# Solitary and Periodic Wave Solutions of Sasa–Satsuma Equation and Their Relationship with Hamilton Energy

Authors: Weiguo Zhang; Xingqian Ling; Bei-Bei Wang; Shaowei Li
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study the exact solitary wave solutions and periodic wave solutions of the S-S equation and give the relationships between solutions and the Hamilton energy of their amplitudes. First, on the basis of the theory of dynamical system, we make qualitative analysis on the amplitudes of solutions. Then, by using undetermined hypothesis method, the first integral method, and the appropriate transformation, two bell-shaped solitary wave solutions and six exact periodic wave solutions are obtained. Furthermore, we discuss the evolutionary relationships between these solutions and find that the appearance of these solutions for the S-S equation is essentially determined by the value which the Hamilton energy takes. Finally, we give some diagrams which show the changing process from the periodic wave solutions to the solitary wave solutions when the Hamilton energy changes.


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## Body

## 1. Introduction

Sasa–Satsuma equation (called S-S equation for short)(1)izT+12zXX+z2z+iεzXXX+6z2zX+3zz2X=0was proposed by Sasa and Satsuma in studying the integrability of the Schrödinger equation with higher-order nonlinear terms, and it is a typical equation which describes the propagation of short pulses in an optical fiber [1–4]. The 1-soliton solutions and the N-soliton solutions of equation (1) were also given by Sasa and Satsuma in [1].In the past 30 years, with the development of nonlinear system, different methods have been found and developed, such as the inverse scattering method, the bilinear method, the Darboux transformation, the Backlünd transformation,G/G′ expansion method, and the auxiliary equation method [4–28]. At the same time, the S-S equation has attracted interests of many scholars because of its importance [4–16]. They have worked out some solitary wave solutions and periodic wave solutions of the form(2)zX,T=uξeik1X−c1T,ξ=X−cT,for equation (1) by the inverse scattering method [4], the bilinearization method [5], the Darboux transformation [7–10], the Backlünd transformation [11], and some other methods, where uξ, eik1X−c1T, and c denote the amplitude, envelope, and wave velocity, respectively.By using transformationszX,T=ux,teiεx+εt/6, T=t, and X=x+ε/2t, [5–10] simplified the S-S equation into the following form:(3)ut+16εuxxx+6u2ux+3u2xu=0.In [5, 6, 11, 12], the solitary wave solutions for the S-S equation in the form of(4)zX,T=A1D+coshBmX+cTeiεkX−cTwere obtained by the Hirota bilinear method, the Backlünd transformation, the Darboux transformation, and the extended exploratory equation method, respectively.In [12], they obtained the dark soliton solution of equation (1) by the generalized Kudryashov method. And by using the extended exploratory equation method, they obtained the unbounded solution for equation (1) with amplitude(5)uX,T=4A2α2−α1τ14A2−α1−α2mx+nt2,and periodic solutions for equation (1) with amplitude(6)uX,T=A2M+Nsn2φ,l.In recent literature [16], they studied the nonlocal S-S equation in the case of inverse space-time and obtained its solitary wave solutions.Although the S-S equation has been solved by various methods in many literatures, there are still some solitary wave solutions and periodic wave solutions for the S-S equation that have not been solved. In this paper, we will study solutions of the form (2) for the S-S equation (1) and expect to find its new periodic wave solutions or solitary wave solutions. At the same time, we will study the relationships between solutions and Hamilton energy for the S-S equation.First, by employing the theory and method of planar dynamical system, we make detailed qualitative analysis on the amplitudes of solutions for the S-S equation and give the global phase portraits under different parameters. We also obtain the conditions for the existence of solitary wave solutions and periodic wave solutions. In Section3, according to the conclusions of the qualitative analysis, we use the undetermined hypothesis method to obtain two bell-shaped solitary wave solutions for the S-S equation, where one is new, bounded, and rational [17, 18]. In Section 4, by the first integral method and appropriate transformation, we obtain six periodic wave solutions for equation (1), and the amplitudes of two solutions are in the fractional form of the Jacobi elliptic function sn2φ,ξ, which are similar as the amplitudes of the periodic wave solutions found in [13]. However, for the rest four kinds of periodic wave solutions, their amplitudes are composed by dnφ,k, cnφ,k, and cosφ, which have not been obtained in the previous literatures. In Section 5, we study the evolutionary relationships between the solitary wave solutions, periodic wave solutions, and the Hamilton energy for the S-S equation and show the reason of the appearance of different kinds of solutions is that the corresponding Hamilton energy h takes different values. When h tends to be Hx2,0, the periodic wave solutions will take the solitary wave solutions as the limit. At last, we will show this evolutionary process in the diagrams.It is worth to point out that (1) by combining the qualitative analysis with the first integral method, we not only obtain the new bell-shaped solitary wave solutions (unimodal) and periodic wave solutions for equation (1), but also show the mathematical meaning for the solutions by establishing the corresponding relationships between the solutions and the bounded trajectories in the phase portraits. (2) In the previous literatures [12, 19–21] for solving the solitary wave and periodic wave solutions of the nonlinear evolutionary equations, the relationships between solutions and the energy of system are rarely described. In this paper, we not only obtain the solutions of equation (1) but also show the relationships between the amplitudes of the solutions and the Hamilton energy for the corresponding system. Then, we have the conclusion that when Hamilton energy h takes different values, equation (1) has different solutions in the form of (2). For a nonlinear complex system, it is important to understand the essential reasons for the various and complex phenomena in this system and only by this way can people really control and apply the system which corresponds to the S-S equation.


## 2. Qualitative Analysis

In this section, we will employ the theory and method of planar dynamical system [29, 30] to make qualitative analysis on the traveling wave solutions of the form (2) for the S-S equation (1) and give the global phase portraits for the amplitudes of these solutions under different parameters. Then, several conclusions for the existence of these solutions can be obtained.Assuming that the S-S equation (1) has solutions in the form of (2), by substituting (2) into equation (1), we have(7)−icu′+c1u+12u′+2ik1u−k12u+u2u+iεu″+3ik1u′−3k12u−ik13u+iε6u2u′+ik1u+3uu2=0.Then, taking the real and imaginary parts of (7) as zero, we can get(8)c1−12k12+εk13u+12−3εk1u″+1−6εk1u2u=0,(9)k1−c−3εk12u′+εu‴+6εu2u′+3εuu2′=0.In (8), we set the coefficients of u, u″, and u2u as zero and obtain(10)c1−12k12+εk13=0,12−3εk1=0,1−6εk1=0.According to (10), we have(11)k1=16ε,c1=1108ε2.By substituting (11) into (9), we have(12)112ε−cu′+εu‴+6εu2u′+3εuu2′=0.Then, integrating on both sides of equation (12) with respect to ξ, we can obtain(13)112ε−cu+εu″+4εu3=g0,where g0 is a constant. According to (13), we have(14)u″+4u3+pu+q=0,where p=1−12εc/48ε2 and q=−g0/4ε.From the above discussions, we can deduce that for studying the solutions of equation (1), we can start from equation (14).Letx=uξ and y=u′ξ, equation (14) is equivalent to the following planar dynamic system:(15)dxdξ=Px,y,dydξ=−4x3+px+q=Qx,y.Since system (15) is a Hamiltonian system, it has the following first integral:(16)Hx,y=y22+x4+2px2+qx=h,where h=Hx,y is the Hamilton energy at the point x,y for system (15).On the planex,y, the number of finite singularities of system (15) depends on the number of the real roots for equation fx=x3+px+q=0. Here, we note the discriminant of fx=0 as(17)Δ=q22+p33=146272ε627243ε4c2+1−12εc3,and then we can get the following conclusions:(a).
IfΔ>0, fx=0 has one real root and two complex roots.(b).
IfΔ=0, fx=0 has three real roots (two of them are equal):(18)x1=−2−p3,x2=x3=−p3.(c).
IfΔ<0, fx=0 has three different real roots:(19)x1=2−q33cosθ3,x2=2−q33cosθ3−2π3,x3=2−q33cosθ3−4π3,wherecosθ=33q/2p−p.In this paper, we mainly discuss the bounded traveling wave solutions of equation (1), so we always assume that p<0 and Δ≤0 in the following. Under this assumption, there exist three finite singular points Pixi,0i=1,2,3 in system (15), where xii=1,2,3 is the real root of fx=0 and satisfies(20)x1+x2+x3=0,x1x2+x1x3+x2x3=p,x1x2x3=−q.Here we assumex1≤x2≤x3, and the Jacobi matrix at Pi for system (15) can be expressed as follows:(21)Jxi,0=01−4f′xi0,where f′xi=3xi2+pi=1,2,3.(1)
In the case ofq=0, we have Δ<0 now, and equation fx=0 has three real roots x1=−−p, x2=0, and x3=−p, which correspond to the three different finite singular points Pixi,0i=1,2,3 of system (15). From (21), we have the determinants of the Jacobi matrix Jxi,0i=1,2,3 as follows:(22)detJx1,0=43x12+p=4−3p+p=−8p>0,detJx2,0=43x22+p=4p<0,detJx3,0=43x32+p=4−3p+p=−8p>0.SoP1 and P3 are centers, and P2 is a saddle point.(2)
In the case ofq<0.(I) WhenΔ=0, the three real roots x1=x2<0<x3 for the equation fx=0 correspond to two different finite singular points P1x1,2,0 and P3x3,0 of system (15). From (20), we have(23)2x1+x3=0,x12+2x1x3=p,x12x3=−q.Then, we obtain the determinants of the Jacobi matrixJxi,0 at the singular points P1,2 and P3 as follows:(24)detJx1,0=43x12+p=44x12+2x1x3=8x12x1+x3=0,detJx3,0=43x32+p>0.SoP1,2 is a cusp and P3 is a center.(II) WhenΔ<0, there exist three different real roots in the equation fx=0 which satisfy x1<x2<0<−x2<−x1<x3, and the corresponding system (15) has three different finite singular points Pixi,0i=1,2,3. Then, from (21), we get the determinants of the Jacobi matrix Jxi,0i=1,2,3 at Pixi,0i=1,2,3 as follows:(25)detJx1,0=43x12+p>4x12+x1x2+x2x3=0,detJx2,0=43x22+p=42x22+x1x3<4x1x2+x22+x2x3=0,detJx3,0=43x32+p=42x32+x1x2>0.So,P1 and P3 are centers, and P2 is a saddle point.(3)
In the case ofq>0.(I) WhenΔ=0, the three real roots x1=−2−p/3 and x2=x3=−p/3 of the equation fx=0 correspond to the finite singular points P1x1,0, P2x2,0=P3x3,0 of system (15), so we get the determinants of the Jacobi matrix Jxi,0i=1,2,3 at Pixi,0i=1,2,3 as follows:(26)detJx1,0=43x12+p=4−4p+p=−12p>0,detJx2,0=detJx3,0=4−p+p=0.SoP1 is a center and P2,3 is a cusp.(II) WhenΔ<0, the three different real roots of the equation fx=0 satisfy x1<−x3<−x2<0<x2<x3, and corresponding system (15) has three different finite singular points Pixi,0i=1,2,3. From (21), we get the determinants of the Jacobi matrix Jxi,0i=1,2,3 at Pixi,0i=1,2,3 as follows:(27)detJx1,0=43x12+p=42x12+x2x3>0,detJx2,0=43x22+p=42x22+x1x3<4x2x1+x2+x3=0,detJx3,0=43x32+p=42x32+x1x2>4x32+x1x3+x2x3=0.So,P1 and P3 are centers and P2 is a saddle point.According to the above qualitative analysis, we can obtain five global phase portraits of system (15) under different parameters (see Figures 1(a)–1(e)).Figure 1
(a)q=0,Δ<0, (b)q>0,Δ<0, (c)q<0,Δ<0, (d)q>0,Δ=0, and (e)q<0,Δ=0.
(a)
(b)
(c)
(d)
(e)Then, we can get the following propositions from the above five global phase portraits.Proposition 1.
LetΔ≤0 and p<0.(1)
WhenΔ<0, no matter what value q takes, there are two homoclinic orbits and numerous closed orbits in system (15) (see Figures 1(a)–1(c)).(2)
WhenΔ=0, there exist one homoclinic orbit and numerous closed orbits in system (15) in the case of q<0 or q>0 (see Figures 1(d) and 1(e)).Because the homoclinic orbits correspond to the bell-shaped solutions of equation (14), and the periodic trajectories correspond to the periodic solutions of equation (14); on the basis of Proposition 1, we can obtain the following theorem.Theorem 1.
Assume Δ ≤ 0 andp<0.(1)
When Δ < 0 in the case ofq<0, q=0, or q>0, there exist two bell-shaped solitary wave solutions (corresponding to the homoclinic orbits in Figures 1(a)–1(c)) and numerous periodic traveling wave solutions (corresponding to closed orbits in Figures 1(a)–1(c)) of the form (2) for equation (1)(2)
When Δ = 0, in the case ofq<0 or q>0, there exist one bell-shaped solitary wave solution (corresponding to the homoclinic orbit in Figures 1(d) and 1(e)) and numerous periodic traveling wave solutions (corresponding to closed orbits in Figures 1(d) and 1(e)) of the form (2) for equation (1)


