01952648653, 𝛽₃≈−1.978305875 when 𝑐=−0.5.

Substituting (35) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝛽1_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₁𝛽₁−𝑠𝑠−𝜙₁𝑠−𝛽₂𝑠−𝛽₃1=−2√𝜉.3𝑐 (37)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (38)𝜙 is confirmed by 𝜙𝜒=𝛽₁𝜙₁−𝛽₃+𝛽₃𝛽₁−𝜙₁sn²𝜔𝜒,𝑚𝜙₁−𝛽₃+𝛽₁−𝜙₁sn²,𝜉𝜒=Ω𝜔𝜒,𝑚1−𝛼²₁Π𝛼arcsinsn𝜔𝜒,𝑚,1,𝑚+𝜔²₁−𝛽₁−𝛽₃𝜙_𝑠−𝜙₁𝜒, (39) where Ω=2√3(𝜙_𝑠−𝜙₁)𝑐/𝜔(𝛽₁−𝛽₃), 𝛼²₁=(𝜙₁−𝛽₁)/(𝜙₁−𝛽₃), 𝜔=(1/2)(𝛽₁−𝛽₂)(𝜙₁−𝛽₃), and 𝑚=(𝛽₁−𝜙₁)(𝛽₂−𝛽₃)/(𝛽₁−𝛽₂)(𝜙₁−𝛽₃).

(iii) For the given ℎ=ℎ₂ in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is a homoclinic orbit connecting with a cusp (𝜙₂,0) of system (13) and passing point (𝛽₁,0) when (𝑐,𝑘)∈𝐵₇, and its expression is 𝑦=±675Ω²−9450Ω−8437524300𝑐²Ω×𝜙₂−𝜙𝜙₂−𝜙𝜙−𝛽₁𝜙−𝛽₂𝜙−𝛽₃𝜙−𝜙_𝑠,𝛽₁≤𝜙<𝜙₂, (40) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of 𝑏₀+𝑏₁𝜓+𝑏₂𝜓²−𝜓³=0, (41) which are obtained by the Cardan formula, and 𝑏₀=(𝑐³((486√14−1663)Ω²−22375Ω+60750√14+207875))/(675(Ω²−14Ω−125)), 𝑏₁=(𝑐²((1458√14−6489)Ω²−31950Ω+182250√14+811125))/(675(Ω²−14Ω−125)), and 𝑏₂=(𝑐((729√14−4482)Ω²+5400Ω+91125√14+560250))/(675(Ω²−14Ω−125)). For example, 𝛽₁≈1.264234987, 𝛽₂≈0.03471643922, 𝛽₃≈−3.882082755 when 𝑐=−1; 𝛽₁≈6.321174982, 𝛽₂≈0.1735822117, 𝛽₃≈−19.41041364 when 𝑐=−5; and 𝛽₁≈19.41041364, 𝛽₂≈−0.1735822117, 𝛽₃≈−6.321174982 when 𝑐=5.

Substituting (40) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝜙_𝛽1𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₂𝜙₂−𝑠𝑠−𝛽₁𝑠−𝛽₂𝑠−𝛽₃=−675Ω²−9450Ω−8437524300𝑐²Ω𝜉. (42)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (43)𝜙 is confirmed by 𝜙𝜒=𝛽₁𝜙₂−𝛽₂+𝛽₂𝛽₁−𝜙₂sn²𝜔𝜒,𝑚𝜙₂−𝛽₂+𝛽₁−𝜙₂sn²,𝜉𝜒=Ω𝜔𝜒,𝑚1−𝛼²₁Π𝛼arcsinsn𝜔𝜒,𝑚,1,𝑚+𝜔²₁−𝜙₂−𝛽₁𝜙₂−𝜙_𝑠𝜒, (44) where Ω=((𝜙₂−𝜙_𝑠)/𝜔(𝜙₂−𝛽₁))24300𝑐²Ω/(675Ω²−9450Ω−84375), 𝛼²₁=(𝜙₂−𝛽₁)/(𝜙₂−𝛽₂), 𝜔=(1/2)(𝜙₂−𝛽₂)(𝛽₁−𝛽₃), and 𝑚=(𝜙₂−𝛽₁)(𝛽₂−𝛽₃)/(𝜙₂−𝛽₂)(𝛽₁−𝛽₃).

(iv) For the given ℎ=ℎ₁ in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is a homoclinic orbit connecting with a saddle point (𝜙₁,0) of system (13) and passing point (𝛽₄,0), and at the same time, there is a homoclinic orbit connecting with the saddle point (𝜙₁,0) and passing point (𝛽₃,0) when (𝑐,𝑘)∈𝐵₁₂, and their expressions are 𝑦=±𝑘6×𝜙₁−𝜙𝛽₁−𝜙𝛽₂−𝜙𝛽₃−𝜙𝜙−𝛽₄𝜙_𝑠,𝛽−𝜙₄≤𝜙<𝜙₁,𝑦=±𝑘6×𝜙−𝜙₁𝛽₁−𝜙𝛽₂−𝜙𝛽₃−𝜙𝜙−𝛽₄𝜙−𝜙_𝑠,𝜙₁<𝜙≤𝛽₃, (45)(46) respectively, where 𝛽₁, 𝛽₂, 𝛽₃, and 𝛽₄(𝛽₄<𝛽₃<𝛽₂<𝛽₁) are four real roots of (28).

Substituting (45) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝜙_𝛽4𝜙_𝑠−𝑠𝑑𝑠𝜙₁−𝑠𝛽₁−𝑠𝛽₂−𝑠𝛽₃−𝑠𝑠−𝛽₄=𝑘6𝜉. (47)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (48)𝜙 is confirmed by 𝜙𝜒=𝛽₄𝛽₁−𝛽₃+𝛽₁𝛽₃−𝛽₄sn²𝜔𝜒,𝑚𝛽₁−𝛽₃+𝛽₃−𝛽₄sn²,𝜉𝜒=Ω𝛼𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝛼²𝛼,𝑚+𝜔²₁−𝛼²𝛽₄−𝜙₁𝜙_𝑠−𝜙₁𝜒, (49) where Ω=((𝜙_𝑠−𝜙₁)/𝜔𝛼²(𝜙₁−𝛽₄))6/𝑘, 𝛼²=(𝜙₁−𝛽₁)(𝛽₄−𝛽₃)/(𝜙₁−𝛽₄)(𝛽₁−𝛽₃), 𝛼²₁=(𝛽₄−𝛽₃)/(𝛽₁−𝛽₃), 𝜔=(1/2)(𝛽₁−𝛽₃)(𝛽₂−𝛽₄), and 𝑚=(𝛽₁−𝛽₂)(𝛽₃−𝛽₄)/(𝛽₁−𝛽₃)(𝛽₂−𝛽₄).

Substituting (46) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝛽3_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₁𝛽₁−𝑠𝛽₂−𝑠𝛽₃−𝑠𝑠−𝛽₄=−𝑘6𝜉. (50)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (51)𝜙 is confirmed by 𝜙𝜒=𝛽₃𝛽₄−𝛽₂+𝛽₂𝛽₃−𝛽₄sn²𝜔𝜒,𝑚𝛽₄−𝛽₂+𝛽₃−𝛽₄sn²,𝜉𝜒=Ω𝛼𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝛼²𝛼,𝑚+𝜔²₁−𝛼²𝛽₃−𝜙₁𝜙_𝑠−𝜙₁𝜒, (52) where Ω=((𝜙₁−𝜙_𝑠)/𝜔𝛼²(𝜙₁−𝛽₃))6/𝑘,𝛼²=(𝜙₁−𝛽₂)(𝛽₃−𝛽₄)/(𝜙₁−𝛽₃)(𝛽₂−𝛽₄), 𝛼²₁=(𝛽₃−𝛽₄)/(𝛽₂−𝛽₄), 𝜔=(1/2)(𝛽₁−𝛽₃)(𝛽₂−𝛽₄), and 𝑚=(𝛽₁−𝛽₂)(𝛽₃−𝛽₄)/(𝛽₁−𝛽₃)(𝛽₂−𝛽₄).