## 3. Solitary Wave Solutions of Equation (1)

In this section, we use the undetermined hypothesis method to obtain two kinds of solitary wave solutions of the form (2) for equation (1). Since equation (1) can be transformed into equation (14), we only need to consider the bounded solutions for equation (14) [31, 32]. Here, we assume that equation (14) has solutions with the following forms:(28)uξ=Aemξ1+emξ2+Bemξ+D,(29)uξ=AB+mξ2+D,where A,B,D,m are undetermined parameters. Substituting (28) and (29) into (14), we can get the solutions of equation (14), and then, we have the solitary wave solutions of equation (1). So the following theorem can be obtained.Theorem 2.
Assumingp<0 and Δ≤0, here we set x0 as the real solution for the equation fx0=x03+px0+q=0.(1)
Whenp+3x02<0, equation (1) has bell-shaped solitary wave solution:(30)zS1±ξ=uS1±ξei1/6εX−1/108ε2T,where the bounded solutionsuS1±ξ of equation (14) can be expressed as follows:(31)uS1±ξ=x0−±2p+3x02/−p−x02sech2−p−3x02ξ2+−1±2x0/−p−x02sech2−p−3x02ξ.Besides, (31) can be equivalently expressed as(32)uS1±ξ=x0−2p+3x022x0±−p−x02cosh2−p−3x02ξ.(2)
Whenp<0 and Δ=0, equation (1) has bell-shaped solutions:(33)zS2±ξ=uS2±ξeiX/6ε−T/108ε2,where the bounded solutionsuS2±ξ of equation (14) can be expressed as follows:(34)uS2±ξ=±−p3−4−3p3−8pξ2.Note 1.
(1)
u
S
i
− and uSi+i=1,2 represent the solutions with − and + before radical sign, respectively.
(2)
Whenq=0 and x0=0, us1±ξ can be simplified as u˜S1±ξ, which can be expressed as follows:
(35)
u
˜
S
1
±
ξ
=
±
−
2
p
sec
h
−
4
p
ξ
.
So whenq=0, equation (1) has bell-shaped solutions:
(36)
z
˜
S
1
±
ξ
=
u
˜
S
1
±
ξ
e
X
/
6
ε
−
T
/
108
ε
2
.
(3)
Equationfx=x3+px+q=0 has three real roots which satisfy x1<x2<x3, since we set x0 as the real root for equation fx=0 and it satisfies p+3x02<0, according to the discussions in Section 3, we know that here x0 is x2.
Here, we point out that(1).
Whenq≠0andΔ<0, the amplitude uS1+ξ of bell-shaped solution zS1+ξ for equation (1) corresponds to the homoclinic trajectory with P2 as a saddle point and P3 as a center in the global phase portraits Figures 1(b) and 1(c). The amplitude uS1−ξ of bell-shaped solution zS1−ξ of equation (1) corresponds to the homoclinic trajectory with P2 as a saddle point and P1 as a center in the global phase portraits Figures 1(b) and 1(c).(2).
Whenq=0 and Δ<0, the amplitudes u˜S1±ξ of bell-shaped solutions z˜S1±ξ for equation (1) correspond to the homoclinic trajectories in the global phase portrait—Figure 1(a).(3).
The amplitudesuS2±ξ of bell-shaped solution zS2+ξ for equation (1) correspond to the homoclinic trajectories in the global phase portraits Figures 1(d) and 1(e)),respectively. Here, uS2±ξ are bounded and rational [17, 18].


## 4. Periodic Wave Solutions of Equation (1)

Similar to the solving process in the last section, we need to solve the periodic wave solutions for equation (14) if we want to get the periodic wave solutions in the form of (2) for equation (1).Equation (14) corresponds to the planar dynamic system (15), and its first integral (16) is the general expression of the trajectories which correspond to the solutions for equation (14). Therefore, we can use the first integral (16) to obtain the periodic wave solutions for equation (14), and then, we can get the periodic wave solutions for the S-S equation (1). In this section, we always assume that p<0 and Δ≤0 and three real roots x1,2,3 of fx=0 satisfy x1<x2<x3.From the first integral (16) of system (15), we can obtain(37)y2=−2x4+2px2+4qx−h.Fromy=x˙, we have(38)dxdξ=x˙=y=±−2x4+2px2+4qx−h.By the separate variable method, we can translate the problem of solving equation (14) into the following integral:(39)∫dx−Fhx=±2ξ,where Fhx=x4+2px2+4qx−h.Note 2.
In Theorem2, by taking the Hamilton energy h=Hx2,0 in (39) and integrating on it, the amplitude uS1±ξ of the bell-shaped solutions zS1±ξ for equation (1) can be obtained. Similarly, we can obtain the amplitude uS2±ξ of the bell-shaped solutions zS2±ξ for equation (1) in the case of Δ=0 and q≠0.


### 4.1. Periodic Wave Solutions Corresponding to Periodic Trajectories Surrounded by Symmetric Homogeneous Orbits

Whenq=0 and Δ<0, there exist symmetric homoclinic orbits in system (15). Since the Hamilton energy on the same periodic trajectory is equal, we can set the Hamiltonian energy on the same periodic trajectory as follows:(40)Hx,y=Hη1,0=η14+2pη12=η12+p−p2=h1,where η1,0 is the intersection of the periodic trajectory and the x-axis and satisfies η1<−p. According to (16), we get(41)Hx,y=12y2+x2+p2−p2=h1.Substituting three solutionsx1=−−p, x2=0, and x3=−p of the equation fx=x3+px=xx2+p=0 into Hx,0, we have Hx1,0=Hx3,0<Hx2,0=0. Then, we have the −Fhx function curve and periodic trajectories when h1∈−p2,0 as shown in Figure 2.Then, (39) can be transformed into(42)±2dξ=dxp2+h1−x2+p2.(43)p2+h1−x2+p2=−x2+p+p2+h1x2+p−p2+h1,in (42), we let A12=p2+h1−p and B12=−p2+h1−p and make transformation t=x/A1=uξ/A1; then, we have(44)±2dξ=dx−x2−A12x2−B12=dx/A1A11−x2/A12x2/A12−B12/A12=dtA11−t2t2−B12/A12.Figure 2
−
F
h
x function curve and periodic trajectories when q=0 and Δ<0.According to the properties of the elliptic function integral [33, 34], the following lemma can be obtained.Lemma 1.
Ifdt/1−t2t2−k′2=u, then we have t=dnu,k1, k12+k′2=1 (k is the mold). Set k′=B1/A1, A12=−2p/2−k12, and B12=−2p1−k12/2−k12; then by integrating on the both sides of (42), we have(45)uP1±ξ=±−2p2−k12dn−4p2−k12ξ,k1,where k12=2+2p/A12=2p2+h1/p2+h1−p and h1∈−p2,0.
Then, we can obtain the following theorem.Theorem 3.
Whenq=0 and Δ < 0, equation (1) has periodic solutions:(46)zP1±ξ=uP1±ξeiX/6ε+T/108ε2,where uP1±ξ are bounded solutions for equation (14).Here, we point out that the amplitudesuP1±ξ of the periodic wave solutions zP1±ξ for equation (1) correspond to the periodic trajectories, which are contained in the homoclinic trajectory centered on P1 and P3 in the global phase portrait (Figure 1(a)), respectively.


### 4.2. Exact Periodic Wave Solutions Corresponding to Closed Trajectories Contained in Asymmetric Homoclinic Orbits in Figures1(b) and 1(c)

Whenq≠0 and Δ<0, there exist asymmetric homoclinic orbits in system (15). In global phase portraits Figures 1(b) and 1(c), we set the Hamilton energy at the point x,y on the periodic trajectory surrounded by the homoclinic orbits as(47)Hx,y=Hη2,0=h2,where Hx,y is given by (16) and η2,0 is the intersection of the periodic trajectory and the x-axis. From Figures 1(b) and 1(c), we can know that Pixi,0i=1,2,3 are three singular points of system (15), and their abscissas xii=1,2,3 satisfy x1<x2<x3. It can be verified that when q<0, Hx3,0<Hx1,0<Hx2,0; when q>0, Hx1,0<Hx3,0<Hx2,0.When Hamilton energyh2 satisfies(48)maxHx1,0,Hx3,0<h2<Hx2,0,its corresponding −Fhx function curve and periodic trajectories are shown in Figure 3. Set the four different real roots for Fhx=x4+2px2+4qx−h=0 as x1′i=1,2,3,4, and the roots satisfy x1′<x1<x2′<x2<x3′<x3<x4′. So −Fhx can be expressed as follows:(49)Fhx=x−x1′x−x2′x−x3′x−x4′.Figure 3
−
F
h
x function curve and periodic trajectories when q<0,Δ<0.Sincex=uξ,y=u′ξ, the bounded solutions of equation (14) satisfy(50)dudξ2=−2Fhx.(1)
Whenu∈x1′,x2′, let u=x4′−x4′−x1′x4′−x2′/x4′−x2′+x2′−x1′sin2θ, we get(51)Fhx=x−x1′x−x2′x−x3′x−x4′=−x2′−x1′2x4′−x1′2x4′−x2′2sin2θcos2θ⋅x3′−x1′x4′−x2′−x2′−x1′x4′−x3′sin2θx4′−x2′+x2′−x1′sin2θ4.From (39) and (51), we have(52)∫du−Fhx=−2∫2dθx3′−x1′x4′−x2′−x2′−x1′x4′−x3′sin2θ=−2x3′−x1′x4′−x2′∫2dθ1−k2sin2θ=2ξ,wherek2=x2′−x1′x4′−x3′/x1′−x3′x2′−x4′. According to the definition of Jacobi elliptic function [33, 34], we get(53)sinθ=−sn122x1′−x3′x2′−x4′ξ,k2.Then, by substituting (53) into u=x4′−x4′−x1′x4′−x2′/x4′−x2′+x2′−x1′sin2θ, we get the bounded solution of equation (14):(54)uP2ξ=xξ=x4′−x4′−x1′x4′−x2′x4′−x2′+x2′−x1′sn2−1/22x1′−x3′x2′−x4′ξ,k2.(2)
Whenx∈x3′,x4′, let u=x1′+x4′−x1′x3′−x1′/x3′−x1′+x4′−x3′sin2θ, we have(55)Fhx=x−x1′x−x2′x−x3′x−x4′=x4′−x1′2x3′−x1′2x4′−x3′2sin2θcos2θ⋅x2′−x1′x4′−x3′sin2θ−x3′−x1′x4′−x2′x3′−x1′+x4′−x2′sin2θ4.From (39) and (55), we have(56)∫du−Fhx=∫2dθ−x2′−x1′x4′−x3′sin2θ+x3′−x1′x4′−x2′=2ξ,where k2=x2′−x1′x4′−x3′/x1′−x3′x2′−x4′. Then we get(57)sinθ=sn122x3′−x1′x4′−x2′ξ,k2.By substituting (57) into u=x1′+x4′−x1′x3′−x1′/x3′−x1′+x4′−x3′sin2θ, we obtain the bounded solution of equation (14):(58)uP3ξ=x1′+x4′−x1′x3′−x1′x3′−x1′+x4′−x3′sn21/22x3′−x1′x4′−x2′ξ,k2.From the above discussions, we have the following lemma.Lemma 2.
Assumingq≠0 and Δ<0, when Hamilton energy h satisfies (48), equation (14) has two periodic wave solutions uP2ξ and uP3ξ, where uP2ξ∈x1′,x2′ and uP3ξ∈x3′,x4′.Then, according to Lemma2 and equation (11), we can get the following theorem.Theorem 4.
Letq≠0 and Δ < 0, when Hamilton energy h satisfies (48), equation (1) has two periodic wave solutions:(59)zP2ξ=uP2ξeiX/6ε−T/108ε2,(60)zP3ξ=uP3ξeiX/6ε−T/108ε2,where uP2ξ and uP3ξ are given by (54) and (58), respectively.Here, we point out that, in Theorem4, uP2ξ corresponds to the periodic trajectory contained in the homoclinic trajectory with P1 as the center, and uP3ξ corresponds to the periodic trajectory contained in the homoclinic trajectory with P3 as the center in Figures 1(b) and 1(c).


### 4.3. Periodic Wave Solutions Corresponding to the Remaining Periodic Trajectories

Except the periodic solutionszPiξi=1,2,3 found in Sections 4.1 and 4.2, there exist other periodic solutions for equation (1), which correspond to the periodic trajectories surrounded by the homoclinic trajectory in Figures 1(b)–1(e). Here, we set(61)Hx,y=Hη3,0=h3,where η3,0 is the intersection of the periodic trajectory and the x-axis.In the case ofΔ<0 and q<0, when h3 satisfies h3>Hx2,0 or Hx3,0<h3<Hx1,0, the corresponding Y=−Fhx function curve and periodic trajectories are shown in Figure 4. In the case of Δ<0 and q>0, when h3>Hx2,0 or Hx1,0<h3<Hx3,0, the corresponding Y=−Fhx function curve and periodic trajectories have the relationships which are similar to Figure 4.Figure 4
q
<
0 and Δ<0.In addition, in the case ofΔ=0 and q>0, we have Hx1,0<Hx2,0=Hx3,0. And we can give the corresponding Y=−Fhx function curves and periodic trajectories when the Hamilton energy h3 satisfies Hx1,0<h3<Hx2,0, which are shown in Figure 5 (note: in the case of Δ=0 and q<0, we can give the similar diagrams).Figure 5
q
>
0 and Δ=0.As shown in Figures4 and 5, there is a common feature for the above four cases, i.e., the function(62)±Fhx=±x4+2px2+4qx−hhas only two different intersections s1, s2s1<s2 with the x-axis. Then, we can express Fhx as(63)Fhx=x−s1x−s2x−α2+β2,where 2α+s1+s2=0, β>0. Then, we divide (63) into the following parts:(64)x−α2+β2=ε1x−r12+1−ε1x−r22,x−s1x−s2=ε2x−r12+1−ε2x−r22,where ri=α1−λi/1+λi,i=1,2, ε1=λ21+λ1/λ2−λ1, ε2=1+λ1/λ2−λ1, and λ1,λ2 are two real roots of the equation(65)s1−s22λ2−4β2+α−s1α−s2λ−4β2=0.Sincex=uξ,y=u′ξ, by substituting s=x−r1/x−r2=uξ−r1/uξ−r2 into (39), we get(66)±2ξ=1r1−r2ε1ε2∫dss2+1−ε1/ε2s2−1+ε2/ε2.Then, lets2=1+ε2/ε21−t2 and m2=ε21−ε1/ε1+ε2, we have(67)±2ε1+ε2r1−r2ξ=∫dt1−t21−m2t2.By using the Jacobi elliptic function integral [33, 34], we get(68)uξ=r1−r1−r21±ε2/1+ε2cn2ε1+ε2r1−r2ξ,m.After simplification, we have the bounded solutions of equation (14) as follows:(69)uP4±ξ=1−λ11+λ1α−σ2α±2α1+λ1/1+λ2cn2σξ,m,where σ=β2+α−s12β2+α−s22, λ1<λ2, and m satisfies(70)m2=12−β2+α−s1α−s22β2+α−s12β2+α−s22.Here, we point out that(1).
In the case ofΔ≤0 and q≠0, we have(71)x4+2px2+4qx−h3=x−s1x−s2x−α2+β2,from (63). Then we expand the right side of the above equation, and let the coefficients of x4,x3,x2,x be the same, we have(72)s1+s2+2α=0,s1s2+2s1α+2s2α+α2+β2=2p,2s1s2α+s1α2+s2α2+s1β2+s2β2=4q.(a)
Whenm2>1/2, which is equivalent to β2+α−s1α−s2<0, since β2+α−s1α−s2=β2+3α2+s1s2=2p+6α2<0, we have α2=s1+s2/22<−p/3, where sii=1,2 and xii=1,2,3 satisfy(73)s1<x1=−x0−−3x0−4p2<x2=x0<x3=−x0+−3x0−4p2<s2.At this time, there exist periodic trajectories surrounding the homoclinic trajectories in system (15).(b)
Whenm2<1/2, which is equivalent to β2+α−s1α−s2>0, we can get s1+s2/22>−p/3. Now sii=1,2 and xii=1,2,3 have the following relationship:(74)x1=x0<x2=−x0−−3x0−4p2<s1<s2<x3=−x0+−3x0−4p2,and there exist periodic trajectories contained in the homoclinic trajectories in system (15).(2).
In the case ofΔ<0 and q=0, when h3>Hx2,0=0, we have(75)−Fhx=x2+p+p2+h3x2−p2+h3−p.Sincep+p2+h3>0, equation x2+p+p2+h3=0 has no real root. By substituting (75) into (39), we get(76)2ξ=∫dxx2+p+p2+h3x2−p2+h3−p.Set−p2+h3−p=p1, p2+h3−p=p2, by using transformation x2=p2/1−t2, we have(77)2ξ=∫dtp1+p21−t21−p1/p1+p2t2.According to the Jacobi elliptic function integral [33, 34], we have the bounded periodic solution of equation (14) as follows:(78)u˜P4±=±−2pk22k2−1cn−4p2k2−1ξ,k,wherek2=−p+p2+h3/2p2+h3∈1/2,1. Here, u˜P4±ξ correspond to the periodic trajectories surrounding the homoclinic trajectory in the global phase portrait Figure 1(a). At this time, equation (1) has periodic wave solutions of the form (2) as follows:(79)z˜P4±ξ=u˜P4±ξeiX/6ε+T/108ε2.(3).
In the case ofΔ<0 and q<0, it can be known from Figure 4 that there exist two periodic trajectories centered on P3 inside the homoclinic trajectories (correspond to the periodic wave solutions uP3 and uP4 of equation (14), respectively), and these two trajectories are divided by the line which corresponds to the Hamilton energy h3=Hx1,0. When Hamilton energy h=Hx1,0, since Hx1,0=x14+2px12+4qx1, and the abscissa of singularity Px1,0 of system (15) satisfies x13+px1+q=0, −Fhx in (39) can be expressed as follows:(80)−Fhx=−x−x12x−s1x−s2,wheres1=−x1−−2p−2x12 and s2=−x1+−2p−2x12. Substituting (80) into (39), we can obtain the periodic wave solution, which corresponds to the periodic trajectory with Hamilton energy h3=Hx1,0 as follows:(81)uP5=x1−2p+3x122x1+−p−x12cos2p+3x12ξ.(4).
In the case ofΔ<0 and q>0, similar to the analysis in (3), there exist two periodic trajectories centered on P1 inside the homoclinic trajectories in Figure 5, and these two periodic trajectories are divided by the line which corresponds to the Hamilton energy h3=Hx3,0; then, we get the periodic wave solution corresponding to the periodic trajectory with Hamilton energy h3=Hx3,0 as follows:(82)uP6=x3−2p+3x322x3−−p−x32cos2p+3x32ξ.Combining with the above discussions, and according to the relationships between the periodic wave solutions of equations (14) and (20), we can obtain the following theorem.Theorem 5.
(1)
Set Δ ≤ 0 andq≠0. (i) When Δ < 0, the Hamilton energy h satisfies h∈Hx3,0,Hx1,0 or h∈Hx2,0,+∞; (ii) when Δ=0, the Hamilton energy h satisfies h≠Hx2,0 and h≠Hx1,0. If (i) or (ii) establishes, there will exist periodic wave solutions of equation (1) as follows:
(83)
q
P
4
±
ξ
=
u
P
4
±
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
,
whereuP4±ξ are given by (69).
(2)
When Δ < 0 andq=0, the Hamilton energy h>Hx2,0=0, equation (1) has periodic wave solutions:
(84)
z
˜
P
4
±
ξ
=
u
˜
P
4
±
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
,
whereu˜P4±ξ are given by (78).
(3)
When Δ < 0,q<0, and h3=Hx1,0, equation (1) has periodic wave solution:
(85)
z
P
5
ξ
=
u
P
5
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
.
When Δ < 0,q>0, and h3=Hx3,0, equation (1) has periodic wave solution:
(86)
z
P
6
ξ
=
u
P
6
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
,
where bounded periodic solutionsuP5ξ and uP6ξ of equation (14) are given by (81) and (82), respectively.


## 4.1. Periodic Wave Solutions Corresponding to Periodic Trajectories Surrounded by Symmetric Homogeneous Orbits

Whenq=0 and Δ<0, there exist symmetric homoclinic orbits in system (15). Since the Hamilton energy on the same periodic trajectory is equal, we can set the Hamiltonian energy on the same periodic trajectory as follows:(40)Hx,y=Hη1,0=η14+2pη12=η12+p−p2=h1,where η1,0 is the intersection of the periodic trajectory and the x-axis and satisfies η1<−p. According to (16), we get(41)Hx,y=12y2+x2+p2−p2=h1.Substituting three solutionsx1=−−p, x2=0, and x3=−p of the equation fx=x3+px=xx2+p=0 into Hx,0, we have Hx1,0=Hx3,0<Hx2,0=0. Then, we have the −Fhx function curve and periodic trajectories when h1∈−p2,0 as shown in Figure 2.Then, (39) can be transformed into(42)±2dξ=dxp2+h1−x2+p2.(43)p2+h1−x2+p2=−x2+p+p2+h1x2+p−p2+h1,in (42), we let A12=p2+h1−p and B12=−p2+h1−p and make transformation t=x/A1=uξ/A1; then, we have(44)±2dξ=dx−x2−A12x2−B12=dx/A1A11−x2/A12x2/A12−B12/A12=dtA11−t2t2−B12/A12.Figure 2
−
F
h
x function curve and periodic trajectories when q=0 and Δ<0.According to the properties of the elliptic function integral [33, 34], the following lemma can be obtained.Lemma 1.
Ifdt/1−t2t2−k′2=u, then we have t=dnu,k1, k12+k′2=1 (k is the mold). Set k′=B1/A1, A12=−2p/2−k12, and B12=−2p1−k12/2−k12; then by integrating on the both sides of (42), we have(45)uP1±ξ=±−2p2−k12dn−4p2−k12ξ,k1,where k12=2+2p/A12=2p2+h1/p2+h1−p and h1∈−p2,0.
Then, we can obtain the following theorem.Theorem 3.
Whenq=0 and Δ < 0, equation (1) has periodic solutions:(46)zP1±ξ=uP1±ξeiX/6ε+T/108ε2,where uP1±ξ are bounded solutions for equation (14).Here, we point out that the amplitudesuP1±ξ of the periodic wave solutions zP1±ξ for equation (1) correspond to the periodic trajectories, which are contained in the homoclinic trajectory centered on P1 and P3 in the global phase portrait (Figure 1(a)), respectively.


## 4.2. Exact Periodic Wave Solutions Corresponding to Closed Trajectories Contained in Asymmetric Homoclinic Orbits in Figures1(b) and 1(c)