(v) For the given ℎ=(128/243)𝑐⁴ in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is a homoclinic orbit connecting with a cusp (𝜙₁,0) of system (13) and passing point (𝛽₃,0) when (𝑐,𝑘)∈𝐵₁₃, and its expression is 1𝑦=±2√×𝜙3𝑐₁−𝜙𝛽₁−𝜙𝛽₂−𝜙𝜙₁−𝜙𝜙−𝛽₃𝜙_𝑠,𝛽−𝜙₃≤𝜙<𝜙₁, (53) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of (36).

Substituting (53) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝜙_𝛽3𝜙_𝑠−𝑠𝑑𝑠𝜙₁−𝑠𝛽₁−𝑠𝛽₂−𝑠𝜙₁−𝑠𝑠−𝛽₃=12√𝜉.3𝑐 (54)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=1𝑐,𝑥,𝑡𝜙(𝑥−𝑐𝑡)+3𝑐². (55)𝜙 is confirmed by 𝜙𝜒=𝛽₃𝜙₁−𝛽₁+𝛽₁𝛽₃−𝜙₁sn²𝜔𝜒,𝑚𝜙₁−𝛽₁+𝛽₃−𝜙₁sn²,𝜉𝜒=Ω𝜔𝜒,𝑚1−𝛼²₁Π𝛼arcsinsn𝜔𝜒,𝑚,1,𝑚+𝜔²₁−𝛽₃−𝜙₁𝜙_𝑠−𝜙₁𝜒, (56) where Ω=2√3(𝜙_𝑠−𝜙₁)𝑐/𝜔(𝜙₁−𝛽₃), 𝛼²₁=(𝜙₁−𝛽₃)/(𝜙₁−𝛽₁), 𝜔=(1/2)(𝛽₂−𝛽₃)(𝛽₁−𝜙₁), and 𝑚=(𝛽₃−𝜙₁)(𝛽₂−𝛽₁)/(𝛽₃−𝛽₂)(𝜙₁−𝛽₁).

(vi) For the given ℎ=ℎ₂ in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is a homoclinic orbit connecting with a cusp (𝜙₂,0) of system (13) and passing point (𝛽₃,0) when (𝑐,𝑘)∈𝐵₁₅, and its expression is 𝑦=±675Ω²−9450Ω−8437524300𝑐²Ω×𝜙−𝜙₂𝜙−𝜙₂𝛽₁−𝜙𝜙−𝛽₂𝜙−𝛽₃𝜙−𝜙_𝑠,𝜙₂<𝜙≤𝛽₃, (57) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of (41).

Substituting (57) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield ^𝛽3_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₂𝛽₁−𝑠𝛽₂−𝑠𝛽₃−𝑠𝑠−𝜙₂=−675Ω²−9450Ω−8437524300𝑐²Ω𝜉. (58)

Completing the above integral, we can get the parametric representation of solitary wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (59)𝜙 is confirmed by 𝜙𝜒=𝛽₃𝜙₂−𝛽₂+𝛽₂𝛽₃−𝜙₂sn²𝜔𝜒,𝑚𝜙₂−𝛽₂+𝛽₃−𝜙₂sn²,𝜉𝜒=Ω𝜔𝜒,𝑚1−𝛼²₁Π𝛼arcsinsn𝜔𝜒,𝑚,1,𝑚+𝜔²₁−𝜙₂−𝛽₃𝜙₂−𝜙_𝑠𝜒, (60) where Ω=((𝜙₂−𝜙_𝑠)/𝜔(𝜙₂−𝛽₃))24300𝑐²Ω/(675Ω²−9450Ω−84375), 𝛼²₁=(𝜙₂−𝛽₃)/(𝜙₂−𝛽₂), 𝜔=(1/2)(𝜙₂−𝛽₂)(𝛽₃−𝛽₁), and 𝑚=(𝜙₂−𝛽₃)(𝛽₂−𝛽₁)/(𝜙₂−𝛽₂)(𝛽₃−𝛽₁).