Whenq≠0 and Δ<0, there exist asymmetric homoclinic orbits in system (15). In global phase portraits Figures 1(b) and 1(c), we set the Hamilton energy at the point x,y on the periodic trajectory surrounded by the homoclinic orbits as(47)Hx,y=Hη2,0=h2,where Hx,y is given by (16) and η2,0 is the intersection of the periodic trajectory and the x-axis. From Figures 1(b) and 1(c), we can know that Pixi,0i=1,2,3 are three singular points of system (15), and their abscissas xii=1,2,3 satisfy x1<x2<x3. It can be verified that when q<0, Hx3,0<Hx1,0<Hx2,0; when q>0, Hx1,0<Hx3,0<Hx2,0.When Hamilton energyh2 satisfies(48)maxHx1,0,Hx3,0<h2<Hx2,0,its corresponding −Fhx function curve and periodic trajectories are shown in Figure 3. Set the four different real roots for Fhx=x4+2px2+4qx−h=0 as x1′i=1,2,3,4, and the roots satisfy x1′<x1<x2′<x2<x3′<x3<x4′. So −Fhx can be expressed as follows:(49)Fhx=x−x1′x−x2′x−x3′x−x4′.Figure 3
−
F
h
x function curve and periodic trajectories when q<0,Δ<0.Sincex=uξ,y=u′ξ, the bounded solutions of equation (14) satisfy(50)dudξ2=−2Fhx.(1)
Whenu∈x1′,x2′, let u=x4′−x4′−x1′x4′−x2′/x4′−x2′+x2′−x1′sin2θ, we get(51)Fhx=x−x1′x−x2′x−x3′x−x4′=−x2′−x1′2x4′−x1′2x4′−x2′2sin2θcos2θ⋅x3′−x1′x4′−x2′−x2′−x1′x4′−x3′sin2θx4′−x2′+x2′−x1′sin2θ4.From (39) and (51), we have(52)∫du−Fhx=−2∫2dθx3′−x1′x4′−x2′−x2′−x1′x4′−x3′sin2θ=−2x3′−x1′x4′−x2′∫2dθ1−k2sin2θ=2ξ,wherek2=x2′−x1′x4′−x3′/x1′−x3′x2′−x4′. According to the definition of Jacobi elliptic function [33, 34], we get(53)sinθ=−sn122x1′−x3′x2′−x4′ξ,k2.Then, by substituting (53) into u=x4′−x4′−x1′x4′−x2′/x4′−x2′+x2′−x1′sin2θ, we get the bounded solution of equation (14):(54)uP2ξ=xξ=x4′−x4′−x1′x4′−x2′x4′−x2′+x2′−x1′sn2−1/22x1′−x3′x2′−x4′ξ,k2.(2)
Whenx∈x3′,x4′, let u=x1′+x4′−x1′x3′−x1′/x3′−x1′+x4′−x3′sin2θ, we have(55)Fhx=x−x1′x−x2′x−x3′x−x4′=x4′−x1′2x3′−x1′2x4′−x3′2sin2θcos2θ⋅x2′−x1′x4′−x3′sin2θ−x3′−x1′x4′−x2′x3′−x1′+x4′−x2′sin2θ4.From (39) and (55), we have(56)∫du−Fhx=∫2dθ−x2′−x1′x4′−x3′sin2θ+x3′−x1′x4′−x2′=2ξ,where k2=x2′−x1′x4′−x3′/x1′−x3′x2′−x4′. Then we get(57)sinθ=sn122x3′−x1′x4′−x2′ξ,k2.By substituting (57) into u=x1′+x4′−x1′x3′−x1′/x3′−x1′+x4′−x3′sin2θ, we obtain the bounded solution of equation (14):(58)uP3ξ=x1′+x4′−x1′x3′−x1′x3′−x1′+x4′−x3′sn21/22x3′−x1′x4′−x2′ξ,k2.From the above discussions, we have the following lemma.Lemma 2.
Assumingq≠0 and Δ<0, when Hamilton energy h satisfies (48), equation (14) has two periodic wave solutions uP2ξ and uP3ξ, where uP2ξ∈x1′,x2′ and uP3ξ∈x3′,x4′.Then, according to Lemma2 and equation (11), we can get the following theorem.Theorem 4.
Letq≠0 and Δ < 0, when Hamilton energy h satisfies (48), equation (1) has two periodic wave solutions:(59)zP2ξ=uP2ξeiX/6ε−T/108ε2,(60)zP3ξ=uP3ξeiX/6ε−T/108ε2,where uP2ξ and uP3ξ are given by (54) and (58), respectively.Here, we point out that, in Theorem4, uP2ξ corresponds to the periodic trajectory contained in the homoclinic trajectory with P1 as the center, and uP3ξ corresponds to the periodic trajectory contained in the homoclinic trajectory with P3 as the center in Figures 1(b) and 1(c).


## 4.3. Periodic Wave Solutions Corresponding to the Remaining Periodic Trajectories

Except the periodic solutionszPiξi=1,2,3 found in Sections 4.1 and 4.2, there exist other periodic solutions for equation (1), which correspond to the periodic trajectories surrounded by the homoclinic trajectory in Figures 1(b)–1(e). Here, we set(61)Hx,y=Hη3,0=h3,where η3,0 is the intersection of the periodic trajectory and the x-axis.In the case ofΔ<0 and q<0, when h3 satisfies h3>Hx2,0 or Hx3,0<h3<Hx1,0, the corresponding Y=−Fhx function curve and periodic trajectories are shown in Figure 4. In the case of Δ<0 and q>0, when h3>Hx2,0 or Hx1,0<h3<Hx3,0, the corresponding Y=−Fhx function curve and periodic trajectories have the relationships which are similar to Figure 4.Figure 4
q
<
0 and Δ<0.In addition, in the case ofΔ=0 and q>0, we have Hx1,0<Hx2,0=Hx3,0. And we can give the corresponding Y=−Fhx function curves and periodic trajectories when the Hamilton energy h3 satisfies Hx1,0<h3<Hx2,0, which are shown in Figure 5 (note: in the case of Δ=0 and q<0, we can give the similar diagrams).Figure 5
q
>
0 and Δ=0.As shown in Figures4 and 5, there is a common feature for the above four cases, i.e., the function(62)±Fhx=±x4+2px2+4qx−hhas only two different intersections s1, s2s1<s2 with the x-axis. Then, we can express Fhx as(63)Fhx=x−s1x−s2x−α2+β2,where 2α+s1+s2=0, β>0. Then, we divide (63) into the following parts:(64)x−α2+β2=ε1x−r12+1−ε1x−r22,x−s1x−s2=ε2x−r12+1−ε2x−r22,where ri=α1−λi/1+λi,i=1,2, ε1=λ21+λ1/λ2−λ1, ε2=1+λ1/λ2−λ1, and λ1,λ2 are two real roots of the equation(65)s1−s22λ2−4β2+α−s1α−s2λ−4β2=0.Sincex=uξ,y=u′ξ, by substituting s=x−r1/x−r2=uξ−r1/uξ−r2 into (39), we get(66)±2ξ=1r1−r2ε1ε2∫dss2+1−ε1/ε2s2−1+ε2/ε2.Then, lets2=1+ε2/ε21−t2 and m2=ε21−ε1/ε1+ε2, we have(67)±2ε1+ε2r1−r2ξ=∫dt1−t21−m2t2.By using the Jacobi elliptic function integral [33, 34], we get(68)uξ=r1−r1−r21±ε2/1+ε2cn2ε1+ε2r1−r2ξ,m.After simplification, we have the bounded solutions of equation (14) as follows:(69)uP4±ξ=1−λ11+λ1α−σ2α±2α1+λ1/1+λ2cn2σξ,m,where σ=β2+α−s12β2+α−s22, λ1<λ2, and m satisfies(70)m2=12−β2+α−s1α−s22β2+α−s12β2+α−s22.Here, we point out that(1).
In the case ofΔ≤0 and q≠0, we have(71)x4+2px2+4qx−h3=x−s1x−s2x−α2+β2,from (63). Then we expand the right side of the above equation, and let the coefficients of x4,x3,x2,x be the same, we have(72)s1+s2+2α=0,s1s2+2s1α+2s2α+α2+β2=2p,2s1s2α+s1α2+s2α2+s1β2+s2β2=4q.(a)
Whenm2>1/2, which is equivalent to β2+α−s1α−s2<0, since β2+α−s1α−s2=β2+3α2+s1s2=2p+6α2<0, we have α2=s1+s2/22<−p/3, where sii=1,2 and xii=1,2,3 satisfy(73)s1<x1=−x0−−3x0−4p2<x2=x0<x3=−x0+−3x0−4p2<s2.At this time, there exist periodic trajectories surrounding the homoclinic trajectories in system (15).(b)
Whenm2<1/2, which is equivalent to β2+α−s1α−s2>0, we can get s1+s2/22>−p/3. Now sii=1,2 and xii=1,2,3 have the following relationship:(74)x1=x0<x2=−x0−−3x0−4p2<s1<s2<x3=−x0+−3x0−4p2,and there exist periodic trajectories contained in the homoclinic trajectories in system (15).(2).
In the case ofΔ<0 and q=0, when h3>Hx2,0=0, we have(75)−Fhx=x2+p+p2+h3x2−p2+h3−p.Sincep+p2+h3>0, equation x2+p+p2+h3=0 has no real root. By substituting (75) into (39), we get(76)2ξ=∫dxx2+p+p2+h3x2−p2+h3−p.Set−p2+h3−p=p1, p2+h3−p=p2, by using transformation x2=p2/1−t2, we have(77)2ξ=∫dtp1+p21−t21−p1/p1+p2t2.According to the Jacobi elliptic function integral [33, 34], we have the bounded periodic solution of equation (14) as follows:(78)u˜P4±=±−2pk22k2−1cn−4p2k2−1ξ,k,wherek2=−p+p2+h3/2p2+h3∈1/2,1. Here, u˜P4±ξ correspond to the periodic trajectories surrounding the homoclinic trajectory in the global phase portrait Figure 1(a). At this time, equation (1) has periodic wave solutions of the form (2) as follows:(79)z˜P4±ξ=u˜P4±ξeiX/6ε+T/108ε2.(3).
In the case ofΔ<0 and q<0, it can be known from Figure 4 that there exist two periodic trajectories centered on P3 inside the homoclinic trajectories (correspond to the periodic wave solutions uP3 and uP4 of equation (14), respectively), and these two trajectories are divided by the line which corresponds to the Hamilton energy h3=Hx1,0. When Hamilton energy h=Hx1,0, since Hx1,0=x14+2px12+4qx1, and the abscissa of singularity Px1,0 of system (15) satisfies x13+px1+q=0, −Fhx in (39) can be expressed as follows:(80)−Fhx=−x−x12x−s1x−s2,wheres1=−x1−−2p−2x12 and s2=−x1+−2p−2x12. Substituting (80) into (39), we can obtain the periodic wave solution, which corresponds to the periodic trajectory with Hamilton energy h3=Hx1,0 as follows:(81)uP5=x1−2p+3x122x1+−p−x12cos2p+3x12ξ.(4).
In the case ofΔ<0 and q>0, similar to the analysis in (3), there exist two periodic trajectories centered on P1 inside the homoclinic trajectories in Figure 5, and these two periodic trajectories are divided by the line which corresponds to the Hamilton energy h3=Hx3,0; then, we get the periodic wave solution corresponding to the periodic trajectory with Hamilton energy h3=Hx3,0 as follows:(82)uP6=x3−2p+3x322x3−−p−x32cos2p+3x32ξ.Combining with the above discussions, and according to the relationships between the periodic wave solutions of equations (14) and (20), we can obtain the following theorem.Theorem 5.
(1)
Set Δ ≤ 0 andq≠0. (i) When Δ < 0, the Hamilton energy h satisfies h∈Hx3,0,Hx1,0 or h∈Hx2,0,+∞; (ii) when Δ=0, the Hamilton energy h satisfies h≠Hx2,0 and h≠Hx1,0. If (i) or (ii) establishes, there will exist periodic wave solutions of equation (1) as follows:
(83)
q
P
4
±
ξ
=
u
P
4
±
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
,
whereuP4±ξ are given by (69).
(2)
When Δ < 0 andq=0, the Hamilton energy h>Hx2,0=0, equation (1) has periodic wave solutions:
(84)
z
˜
P
4
±
ξ
=
u
˜
P
4
±
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
,
whereu˜P4±ξ are given by (78).
(3)
When Δ < 0,q<0, and h3=Hx1,0, equation (1) has periodic wave solution:
(85)
z
P
5
ξ
=
u
P
5
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
.
When Δ < 0,q>0, and h3=Hx3,0, equation (1) has periodic wave solution:
(86)
z
P
6
ξ
=
u
P
6
ξ
e
i
X
/
6
ε
−
T
/
108
ε
2
,
where bounded periodic solutionsuP5ξ and uP6ξ of equation (14) are given by (81) and (82), respectively.


## 5. Relationships between the Solitary Wave Solutions and Periodic Wave Solutions of Equation (1)

In this paper, the two forms of bell-shaped solitary wave solutions for equation (1) have been given in Section 3, and the exact periodic wave solutions of equation (1) under different parameters are obtained in Section 4. In this section, we will study the evolutionary relationships between the solitary wave solutions and the periodic wave solutions for equation (1) and reveal the influence by the energy variation of the Hamilton system on the waveform of the solutions.