3.3. Exact Parametric Representations of Smooth Periodic Wave Solutions

(i) For the given ℎ=ℎ₁ in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is one periodic orbit passing points (𝛽₃,0) and (𝛽₄,0) when (𝑐,𝑘)∈𝐵₄, and its expression is 𝑦=±𝑘6×𝜙−𝜙₁𝛽₁−𝜙𝛽₂−𝜙𝛽₃−𝜙𝜙−𝛽₄𝜙−𝜙_𝑠,𝛽₄≤𝜙≤𝛽₃, (61) where 𝛽₁, 𝛽₂, 𝛽₃, and 𝛽₄(𝛽₄<𝛽₃<𝛽₂<𝛽₁) are four real roots of (28).

Substituting (61) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield ^𝛽3_𝜙𝜙_𝑠−𝑠𝑑𝑠𝜙₁−𝑠𝛽₁−𝑠𝛽₂−𝑠𝛽₃−𝑠𝑠−𝛽₄=𝑘6𝜉. (62)

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (63)𝜙 is confirmed by 𝜙𝜒=𝛽₃𝛽₄−𝛽₂+𝛽₂𝛽₃−𝛽₄sn²𝜔𝜒,𝑚𝛽₄−𝛽₂+𝛽₃−𝛽₄sn²,𝜉𝜒=Ω𝛼𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝛼²𝛼,𝑚+𝜔²₁−𝛼²𝛽₃−𝜙₁𝜙_𝑠−𝜙₁𝜒, (64) where Ω=((𝜙_𝑠−𝜙₁)/𝜔𝛼²(𝜙₁−𝛽₃))6/𝑘, 𝛼²=(𝜙₁−𝛽₂)(𝛽₃−𝛽₄)/(𝜙₁−𝛽₃)(𝛽₂−𝛽₄), 𝛼²₁=(𝛽₃−𝛽₄)/(𝛽₂−𝛽₄), 𝜔=(1/2)(𝛽₁−𝛽₃)(𝛽₂−𝛽₄), and 𝑚=(𝛽₁−𝛽₂)(𝛽₃−𝛽₄)/(𝛽₁−𝛽₃)(𝛽₂−𝛽₄).

(ii) For the given ℎ=(128/243)𝑐⁴ in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is one periodic orbit passing points (𝛽₂,0) and (𝛽₃,0) when (𝑐,𝑘)∈𝐵₅, and its expression is 1𝑦=±2√×𝜙3𝑐₁−𝜙𝛽₁−𝜙𝜙₁−𝜙𝛽₂−𝜙𝜙−𝛽₃𝜙_𝑠,𝛽−𝜙₃≤𝜙≤𝛽₂, (65) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of (36).

Substituting (65) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield ^𝛽2_𝜙𝜙_𝑠−𝑠𝑑𝑠𝜙₁−𝑠𝛽₁−𝑠𝜙₁−𝑠𝛽₂−𝑠𝑠−𝛽₃1=−2√𝜉.3𝑐 (66)

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=1𝑐,𝑥,𝑡𝜙(𝑥−𝑐𝑡)+3𝑐². (67)𝜙 is confirmed by 𝜙𝜒=𝛽₂𝛽₃−𝜙₁+𝜙₁𝛽₂−𝛽₃sn²𝜔𝜒,𝑚𝛽₃−𝜙₁+𝛽₂−𝛽₃sn²,𝜉𝜒=Ω𝑚𝜔𝜒,𝑚₂−𝛼²Farcsinsn𝜔𝜒,𝑚,𝑚+𝛼²Earcsinsn𝜔𝜒,𝑚,𝑚−𝜔𝑚²𝜙₁−𝛽₂𝜙₁−𝜙_𝑠𝜒, (68) where Ω=(2√3(𝜙₁−𝜙_𝑠)𝑐)/𝜔𝑚²(𝜙₁−𝛽₂), 𝛼²=(𝛽₂−𝛽₃)/(𝜙₁−𝛽₃), 𝜔=(1/2)(𝛽₁−𝛽₂)(𝜙₁−𝛽₃), 𝑚=(𝛽₁−𝜙₁)(𝛽₂−𝛽₃)/(𝛽₁−𝛽₂)(𝜙₁−𝛽₃), andF(⋅,⋅) and E(⋅,⋅) are Legendre's incomplete elliptic integrals of the first and second kinds, respectively (see [11]).