### 5.1. The Relationships between Periodic Wave Solutions, Solitary Wave Solutions for Equation (1), and the Value of Hamilton Energy of System (15)

The planar dynamic system (15) is a Hamilton system, and the points on the same trajectory have the same Hamilton energy. When the parameters are fixed, for any h∈R, the trajectory Ch=x,y∈R×RHx,y=h with energy h in the global phase portrait is unique for −Fhx function curve in (39), where −Fhx=−x4−2px2−4qx+h.We take the global phase portrait Figure1(b) as an example and discuss the relationships between the traveling wave solutions of the form (2) and the value which the Hamilton energy takes.We will discuss in the following three cases:h=Hx2,0, h>Hx2,0, and h<Hx2,0.As shown in Figure6, these three cases correspond to the three types of trajectories in system (15) in the case of Δ<0 and q<0: the homoclinic trajectory, the periodic trajectory surrounding the homoclinic trajectory, and the periodic trajectory contained in the same trajectory. From the discussions in Sections 3 and 4, we know that in Figure 6, the homoclinic trajectories correspond to the bell-shaped solutions uS1±ξ of equation (14), and the periodic trajectories surrounding the homoclinic trajectories correspond to the periodic solutions uP4+ξ (modulo m2<1/2) of equation (14), and the periodic trajectories contained in the homoclinic trajectories correspond to the periodic solutions uP2ξ and uP3ξ. Then, we can obtain that, in the case of Δ<0 and q<0, the relationships between the value of Hamilton energy h and the solitary wave solutions, periodic wave solutions of equation (1) are as follows:(1)
Whenh≤Hx3,0, there is no bounded trajectory in system (15). At this time, the S-S equation (1) has no bounded traveling wave solution of the form (2).(2)
Whenh∈Hx3,0,Hx1,0, there is a periodic trajectory centered on P3 and contained in the homoclinic orbit in system (15). At this time, the S-S equation (1) has a cluster of periodic wave solution zP4+ξ=uP4+ξeiX/6ε−T/108ε2 of the form (2) (modulo m2<1/2), where uP4+xi is given by (69).(3)
Whenh∈Hx1,0,Hx2,0, there are periodic trajectories centered on P1 and P3 and contained in the homoclinic trajectory in system (15). And there exist periodic wave solutions zP2ξ=uP2ξeiX/6ε−T/108ε2 and zP3ξ=uP3ξeiX/6ε−T/108ε2 of the form (2) of the S-S equation (1), where uP2ξ and uP3ξ are given by (54) and (58), respectively. And the amplitudes of the solutions are centered on P1 and P3, respectively.(4)
Whenh=Hx2,0, there are two homoclinic trajectories in system (15). At this time, the S-S equation (1) has two bell-shaped solitary wave solutions zS1±ξ=uS1±ξeiX/6ε−T/108ε2 of the form (2), where uS1±ξ are given by (32).(5)
Whenh>Hx2,0, there exist periodic trajectories surrounding the homoclinic trajectory in system (15). And the S-S equation (1) has a cluster of periodic wave solution zP4+ξ=uP4+ξeiX/6ε−T/108ε2 (modulo m2>1/2) of the form (2), where uP4+ξ is given by (69).Figure 6
Diagram of−Fhx function curve and trajectories when Δ<0,q<0.From the above discussions, we obtain that, in the case ofΔ<0 and q<0, when the Hamilton energy h takes different value near Hx2,0, by the first integral (16) of system (15), we can give the bell-shaped solutions or periodic wave solutions for equation (14). Then, the solitary wave solutions or periodic wave solutions of equation (1) can be obtained. For other cases of Δ and q, we can also make similar discussions. From the above analysis, we can get the conclusion that when Hamilton energy h takes different value, equation (1) has different kinds of solutions.


### 5.2. The Relationships between Periodic Wave Solutions, Solitary Wave Solutions of Equation (1), and the Limit of Hamilton Energy in System (15)

#### 5.2.1. The Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of the Symmetric Homoclinic Trajectories

WhenΔ<0 and q=0, the global phase portrait Figure 1(a) with symmetric homoclinic orbit is given by the qualitative analysis in Section 2. In Section 3, the solitary wave solutions z˜S1±ξ of the form (2) of equation (1) which correspond to the symmetric homoclinic orbits are given by (36) and their amplitudes u˜S1±ξ=±−2psech−4pξ correspond to the homoclinic trajectories in the global phase portrait Figure 1(a) (with Hamiltonian energy h1=Hx2,0=0). In Section 4.1, the periodic wave solutions zP1±ξ of the form (2) for equation (1) are given which correspond to the homoclinic trajectories inside the closed orbit. And their amplitudes uP1±ξ are given by (45). In Section 4.3, the periodic wave solutions z˜P4±ξ of equation (1) are given, which correspond to the closed orbits surrounding the homoclinic orbit, and their amplitudes u˜P4±ξ are given by (78).Whenh1 approaches the Hamilton energy Hx2,0=0 which corresponds to the solitary wave solution, we have limh1⟶0−k12=2p2/p2−p=1. According to (45), we can obtain(87)limh1⟶0−uP1±ξ=limk1⟶1±−2p2−k12dn−4p2−k12ξ,k1=±−2pdn−4pξ,1=±−2psech−4pξ=u˜S1±ξ.So we havelimh1⟶0−zP1±ξ=z˜S1±ξ.This indicates that whenh1⟶0−, the periodic wave solutions converge to the solitary wave solutions; here, periodic wave solutions correspond to the closed trajectories contained in the symmetric homoclinic trajectories, and solitary wave solutions correspond to the homoclinic trajectories.According to (78), when h3⟶0+, we have k⟶1. So(88)limh3⟶0+u˜P4±ξ=limk1⟶1u˜P4±ξ=±−2pcn−4pξ,1=±−2psech−4pξ=u˜S1±ξ.Thus, we havelimh3⟶0+zP4±ξ=z˜S1±ξ, the periodic solutions z˜P4±ξ take the solitary wave solutions z˜S1±ξ as the limit, where z˜P4±ξ correspond to the closed trajectories that surround the symmetric homoclinic trajectories and z˜S1±ξ correspond to the homoclinic trajectories. Therefore, we have the following theorem.Theorem 6.
In the case of Δ < 0 andq=0, when the Hamilton energy h1 of system (15) tends to be 0−, the periodic trajectories centered on P1 and P3 in Figure 1(a) will expand into homoclinic trajectories, and the corresponding periodic wave solutions zP1±ξ of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z˜S1±ξ; when the Hamilton energy h3 of the system gradually tends to be 0+, the periodic trajectories surrounding the symmetric homoclinic trajectories in Figure 1(a) will shrink into homoclinic trajectories, and the corresponding periodic wave solutions z˜P4±ξ of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z˜S1±ξ.


#### 5.2.2. The Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of Asymmetric Homoclinic Trajectories

According to the analysis of the global phase portraits under different parameters for system (15) in Section 2, we know that there exist two asymmetric homoclinic trajectories in system (15) when Δ<0,q<0 or Δ<0,q>0. Since the analyses of the two cases are the same, we only need to consider the case of Δ<0,q<0.When the Hamilton energyh=Hx2,0, there exist homoclinic trajectories centered on P1 and P3 in system (15), which correspond to the bell-shaped solutions zS1±ξ of the form (2) for equation (1), respectively; when h∈Hx1,0,Hx2,0, a cluster of periodic trajectories exist inside the homoclinic trajectories centered on P1 or P3, which correspond to the periodic wave solutions zP2ξ and zP3ξ of the form (2) for equation (1); when h>Hx2,0, there exist period trajectories surrounding the homoclinic trajectory in system (15), which correspond to the periodic wave solutions zP4+ξm2>1/2 of the form (2) for equation (1).Here we first consider the limit relationships between the periodic wave solutionszP2ξ, zP3ξ, and the solitary wave solutions zS1±ξ.From qualitative analysis, we know that system (15) has three different singular points Pixi,0i=1,2,3, where x1, x2, and x3 can be expressed as follows:(89)x1=−x0−−3x02−4p2,x2=x0,x3=−x0+−3x22−4p2.As shown in Figure6, when h=Hx2,0, the abscissas eii=1,2,3,4 of the intersections of the function curve Y=−Fhx and the x-axis can be expressed as follows:(90)e1=−x2−−2p−2x22,e2=e3=x2,e4=−x2+−2p−2x22.It is easy to know thateii=1,2,3,4 and xii=1,2,3 satisfy e1<x1<e2=e3=x2<x3<e4, and the abscissas x1′i=1,2,3,4 of the intersections of the periodic trajectory contained in the homoclinic trajectory and the x-axis in Figure 6 can make the equation(91)−Fhx=−x4−2px2−4qx+h=0establish, where e1<x1′<x2′<e2=e3<x3′<x4′<e4.Because the figure of functionY=−Fhx is a continuous polynomial curve with respect to h, the zero point x1′ of −Fhx=0 has continuity about h. When h⟶H−x2,0 and x1′⟶e1, we have x2′⟶e2=x2, x3′⟶e3=x2, x4′⟶e4, and modulo k⟶1. Thus, according to (54) and (32), we have(92)limh⟶H−x2,0uP2ξ=limk⟶1uP2ξ=e4−e4−e1e4−e2e4−e2+e2−e1sn2−e1−e3e2−e4ξ,1=x2−2p+3x222x2−−p−x22cosh−p+3x22ξ=uS1−ξ.Similarly, we can provelimh⟶H−x2,0uP3ξ=uS1+ξ. Therefore, when h⟶H−x2,0, the periodic wave solutions zP2ξ and zP3ξ evolve into the bell-shaped solitary wave solutions zS1−ξ and zS1+ξ, respectively, where zP2ξ and zP3ξ correspond to the periodic trajectories contained in the homoclinic trajectories.Next, we consider the relationships between the periodic wave solutionszP4+ξ and zS1±ξ, which correspond to the periodic trajectories surrounding the asymmetric homoclinic trajectories.From Figure6, we can know that the periodic trajectory corresponding to the Y=−Fhx function curve in the case of h>Hx2,0 surrounds the homoclinic trajectory corresponding to the −Fhx function curve in the case of h=Hx2,0. Set the abscissas of the intersections of the periodic trajectory and the x-axis as s1 and s2 when h>Hx2,0, and set them as eii=1,2,3 when h=Hx2,0. s1, s2, and eii=1,2,3 have the following relationship:(93)s1<e1<e2=e3=x2<e4<s2.Whenh⟶H+x2,0, we have s1⟶−x0−−2p−2x02, s2⟶−x0+−2p−2x02, α=−1/2s1+s2⟶x2, β⟶0, m⟶1, λ1⟶0, λ2⟶−p+3x22/p+x22, and σ⟶−2p−6x22, so we can obtain(94)limh⟶H+x2,0uP4+ξ=limm⟶11−λ11+λ1α−σ2α+2α1+λ1/1+λ2cn2σξ,m=x2−2p+3x222x2±−p−x22cosh2−p+3x22ξ=uS1±ξ,that is, the periodic trajectory surrounding the homoclinic trajectory contracts into the homoclinic trajectory. So we can deduce that the periodic wave solution zP4+ξ of the form (2) for equation (1) evolves into the bell-shaped solitary wave solutions zS1±ξ.Theorem 7.
Assume Δ < 0,q<0, when Hamilton energy h⟶H−x2,0, the periodic wave solutions zP2ξ and zP3ξ of the form (2) corresponding to the periodic trajectories contained in the homoclinic trajectory evolve into bell-shaped solutions zS1−ξ and zS1+ξ, respectively. When Hamilton energy h⟶H+x2,0, the periodic wave solution zP4+ξm2>1/2 of the form (2) corresponding to the periodic trajectory in the homoclinic trajectory evolves into the bell-shaped solutions zS1±ξ.


#### 5.2.3. The Relationships between the Traveling Wave Solutions inside or outside a Single Homoclinic Orbit and the Limit of Hamilton Energy

Now we take the case ofΔ=0 and q>0 as an example. In this case, as shown in Figure 5 and according to the above solving process of the solitary wave solutions and the periodic wave solutions, we know that when h=Hx2,0, homoclinic trajectory with P2,3 as a cusp and P1 as a center in system (15) corresponds to the bell-shape solution zS2+ξ. When h<Hx2,0h>Hx2,0, there exists the periodic trajectory contained in(surrounding) the homoclinic trajectory in system (15), and it corresponds to the periodic wave solution zP4−ξ of the form (2) for equation (1), whose modulo satisfies m2<1/2m2>1/2.Whenh=Hx2,0, according to the Hamilton energy function, we have the following:(1)
The periodic trajectory contained in the homoclinic trajectory has two intersectionss1 and s2 with the x-axis, where s1 and s2 satisfy(95)−3−p3<s1<x1=−2−p3<s2<x2,3=−p3.(2)
The periodic trajectory surrounding the same trajectory has two intersectionss1 and s2 with the x-axis, where s1 and s2 satisfy(96)s1<−3−p3<x1=−2−p3<x2,3=−p3<s2.For the periodic trajectory contained in the homoclinic trajectory, as shown in Figure5, when Hamilton energy h⟶H−x2,0, we have s1⟶−3−p/3, s2⟶−p/3, α⟶−p/3, β⟶0, and m⟶0, so(97)limh⟶H−x2,0uP4−ξ=limm⟶0uP4−ξ=uS2+ξ.At this time, the periodic wave solutionzP4−ξm2<1/2 of the form (2) for equation (1) evolves into the bell-shaped solitary solution z2+ξ.For the periodic trajectory surrounding the homoclinic trajectory, it is known from Figure5 that when the Hamilton energy h⟶H+x2,0, we have s1⟶−3−p/3 and s2⟶−p/3, then α⟶−p/3, β⟶0, and m⟶1. By calculating, we can obtain(98)limh⟶H+x2,0uP4−ξ=limm⟶1uP4−ξ=uS2+ξ.In Figure5, it can be seen that the periodic trajectory surrounding the homoclinic trajectory contracts into a homoclinic trajectory, which corresponds to the evolutionary process from periodic wave solution zP4−ξm2>1/2 to the bell-shaped solitary wave solution zS2+ξ. Therefore, we can obtain the following theorem.Theorem 8.
When Δ = 0,q>0, and Hamilton energy h⟶H−x2,0, the periodic wave solution zP4−ξm2>1/2 of equation (1) corresponding to the periodic trajectory contained in the homoclinic trajectory in system (15) will evolve into the bell-shaped solution zS2+ξ. When the Hamilton energy h⟶H+x2,0, the periodic wave solution zP4−ξm2>1/2 of equation (1), which corresponds to the periodic trajectory surrounding the homoclinic trajectory, will evolve into the bell-shaped solution zS2+ξ.


### 5.3. Diagrams of the Relationships between Hamilton Energy and Traveling Wave Solutions

In the last section, we have discussed the evolutionary relationships between the periodic wave solutions and the solitary wave solutions of equation (1), when the Hamilton energy of system (15) approaches a certain limit. In this section, we will give examples of 3D diagrams to intuitively demonstrate the relationships between these two solutions: When the Hamilton energy changes, the amplitudes uPiξi=1,2,3,4 of the periodic wave solutions for equation (1) will evolve into the amplitudes uSiξi=1,2 of the bell-shaped solutions for equation (1).


#### 5.3.1. The Evolution from Periodic Solutions to Solitary Wave Solutions Caused by the Variation of Hamiltonian Energy in the Case of Δ < 0 andq<0

The case ofΔ<0 and q<0 is shown in the global phase portrait Figure 1(b). When h=Hx2,0, equation (1) has bell-shaped solitary wave solutions zS1±ξ of the form (2). When Hx1,0<h<Hx2,0, equation (1) has periodic wave solutions zP2ξ and zP3ξ, which satisfy limh⟶H−x2,0zP2ξ=zS1−ξ and limh⟶H−x2,0zP3ξ=zS1+ξ. When h>Hx2,0, equation (1) has a periodic wave solution zP4+ξm2>1/2 of the form (2), which satisfies limh⟶H+x2,0zP4+ξ=zS1±ξ. Due to the space limitation, we only give the diagrams of the evolution from the amplitude uP3ξ of the periodic wave solution zP3ξ to the amplitude uS1+ξ of the solitary wave solution zS1+ξ.Here, we takep=−7, q=−6, c=25/12, and x2=−1, and then, we have Hx2,0=11. When Hamilton energy h⟶H−x2,0, the amplitude of the periodic wave solution for equation (1) satisfies uP3ξ⟶uS1+ξ. The corresponding diagrams are shown in Figures 7(a)–7(d).Figure 7
(a)h=10.99,uP3ξ, (b)h=10.9999,uP3ξ, (c)h=10.999999,uP3ξ, and (d)h=Hx2,0,uS1+ξ.
(a)
(b)
(c)
(d)


#### 5.3.2. The Evolution from Periodic Solutions to Solitary Wave Solutions Caused by the Variation of Hamilton Energy in the Case ofΔ=0 and q>0

The case ofp<0,Δ=0,q>0 corresponds to the global phase portrait Figure 1(e). When h=Hx2,0, equation (1) has a bell-shape solution zS2+ξ in the form of (2). When Hx1,0<h<Hx2,0, equation (1) has a periodic wave solution zP4−ξm2>1/2 in the form of (2), which satisfies limh⟶H−x2,0zP4−ξ=zS2+ξ. When h>Hx2,0, equation (1) has a periodic wave solution zP4−ξm2>1/2 of the form (2), which satisfies limh⟶H+x2,0zP4−ξ=zS2+ξ. Due to the space limitation, here we only give the diagrams of the evolution from the amplitude uP4−ξ of the periodic wave solution zP4−ξm2>1/2 to the amplitude uS2+ξ of solitary wave solution zS2+ξ.Takep=−3, q=2, x2=2, and c=4/3, we have Hx2,0=3. When Hamilton energy h⟶H+x2,0, the diagrams of the amplitude uP4−ξm2>1/2⟶uS2+ξ of the traveling wave solution for equation (1) are shown in Figures 8(a)–8(d) (for the sake of beauty, we adjusted the positive and negative axes of the vertical direction of the graphic).Figure 8
(a)h=3.05,uP4−ξ,m2>1/2, (b)h=3.03,uP4−ξ,m2>1/2, and (c)h=3.01,uP4−ξ,m2>1/2 and (d)h=Hx2,0,uS2+ξ.
(a)
(b)
(c)
(d)It can be seen that by the qualitative analysis and the first integral method, we not only obtain all the unimodal bell-shaped solitary wave solutions and the periodic wave solutions of the S-S equation (1), but also show the mathematical meaning of the obtained solutions, i.e., the relationships between solutions and bounded trajectories. Moreover, we reveal the relationships between the solitary wave solutions, the periodic wave solutions of the S-S equation, and the energy h of the Hamilton system.For scholars who want to know more about the application of Hamilton energy in the research of solitary wave solution, refer to literatures [35–37].


## 5.1. The Relationships between Periodic Wave Solutions, Solitary Wave Solutions for Equation (1), and the Value of Hamilton Energy of System (15)

The planar dynamic system (15) is a Hamilton system, and the points on the same trajectory have the same Hamilton energy. When the parameters are fixed, for any h∈R, the trajectory Ch=x,y∈R×RHx,y=h with energy h in the global phase portrait is unique for −Fhx function curve in (39), where −Fhx=−x4−2px2−4qx+h.We take the global phase portrait Figure1(b) as an example and discuss the relationships between the traveling wave solutions of the form (2) and the value which the Hamilton energy takes.We will discuss in the following three cases:h=Hx2,0, h>Hx2,0, and h<Hx2,0.As shown in Figure6, these three cases correspond to the three types of trajectories in system (15) in the case of Δ<0 and q<0: the homoclinic trajectory, the periodic trajectory surrounding the homoclinic trajectory, and the periodic trajectory contained in the same trajectory. From the discussions in Sections 3 and 4, we know that in Figure 6, the homoclinic trajectories correspond to the bell-shaped solutions uS1±ξ of equation (14), and the periodic trajectories surrounding the homoclinic trajectories correspond to the periodic solutions uP4+ξ (modulo m2<1/2) of equation (14), and the periodic trajectories contained in the homoclinic trajectories correspond to the periodic solutions uP2ξ and uP3ξ. Then, we can obtain that, in the case of Δ<0 and q<0, the relationships between the value of Hamilton energy h and the solitary wave solutions, periodic wave solutions of equation (1) are as follows:(1)
Whenh≤Hx3,0, there is no bounded trajectory in system (15). At this time, the S-S equation (1) has no bounded traveling wave solution of the form (2).(2)
Whenh∈Hx3,0,Hx1,0, there is a periodic trajectory centered on P3 and contained in the homoclinic orbit in system (15). At this time, the S-S equation (1) has a cluster of periodic wave solution zP4+ξ=uP4+ξeiX/6ε−T/108ε2 of the form (2) (modulo m2<1/2), where uP4+xi is given by (69).(3)
Whenh∈Hx1,0,Hx2,0, there are periodic trajectories centered on P1 and P3 and contained in the homoclinic trajectory in system (15). And there exist periodic wave solutions zP2ξ=uP2ξeiX/6ε−T/108ε2 and zP3ξ=uP3ξeiX/6ε−T/108ε2 of the form (2) of the S-S equation (1), where uP2ξ and uP3ξ are given by (54) and (58), respectively. And the amplitudes of the solutions are centered on P1 and P3, respectively.(4)
Whenh=Hx2,0, there are two homoclinic trajectories in system (15). At this time, the S-S equation (1) has two bell-shaped solitary wave solutions zS1±ξ=uS1±ξeiX/6ε−T/108ε2 of the form (2), where uS1±ξ are given by (32).(5)
Whenh>Hx2,0, there exist periodic trajectories surrounding the homoclinic trajectory in system (15). And the S-S equation (1) has a cluster of periodic wave solution zP4+ξ=uP4+ξeiX/6ε−T/108ε2 (modulo m2>1/2) of the form (2), where uP4+ξ is given by (69).Figure 6
Diagram of−Fhx function curve and trajectories when Δ<0,q<0.From the above discussions, we obtain that, in the case ofΔ<0 and q<0, when the Hamilton energy h takes different value near Hx2,0, by the first integral (16) of system (15), we can give the bell-shaped solutions or periodic wave solutions for equation (14). Then, the solitary wave solutions or periodic wave solutions of equation (1) can be obtained. For other cases of Δ and q, we can also make similar discussions. From the above analysis, we can get the conclusion that when Hamilton energy h takes different value, equation (1) has different kinds of solutions.


## 5.2. The Relationships between Periodic Wave Solutions, Solitary Wave Solutions of Equation (1), and the Limit of Hamilton Energy in System (15)

### 5.2.1. The Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of the Symmetric Homoclinic Trajectories

WhenΔ<0 and q=0, the global phase portrait Figure 1(a) with symmetric homoclinic orbit is given by the qualitative analysis in Section 2. In Section 3, the solitary wave solutions z˜S1±ξ of the form (2) of equation (1) which correspond to the symmetric homoclinic orbits are given by (36) and their amplitudes u˜S1±ξ=±−2psech−4pξ correspond to the homoclinic trajectories in the global phase portrait Figure 1(a) (with Hamiltonian energy h1=Hx2,0=0). In Section 4.1, the periodic wave solutions zP1±ξ of the form (2) for equation (1) are given which correspond to the homoclinic trajectories inside the closed orbit. And their amplitudes uP1±ξ are given by (45). In Section 4.3, the periodic wave solutions z˜P4±ξ of equation (1) are given, which correspond to the closed orbits surrounding the homoclinic orbit, and their amplitudes u˜P4±ξ are given by (78).Whenh1 approaches the Hamilton energy Hx2,0=0 which corresponds to the solitary wave solution, we have limh1⟶0−k12=2p2/p2−p=1. According to (45), we can obtain(87)limh1⟶0−uP1±ξ=limk1⟶1±−2p2−k12dn−4p2−k12ξ,k1=±−2pdn−4pξ,1=±−2psech−4pξ=u˜S1±ξ.So we havelimh1⟶0−zP1±ξ=z˜S1±ξ.This indicates that whenh1⟶0−, the periodic wave solutions converge to the solitary wave solutions; here, periodic wave solutions correspond to the closed trajectories contained in the symmetric homoclinic trajectories, and solitary wave solutions correspond to the homoclinic trajectories.According to (78), when h3⟶0+, we have k⟶1. So(88)limh3⟶0+u˜P4±ξ=limk1⟶1u˜P4±ξ=±−2pcn−4pξ,1=±−2psech−4pξ=u˜S1±ξ.Thus, we havelimh3⟶0+zP4±ξ=z˜S1±ξ, the periodic solutions z˜P4±ξ take the solitary wave solutions z˜S1±ξ as the limit, where z˜P4±ξ correspond to the closed trajectories that surround the symmetric homoclinic trajectories and z˜S1±ξ correspond to the homoclinic trajectories. Therefore, we have the following theorem.Theorem 6.
In the case of Δ < 0 andq=0, when the Hamilton energy h1 of system (15) tends to be 0−, the periodic trajectories centered on P1 and P3 in Figure 1(a) will expand into homoclinic trajectories, and the corresponding periodic wave solutions zP1±ξ of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z˜S1±ξ; when the Hamilton energy h3 of the system gradually tends to be 0+, the periodic trajectories surrounding the symmetric homoclinic trajectories in Figure 1(a) will shrink into homoclinic trajectories, and the corresponding periodic wave solutions z˜P4±ξ of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z˜S1±ξ.