(iii) For the given ℎ=ℎ₂ in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is one periodic orbit passing points (𝛽₂,0) and (𝛽₃,0) when (𝑐,𝑘)∈𝐵₇, and its expression is 𝑦=±675Ω²−9450Ω−8437524300𝑐²Ω×𝜙₂−𝜙𝜙₂−𝜙𝛽₁−𝜙𝛽₂−𝜙𝜙−𝛽₃𝜙_𝑠,𝛽−𝜙₃≤𝜙≤𝛽₂, (69) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of (41).

Substituting (69) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield ^𝛽2_𝜙𝜙_𝑠−𝑠𝑑𝑠𝜙₂−𝑠𝜙₂−𝑠𝛽₁−𝑠𝛽₂−𝑠𝑠−𝛽₃=675Ω²−9450Ω−8437524300𝑐²Ω𝜉. (70)

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (71)𝜙 is confirmed by 𝜙𝜒=𝛽₂𝛽₃−𝛽₁+𝛽₁𝛽₂−𝛽₃sn²𝜔𝜒,𝑚𝛽₃−𝛽₁+𝛽₂−𝛽₃sn²,𝜉𝜒=Ω𝑚𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝑚²𝛼,𝑚+𝜔²₁−𝑚²𝜙₂−𝛽₂𝜙₂−𝜙_𝑠𝜒, (72) where Ω=((𝜙_𝑠−𝜙₂)/𝜔𝑚²(𝜙₂−𝛽₂))24300𝑐²Ω/(675Ω²−9450Ω−84375), 𝛼²₁=(𝛽₂−𝛽₃)/(𝛽₁−𝛽₃), 𝜔=(1/2)(𝜙₂−𝛽₂)(𝛽₁−𝛽₃), and 𝑚=(𝜙₂−𝛽₁)(𝛽₂−𝛽₃)/(𝜙₂−𝛽₂)(𝛽₁−𝛽₃).

(iv) For the given ℎ=ℎ₁ in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is one periodic orbit passing points (𝛽₁,0) and (𝛽₂,0) when (𝑐,𝑘)∈𝐵₁₂, and its expression is 𝑦=±𝑘6𝜙−𝜙₁𝛽₁−𝜙𝜙−𝛽₂𝜙−𝛽₃𝜙−𝛽₄𝜙−𝜙_𝑠,𝛽₂≤𝜙≤𝛽₁, (73) where 𝛽₁, 𝛽₂, 𝛽₃, and 𝛽₄(𝛽₄<𝛽₃<𝛽₂<𝛽₁) are four real roots of (28).

Substituting (73) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield ^𝛽1_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₁𝛽₁−𝑠𝑠−𝛽₂𝑠−𝛽₃𝑠−𝛽₄=𝑘6𝜉. (74)

Completing the above integral, we can get the parametric representation of smooth periodic wave solution which is the same as (33).

(v) For the givenℎ=(128/243)𝑐⁴ in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is one periodic orbit passing points (𝛽₁,0) and (𝛽₂,0) when (𝑐,𝑘)∈𝐵₁₃, and its expression is 1𝑦=±2√×3𝑐𝜙−𝜙₁𝛽₁−𝜙𝜙−𝛽₂𝜙−𝜙₁𝜙−𝛽₃𝜙−𝜙_𝑠,𝛽₂≤𝜙≤𝛽₁, (75) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of (36).