### 5.2.2. The Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of Asymmetric Homoclinic Trajectories

According to the analysis of the global phase portraits under different parameters for system (15) in Section 2, we know that there exist two asymmetric homoclinic trajectories in system (15) when Δ<0,q<0 or Δ<0,q>0. Since the analyses of the two cases are the same, we only need to consider the case of Δ<0,q<0.When the Hamilton energyh=Hx2,0, there exist homoclinic trajectories centered on P1 and P3 in system (15), which correspond to the bell-shaped solutions zS1±ξ of the form (2) for equation (1), respectively; when h∈Hx1,0,Hx2,0, a cluster of periodic trajectories exist inside the homoclinic trajectories centered on P1 or P3, which correspond to the periodic wave solutions zP2ξ and zP3ξ of the form (2) for equation (1); when h>Hx2,0, there exist period trajectories surrounding the homoclinic trajectory in system (15), which correspond to the periodic wave solutions zP4+ξm2>1/2 of the form (2) for equation (1).Here we first consider the limit relationships between the periodic wave solutionszP2ξ, zP3ξ, and the solitary wave solutions zS1±ξ.From qualitative analysis, we know that system (15) has three different singular points Pixi,0i=1,2,3, where x1, x2, and x3 can be expressed as follows:(89)x1=−x0−−3x02−4p2,x2=x0,x3=−x0+−3x22−4p2.As shown in Figure6, when h=Hx2,0, the abscissas eii=1,2,3,4 of the intersections of the function curve Y=−Fhx and the x-axis can be expressed as follows:(90)e1=−x2−−2p−2x22,e2=e3=x2,e4=−x2+−2p−2x22.It is easy to know thateii=1,2,3,4 and xii=1,2,3 satisfy e1<x1<e2=e3=x2<x3<e4, and the abscissas x1′i=1,2,3,4 of the intersections of the periodic trajectory contained in the homoclinic trajectory and the x-axis in Figure 6 can make the equation(91)−Fhx=−x4−2px2−4qx+h=0establish, where e1<x1′<x2′<e2=e3<x3′<x4′<e4.Because the figure of functionY=−Fhx is a continuous polynomial curve with respect to h, the zero point x1′ of −Fhx=0 has continuity about h. When h⟶H−x2,0 and x1′⟶e1, we have x2′⟶e2=x2, x3′⟶e3=x2, x4′⟶e4, and modulo k⟶1. Thus, according to (54) and (32), we have(92)limh⟶H−x2,0uP2ξ=limk⟶1uP2ξ=e4−e4−e1e4−e2e4−e2+e2−e1sn2−e1−e3e2−e4ξ,1=x2−2p+3x222x2−−p−x22cosh−p+3x22ξ=uS1−ξ.Similarly, we can provelimh⟶H−x2,0uP3ξ=uS1+ξ. Therefore, when h⟶H−x2,0, the periodic wave solutions zP2ξ and zP3ξ evolve into the bell-shaped solitary wave solutions zS1−ξ and zS1+ξ, respectively, where zP2ξ and zP3ξ correspond to the periodic trajectories contained in the homoclinic trajectories.Next, we consider the relationships between the periodic wave solutionszP4+ξ and zS1±ξ, which correspond to the periodic trajectories surrounding the asymmetric homoclinic trajectories.From Figure6, we can know that the periodic trajectory corresponding to the Y=−Fhx function curve in the case of h>Hx2,0 surrounds the homoclinic trajectory corresponding to the −Fhx function curve in the case of h=Hx2,0. Set the abscissas of the intersections of the periodic trajectory and the x-axis as s1 and s2 when h>Hx2,0, and set them as eii=1,2,3 when h=Hx2,0. s1, s2, and eii=1,2,3 have the following relationship:(93)s1<e1<e2=e3=x2<e4<s2.Whenh⟶H+x2,0, we have s1⟶−x0−−2p−2x02, s2⟶−x0+−2p−2x02, α=−1/2s1+s2⟶x2, β⟶0, m⟶1, λ1⟶0, λ2⟶−p+3x22/p+x22, and σ⟶−2p−6x22, so we can obtain(94)limh⟶H+x2,0uP4+ξ=limm⟶11−λ11+λ1α−σ2α+2α1+λ1/1+λ2cn2σξ,m=x2−2p+3x222x2±−p−x22cosh2−p+3x22ξ=uS1±ξ,that is, the periodic trajectory surrounding the homoclinic trajectory contracts into the homoclinic trajectory. So we can deduce that the periodic wave solution zP4+ξ of the form (2) for equation (1) evolves into the bell-shaped solitary wave solutions zS1±ξ.Theorem 7.
Assume Δ < 0,q<0, when Hamilton energy h⟶H−x2,0, the periodic wave solutions zP2ξ and zP3ξ of the form (2) corresponding to the periodic trajectories contained in the homoclinic trajectory evolve into bell-shaped solutions zS1−ξ and zS1+ξ, respectively. When Hamilton energy h⟶H+x2,0, the periodic wave solution zP4+ξm2>1/2 of the form (2) corresponding to the periodic trajectory in the homoclinic trajectory evolves into the bell-shaped solutions zS1±ξ.


### 5.2.3. The Relationships between the Traveling Wave Solutions inside or outside a Single Homoclinic Orbit and the Limit of Hamilton Energy

Now we take the case ofΔ=0 and q>0 as an example. In this case, as shown in Figure 5 and according to the above solving process of the solitary wave solutions and the periodic wave solutions, we know that when h=Hx2,0, homoclinic trajectory with P2,3 as a cusp and P1 as a center in system (15) corresponds to the bell-shape solution zS2+ξ. When h<Hx2,0h>Hx2,0, there exists the periodic trajectory contained in(surrounding) the homoclinic trajectory in system (15), and it corresponds to the periodic wave solution zP4−ξ of the form (2) for equation (1), whose modulo satisfies m2<1/2m2>1/2.Whenh=Hx2,0, according to the Hamilton energy function, we have the following:(1)
The periodic trajectory contained in the homoclinic trajectory has two intersectionss1 and s2 with the x-axis, where s1 and s2 satisfy(95)−3−p3<s1<x1=−2−p3<s2<x2,3=−p3.(2)
The periodic trajectory surrounding the same trajectory has two intersectionss1 and s2 with the x-axis, where s1 and s2 satisfy(96)s1<−3−p3<x1=−2−p3<x2,3=−p3<s2.For the periodic trajectory contained in the homoclinic trajectory, as shown in Figure5, when Hamilton energy h⟶H−x2,0, we have s1⟶−3−p/3, s2⟶−p/3, α⟶−p/3, β⟶0, and m⟶0, so(97)limh⟶H−x2,0uP4−ξ=limm⟶0uP4−ξ=uS2+ξ.At this time, the periodic wave solutionzP4−ξm2<1/2 of the form (2) for equation (1) evolves into the bell-shaped solitary solution z2+ξ.For the periodic trajectory surrounding the homoclinic trajectory, it is known from Figure5 that when the Hamilton energy h⟶H+x2,0, we have s1⟶−3−p/3 and s2⟶−p/3, then α⟶−p/3, β⟶0, and m⟶1. By calculating, we can obtain(98)limh⟶H+x2,0uP4−ξ=limm⟶1uP4−ξ=uS2+ξ.In Figure5, it can be seen that the periodic trajectory surrounding the homoclinic trajectory contracts into a homoclinic trajectory, which corresponds to the evolutionary process from periodic wave solution zP4−ξm2>1/2 to the bell-shaped solitary wave solution zS2+ξ. Therefore, we can obtain the following theorem.Theorem 8.
When Δ = 0,q>0, and Hamilton energy h⟶H−x2,0, the periodic wave solution zP4−ξm2>1/2 of equation (1) corresponding to the periodic trajectory contained in the homoclinic trajectory in system (15) will evolve into the bell-shaped solution zS2+ξ. When the Hamilton energy h⟶H+x2,0, the periodic wave solution zP4−ξm2>1/2 of equation (1), which corresponds to the periodic trajectory surrounding the homoclinic trajectory, will evolve into the bell-shaped solution zS2+ξ.


## 5.2.1. The Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of the Symmetric Homoclinic Trajectories

WhenΔ<0 and q=0, the global phase portrait Figure 1(a) with symmetric homoclinic orbit is given by the qualitative analysis in Section 2. In Section 3, the solitary wave solutions z˜S1±ξ of the form (2) of equation (1) which correspond to the symmetric homoclinic orbits are given by (36) and their amplitudes u˜S1±ξ=±−2psech−4pξ correspond to the homoclinic trajectories in the global phase portrait Figure 1(a) (with Hamiltonian energy h1=Hx2,0=0). In Section 4.1, the periodic wave solutions zP1±ξ of the form (2) for equation (1) are given which correspond to the homoclinic trajectories inside the closed orbit. And their amplitudes uP1±ξ are given by (45). In Section 4.3, the periodic wave solutions z˜P4±ξ of equation (1) are given, which correspond to the closed orbits surrounding the homoclinic orbit, and their amplitudes u˜P4±ξ are given by (78).Whenh1 approaches the Hamilton energy Hx2,0=0 which corresponds to the solitary wave solution, we have limh1⟶0−k12=2p2/p2−p=1. According to (45), we can obtain(87)limh1⟶0−uP1±ξ=limk1⟶1±−2p2−k12dn−4p2−k12ξ,k1=±−2pdn−4pξ,1=±−2psech−4pξ=u˜S1±ξ.So we havelimh1⟶0−zP1±ξ=z˜S1±ξ.This indicates that whenh1⟶0−, the periodic wave solutions converge to the solitary wave solutions; here, periodic wave solutions correspond to the closed trajectories contained in the symmetric homoclinic trajectories, and solitary wave solutions correspond to the homoclinic trajectories.According to (78), when h3⟶0+, we have k⟶1. So(88)limh3⟶0+u˜P4±ξ=limk1⟶1u˜P4±ξ=±−2pcn−4pξ,1=±−2psech−4pξ=u˜S1±ξ.Thus, we havelimh3⟶0+zP4±ξ=z˜S1±ξ, the periodic solutions z˜P4±ξ take the solitary wave solutions z˜S1±ξ as the limit, where z˜P4±ξ correspond to the closed trajectories that surround the symmetric homoclinic trajectories and z˜S1±ξ correspond to the homoclinic trajectories. Therefore, we have the following theorem.Theorem 6.
In the case of Δ < 0 andq=0, when the Hamilton energy h1 of system (15) tends to be 0−, the periodic trajectories centered on P1 and P3 in Figure 1(a) will expand into homoclinic trajectories, and the corresponding periodic wave solutions zP1±ξ of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z˜S1±ξ; when the Hamilton energy h3 of the system gradually tends to be 0+, the periodic trajectories surrounding the symmetric homoclinic trajectories in Figure 1(a) will shrink into homoclinic trajectories, and the corresponding periodic wave solutions z˜P4±ξ of the form (2) for equation (1) will evolve into the bell-shaped solitary wave solutions z˜S1±ξ.


## 5.2.2. The Limit Relationships between Periodic Wave Solutions and Solitary Wave Solutions Corresponding to the Inner and Outer Trajectories of Asymmetric Homoclinic Trajectories

According to the analysis of the global phase portraits under different parameters for system (15) in Section 2, we know that there exist two asymmetric homoclinic trajectories in system (15) when Δ<0,q<0 or Δ<0,q>0. Since the analyses of the two cases are the same, we only need to consider the case of Δ<0,q<0.When the Hamilton energyh=Hx2,0, there exist homoclinic trajectories centered on P1 and P3 in system (15), which correspond to the bell-shaped solutions zS1±ξ of the form (2) for equation (1), respectively; when h∈Hx1,0,Hx2,0, a cluster of periodic trajectories exist inside the homoclinic trajectories centered on P1 or P3, which correspond to the periodic wave solutions zP2ξ and zP3ξ of the form (2) for equation (1); when h>Hx2,0, there exist period trajectories surrounding the homoclinic trajectory in system (15), which correspond to the periodic wave solutions zP4+ξm2>1/2 of the form (2) for equation (1).Here we first consider the limit relationships between the periodic wave solutionszP2ξ, zP3ξ, and the solitary wave solutions zS1±ξ.From qualitative analysis, we know that system (15) has three different singular points Pixi,0i=1,2,3, where x1, x2, and x3 can be expressed as follows:(89)x1=−x0−−3x02−4p2,x2=x0,x3=−x0+−3x22−4p2.As shown in Figure6, when h=Hx2,0, the abscissas eii=1,2,3,4 of the intersections of the function curve Y=−Fhx and the x-axis can be expressed as follows:(90)e1=−x2−−2p−2x22,e2=e3=x2,e4=−x2+−2p−2x22.It is easy to know thateii=1,2,3,4 and xii=1,2,3 satisfy e1<x1<e2=e3=x2<x3<e4, and the abscissas x1′i=1,2,3,4 of the intersections of the periodic trajectory contained in the homoclinic trajectory and the x-axis in Figure 6 can make the equation(91)−Fhx=−x4−2px2−4qx+h=0establish, where e1<x1′<x2′<e2=e3<x3′<x4′<e4.Because the figure of functionY=−Fhx is a continuous polynomial curve with respect to h, the zero point x1′ of −Fhx=0 has continuity about h. When h⟶H−x2,0 and x1′⟶e1, we have x2′⟶e2=x2, x3′⟶e3=x2, x4′⟶e4, and modulo k⟶1. Thus, according to (54) and (32), we have(92)limh⟶H−x2,0uP2ξ=limk⟶1uP2ξ=e4−e4−e1e4−e2e4−e2+e2−e1sn2−e1−e3e2−e4ξ,1=x2−2p+3x222x2−−p−x22cosh−p+3x22ξ=uS1−ξ.Similarly, we can provelimh⟶H−x2,0uP3ξ=uS1+ξ. Therefore, when h⟶H−x2,0, the periodic wave solutions zP2ξ and zP3ξ evolve into the bell-shaped solitary wave solutions zS1−ξ and zS1+ξ, respectively, where zP2ξ and zP3ξ correspond to the periodic trajectories contained in the homoclinic trajectories.Next, we consider the relationships between the periodic wave solutionszP4+ξ and zS1±ξ, which correspond to the periodic trajectories surrounding the asymmetric homoclinic trajectories.From Figure6, we can know that the periodic trajectory corresponding to the Y=−Fhx function curve in the case of h>Hx2,0 surrounds the homoclinic trajectory corresponding to the −Fhx function curve in the case of h=Hx2,0. Set the abscissas of the intersections of the periodic trajectory and the x-axis as s1 and s2 when h>Hx2,0, and set them as eii=1,2,3 when h=Hx2,0. s1, s2, and eii=1,2,3 have the following relationship:(93)s1<e1<e2=e3=x2<e4<s2.Whenh⟶H+x2,0, we have s1⟶−x0−−2p−2x02, s2⟶−x0+−2p−2x02, α=−1/2s1+s2⟶x2, β⟶0, m⟶1, λ1⟶0, λ2⟶−p+3x22/p+x22, and σ⟶−2p−6x22, so we can obtain(94)limh⟶H+x2,0uP4+ξ=limm⟶11−λ11+λ1α−σ2α+2α1+λ1/1+λ2cn2σξ,m=x2−2p+3x222x2±−p−x22cosh2−p+3x22ξ=uS1±ξ,that is, the periodic trajectory surrounding the homoclinic trajectory contracts into the homoclinic trajectory. So we can deduce that the periodic wave solution zP4+ξ of the form (2) for equation (1) evolves into the bell-shaped solitary wave solutions zS1±ξ.Theorem 7.
Assume Δ < 0,q<0, when Hamilton energy h⟶H−x2,0, the periodic wave solutions zP2ξ and zP3ξ of the form (2) corresponding to the periodic trajectories contained in the homoclinic trajectory evolve into bell-shaped solutions zS1−ξ and zS1+ξ, respectively. When Hamilton energy h⟶H+x2,0, the periodic wave solution zP4+ξm2>1/2 of the form (2) corresponding to the periodic trajectory in the homoclinic trajectory evolves into the bell-shaped solutions zS1±ξ.


## 5.2.3. The Relationships between the Traveling Wave Solutions inside or outside a Single Homoclinic Orbit and the Limit of Hamilton Energy

Now we take the case ofΔ=0 and q>0 as an example. In this case, as shown in Figure 5 and according to the above solving process of the solitary wave solutions and the periodic wave solutions, we know that when h=Hx2,0, homoclinic trajectory with P2,3 as a cusp and P1 as a center in system (15) corresponds to the bell-shape solution zS2+ξ. When h<Hx2,0h>Hx2,0, there exists the periodic trajectory contained in(surrounding) the homoclinic trajectory in system (15), and it corresponds to the periodic wave solution zP4−ξ of the form (2) for equation (1), whose modulo satisfies m2<1/2m2>1/2.Whenh=Hx2,0, according to the Hamilton energy function, we have the following:(1)
The periodic trajectory contained in the homoclinic trajectory has two intersectionss1 and s2 with the x-axis, where s1 and s2 satisfy(95)−3−p3<s1<x1=−2−p3<s2<x2,3=−p3.(2)
The periodic trajectory surrounding the same trajectory has two intersectionss1 and s2 with the x-axis, where s1 and s2 satisfy(96)s1<−3−p3<x1=−2−p3<x2,3=−p3<s2.For the periodic trajectory contained in the homoclinic trajectory, as shown in Figure5, when Hamilton energy h⟶H−x2,0, we have s1⟶−3−p/3, s2⟶−p/3, α⟶−p/3, β⟶0, and m⟶0, so(97)limh⟶H−x2,0uP4−ξ=limm⟶0uP4−ξ=uS2+ξ.At this time, the periodic wave solutionzP4−ξm2<1/2 of the form (2) for equation (1) evolves into the bell-shaped solitary solution z2+ξ.For the periodic trajectory surrounding the homoclinic trajectory, it is known from Figure5 that when the Hamilton energy h⟶H+x2,0, we have s1⟶−3−p/3 and s2⟶−p/3, then α⟶−p/3, β⟶0, and m⟶1. By calculating, we can obtain(98)limh⟶H+x2,0uP4−ξ=limm⟶1uP4−ξ=uS2+ξ.In Figure5, it can be seen that the periodic trajectory surrounding the homoclinic trajectory contracts into a homoclinic trajectory, which corresponds to the evolutionary process from periodic wave solution zP4−ξm2>1/2 to the bell-shaped solitary wave solution zS2+ξ. Therefore, we can obtain the following theorem.Theorem 8.
When Δ = 0,q>0, and Hamilton energy h⟶H−x2,0, the periodic wave solution zP4−ξm2>1/2 of equation (1) corresponding to the periodic trajectory contained in the homoclinic trajectory in system (15) will evolve into the bell-shaped solution zS2+ξ. When the Hamilton energy h⟶H+x2,0, the periodic wave solution zP4−ξm2>1/2 of equation (1), which corresponds to the periodic trajectory surrounding the homoclinic trajectory, will evolve into the bell-shaped solution zS2+ξ.


## 5.3. Diagrams of the Relationships between Hamilton Energy and Traveling Wave Solutions

In the last section, we have discussed the evolutionary relationships between the periodic wave solutions and the solitary wave solutions of equation (1), when the Hamilton energy of system (15) approaches a certain limit. In this section, we will give examples of 3D diagrams to intuitively demonstrate the relationships between these two solutions: When the Hamilton energy changes, the amplitudes uPiξi=1,2,3,4 of the periodic wave solutions for equation (1) will evolve into the amplitudes uSiξi=1,2 of the bell-shaped solutions for equation (1).


### 5.3.1. The Evolution from Periodic Solutions to Solitary Wave Solutions Caused by the Variation of Hamiltonian Energy in the Case of Δ < 0 andq<0

The case ofΔ<0 and q<0 is shown in the global phase portrait Figure 1(b). When h=Hx2,0, equation (1) has bell-shaped solitary wave solutions zS1±ξ of the form (2). When Hx1,0<h<Hx2,0, equation (1) has periodic wave solutions zP2ξ and zP3ξ, which satisfy limh⟶H−x2,0zP2ξ=zS1−ξ and limh⟶H−x2,0zP3ξ=zS1+ξ. When h>Hx2,0, equation (1) has a periodic wave solution zP4+ξm2>1/2 of the form (2), which satisfies limh⟶H+x2,0zP4+ξ=zS1±ξ. Due to the space limitation, we only give the diagrams of the evolution from the amplitude uP3ξ of the periodic wave solution zP3ξ to the amplitude uS1+ξ of the solitary wave solution zS1+ξ.Here, we takep=−7, q=−6, c=25/12, and x2=−1, and then, we have Hx2,0=11. When Hamilton energy h⟶H−x2,0, the amplitude of the periodic wave solution for equation (1) satisfies uP3ξ⟶uS1+ξ. The corresponding diagrams are shown in Figures 7(a)–7(d).Figure 7
(a)h=10.99,uP3ξ, (b)h=10.9999,uP3ξ, (c)h=10.999999,uP3ξ, and (d)h=Hx2,0,uS1+ξ.
(a)
(b)
(c)
(d)


### 5.3.2. The Evolution from Periodic Solutions to Solitary Wave Solutions Caused by the Variation of Hamilton Energy in the Case ofΔ=0 and q>0

The case ofp<0,Δ=0,q>0 corresponds to the global phase portrait Figure 1(e). When h=Hx2,0, equation (1) has a bell-shape solution zS2+ξ in the form of (2). When Hx1,0<h<Hx2,0, equation (1) has a periodic wave solution zP4−ξm2>1/2 in the form of (2), which satisfies limh⟶H−x2,0zP4−ξ=zS2+ξ. When h>Hx2,0, equation (1) has a periodic wave solution zP4−ξm2>1/2 of the form (2), which satisfies limh⟶H+x2,0zP4−ξ=zS2+ξ. Due to the space limitation, here we only give the diagrams of the evolution from the amplitude uP4−ξ of the periodic wave solution zP4−ξm2>1/2 to the amplitude uS2+ξ of solitary wave solution zS2+ξ.Takep=−3, q=2, x2=2, and c=4/3, we have Hx2,0=3. When Hamilton energy h⟶H+x2,0, the diagrams of the amplitude uP4−ξm2>1/2⟶uS2+ξ of the traveling wave solution for equation (1) are shown in Figures 8(a)–8(d) (for the sake of beauty, we adjusted the positive and negative axes of the vertical direction of the graphic).Figure 8
(a)h=3.05,uP4−ξ,m2>1/2, (b)h=3.03,uP4−ξ,m2>1/2, and (c)h=3.01,uP4−ξ,m2>1/2 and (d)h=Hx2,0,uS2+ξ.
(a)
(b)
(c)
(d)It can be seen that by the qualitative analysis and the first integral method, we not only obtain all the unimodal bell-shaped solitary wave solutions and the periodic wave solutions of the S-S equation (1), but also show the mathematical meaning of the obtained solutions, i.e., the relationships between solutions and bounded trajectories. Moreover, we reveal the relationships between the solitary wave solutions, the periodic wave solutions of the S-S equation, and the energy h of the Hamilton system.For scholars who want to know more about the application of Hamilton energy in the research of solitary wave solution, refer to literatures [35–37].


## 5.3.1. The Evolution from Periodic Solutions to Solitary Wave Solutions Caused by the Variation of Hamiltonian Energy in the Case of Δ < 0 andq<0

The case ofΔ<0 and q<0 is shown in the global phase portrait Figure 1(b). When h=Hx2,0, equation (1) has bell-shaped solitary wave solutions zS1±ξ of the form (2). When Hx1,0<h<Hx2,0, equation (1) has periodic wave solutions zP2ξ and zP3ξ, which satisfy limh⟶H−x2,0zP2ξ=zS1−ξ and limh⟶H−x2,0zP3ξ=zS1+ξ. When h>Hx2,0, equation (1) has a periodic wave solution zP4+ξm2>1/2 of the form (2), which satisfies limh⟶H+x2,0zP4+ξ=zS1±ξ. Due to the space limitation, we only give the diagrams of the evolution from the amplitude uP3ξ of the periodic wave solution zP3ξ to the amplitude uS1+ξ of the solitary wave solution zS1+ξ.Here, we takep=−7, q=−6, c=25/12, and x2=−1, and then, we have Hx2,0=11. When Hamilton energy h⟶H−x2,0, the amplitude of the periodic wave solution for equation (1) satisfies uP3ξ⟶uS1+ξ. The corresponding diagrams are shown in Figures 7(a)–7(d).Figure 7
(a)h=10.99,uP3ξ, (b)h=10.9999,uP3ξ, (c)h=10.999999,uP3ξ, and (d)h=Hx2,0,uS1+ξ.
(a)
(b)
(c)
(d)


## 5.3.2. The Evolution from Periodic Solutions to Solitary Wave Solutions Caused by the Variation of Hamilton Energy in the Case ofΔ=0 and q>0

The case ofp<0,Δ=0,q>0 corresponds to the global phase portrait Figure 1(e). When h=Hx2,0, equation (1) has a bell-shape solution zS2+ξ in the form of (2). When Hx1,0<h<Hx2,0, equation (1) has a periodic wave solution zP4−ξm2>1/2 in the form of (2), which satisfies limh⟶H−x2,0zP4−ξ=zS2+ξ. When h>Hx2,0, equation (1) has a periodic wave solution zP4−ξm2>1/2 of the form (2), which satisfies limh⟶H+x2,0zP4−ξ=zS2+ξ. Due to the space limitation, here we only give the diagrams of the evolution from the amplitude uP4−ξ of the periodic wave solution zP4−ξm2>1/2 to the amplitude uS2+ξ of solitary wave solution zS2+ξ.Takep=−3, q=2, x2=2, and c=4/3, we have Hx2,0=3. When Hamilton energy h⟶H+x2,0, the diagrams of the amplitude uP4−ξm2>1/2⟶uS2+ξ of the traveling wave solution for equation (1) are shown in Figures 8(a)–8(d) (for the sake of beauty, we adjusted the positive and negative axes of the vertical direction of the graphic).Figure 8
(a)h=3.05,uP4−ξ,m2>1/2, (b)h=3.03,uP4−ξ,m2>1/2, and (c)h=3.01,uP4−ξ,m2>1/2 and (d)h=Hx2,0,uS2+ξ.
(a)
(b)
(c)
(d)It can be seen that by the qualitative analysis and the first integral method, we not only obtain all the unimodal bell-shaped solitary wave solutions and the periodic wave solutions of the S-S equation (1), but also show the mathematical meaning of the obtained solutions, i.e., the relationships between solutions and bounded trajectories. Moreover, we reveal the relationships between the solitary wave solutions, the periodic wave solutions of the S-S equation, and the energy h of the Hamilton system.For scholars who want to know more about the application of Hamilton energy in the research of solitary wave solution, refer to literatures [35–37].



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*Source: 8760179-2020-04-10.xml*