Substituting (75) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield ^𝛽1_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₁𝛽₁−𝑠𝑠−𝛽₂𝑠−𝜙₁𝑠−𝛽₃=12√𝜉.3𝑐 (76)

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=𝜙()+1𝑐,𝑥,𝑡𝑥−𝑐𝑡3𝑐². (77)𝜙 is confirmed by 𝜙𝜒=𝛽₁𝛽₂−𝛽₃+𝛽₃𝛽₁−𝛽₃sn²𝜔𝜒,𝑚𝛽₂−𝛽₃+𝛽₁−𝛽₃sn²,𝜉𝜒=Ω𝑚𝜔𝜒,𝑚₂−𝛼²₁Πarcsinsn𝜔𝜒,𝑚,𝑚²𝛼,𝑚+𝜔²₁−𝑚²𝜙₁−𝛽₁𝜙₁−𝜙_𝑠𝜒, (78) where Ω=2√3(𝜙₁−𝜙_𝑠)𝑐/𝜔𝑚²(𝛽₁−𝜙₁), 𝛼²₁=(𝛽₂−𝛽₁)/(𝛽₂−𝛽₃), 𝜔=(1/2)(𝛽₁−𝜙₁)(𝛽₂−𝛽₃), and 𝑚=(𝛽₁−𝛽₂)(𝜙₁−𝛽₃)/(𝛽₁−𝜙₁)(𝛽₂−𝛽₃).

(vi) For the given ℎ=ℎ₂ in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is one periodic orbit passing points (𝛽₁,0) and (𝛽₂,0) when (𝑐,𝑘)∈𝐵₁₅, and its expression is 𝑦=±675Ω²−9450Ω−8437524300𝑐²Ω×𝜙−𝜙₂𝛽₁−𝜙𝜙−𝛽₂𝜙−𝛽₃𝜙−𝜙₂𝜙−𝜙_𝑠,𝛽₂≤𝜙≤𝛽₁, (79) where 𝛽₁, 𝛽₂, and 𝛽₃(𝛽₃<𝛽₂<𝛽₁) are three real roots of (41).

Substituting (79) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield ^𝛽1_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₂𝛽₁−𝑠𝑠−𝛽₂𝑠−𝛽₃𝑠−𝜙₂=675Ω²−9450Ω−8437524300𝑐²Ω𝜉. (80)

Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows: 𝑢()()+4𝑥,𝑡=𝜙𝑥−𝑐𝑡3𝜌()=1𝑐,𝑥,𝑡𝜙(𝑥−𝑐𝑡)+3𝑐². (81)𝜙 is confirmed by 𝜙𝜒=𝛽₁𝛽₂−𝜙₂+𝜙₂𝛽₁−𝛽₂sn²𝜔𝜒,𝑚𝛽₂−𝜙₂+𝛽₁−𝛽₂sn²,𝜉𝜒=Ω𝑚𝜔𝜒,𝑚₂−𝛼²Farcsinsn𝜔𝜒,𝑚,𝑚+𝛼²Earcsinsn𝜔𝜒,𝑚,𝑚−𝜔𝑚²𝛽₂−𝛽₁𝜙₂−𝜙_𝑠𝜒, (82) where Ω=((𝜙₂−𝜙_𝑠)/𝜔𝑚²(𝛽₁−𝛽₂))24300𝑐²Ω/(675Ω²−9450Ω−84375), 𝛼²=(𝛽₂−𝛽₁)/(𝛽₂−𝜙₂), 𝜔=(1/2)(𝛽₁−𝛽₃)(𝛽₂−𝜙₂), and 𝑚=(𝛽₁−𝛽₂)(𝛽₃−𝜙₂)/(𝛽₁−𝛽₃)(𝛽₂−𝜙₂).

Remark 2. From Figures 3(f) and 3(n), we see that (2) has two solitary wave solutions when (𝑐,𝑘)∈𝐵₆ and two solitary wave solutions when (𝑐,𝑘)∈𝐵₁₄, but the exact parametric representations are very complex, so here we omit them.

4. Conclusion

In this paper, using the bifurcation theory and the method of phase portraits analysis, we investigated the bifurcations of (2) (which is called a two-component Degasperis-Procesi equation) and obtained some exact parametric representations of peakon, solitary wave and smooth periodic wave solutions (see (23), (30), and (63), etc.). We can say that we obtained some new results of (2) in this paper.

Acknowledgments

The authors thank the referees very much for their perceptive comments and suggestions. This work is supported by the Natural Science Foundation of Yunnan Province, China (no. 2013FZ117) and the National Natural Science Foundation of China (no. 11161020).