# Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

Authors: Qing Meng; Bin He
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We employ the approaches of both dynamical system and numerical simulation to investigate a generalized KdV equation, which is presented by Yin (2012). Some peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions are obtained, and the planar graphs of the compactons and the periodic cusp waves are simulated.


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## Body

## 1. Introduction

To study the role of nonlinear dispersion in the formation of patterns in the liquid drop, Rosenau and Hyman [1] showed in a particular generalization of the KdV equation
(1)ut+(u2)x+(u2)xxx=0,
which is called K(2,2) equation. They found some solitary waves with compact support in it, which they called compactons. These compactons had the property that the width was independent of the amplitude. Equation (1) has been studied successfully by some authors [2–8]. However, (1) does not exhibit the usual energy conservation law. Instead of (1), Cooper et al. considered the corresponding generalized KdV equation [9]
(2)ut+uux+α(2uuxxx+4uxuxx)=0,
which can be derived from a Lagrangian. Equation (2) has the same terms as (1), except the relative weights of the terms. Equation (2) also admits compacton solutions. The stability of the compacton solutions to (2) was studied in [10]. In the presence of a linear dispersion term, (2) turns into a generalized KdV equation with combined dispersion [11]
(3)ut+uux+α(2uuxxx+4uxuxx)+βuxxx=0.
Obviously, for β=0, (3) turns into (2). Yin [11] indicated that (3) has two conservative laws and showed that the smooth solitary waves are stable for any speed of wave propagation.In this paper, we investigated (3) using the approaches of dynamical system and numerical simulation [12–14], we will give some new exact travelling wave solutions and simulate the compactons and the periodic cusp waves.Next, we always suppose thatα≠0. Using the following independent variable transformation:
(4)u(x,t)=ϕ(ξ),ξ=x-ct,
where c(c≠0) is the wave speed, and substituting (4) into (3), we obtain
(5)-cϕ′+ϕϕ′+α(2ϕϕ′′′+4ϕ′ϕ′′)+βϕ′′′=0,
where “′” is the derivative with respect to ξ.Integrating (5) once with respect to ξ, we have
(6)-cϕ+12ϕ2+α(2ϕϕ′′+(ϕ′)2)+βϕ′′=g,
where g is the integral constant.Lettingy=dϕ/dξ, we get the following planar dynamical system:
(7)dϕdξ=y,dydξ=g+cϕ-(1/2)ϕ2-αy22αϕ+β.The rest of this paper is organized as follows. In Section2, we discuss the bifurcation sets and phase portraits of system (7), where explicit parametric conditions will be derived. In Section 3, we give some exact travelling wave solutions which include peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions of (3). In Section 4, the numerical simulations of the compactons and the periodic cusp waves are given. A short conclusion will be given in Section 5.


## 2. Bifurcation Sets and Phase Portraits of System (7)

Using the transformationdξ=(2αϕ+β)dτ, it carries (7) into the Hamiltonian system
(8)dϕdτ=(2αϕ+β)y,dydτ=g+cϕ-12ϕ2-αy2.
System (8) has the following first integral:
(9)H(ϕ,y)=(2αϕ+β)y2-(2gϕ+cϕ2-13ϕ3)=h.For a fixedh, the level curve H(ϕ,y)=h defined by (9) determines a set of invariant curves of system (8) which contains different branches of curves. As h is varied, it defines different families of orbits of (8) with different dynamical behaviors.WriteΔ=-2α(β2+4αβc-8gα2), ϕs=-β/2α. Clearly, when g>-c2/2, system (8) has two equilibrium points at (ϕ1,0) and (ϕ2,0) in ϕ-axis, where ϕ1,2=c±2g+c2. When g=-c2/2, system (8) has only one equilibrium point at (c,0) in ϕ-axis. When Δ>0, there exist two equilibrium points of system (8) in line ϕ=ϕs at (ϕs,Y±), where Y±=±(1/4α2)Δ. When Δ<0, there is no equilibrium point of system (8) in line ϕ=ϕs.LetM(ϕe,ye) be the coefficient matrix of the linearized system of (8) at equilibrium point (ϕe,ye); then we have Trace M(ϕe,0)=0 and
(10)J(ϕ1,0)=detM(ϕ1,0)=2g+c2(2α2g+c2+(2αc+β)),J(ϕ2,0)=detM(ϕ2,0)=2g+c2(2α2g+c2-(2αc+β)),J(ϕs,Y±)=detM(ϕs,Y±)=-4α2Y±2.For an equilibrium point(ϕe,ye) of a planar integrable system, we know that (ϕe,ye) is a saddle point if J(ϕe,ye)<0, a center point if J(ϕe,ye)>0 and Trace M(ϕe,ye)=0, and a cusp if J(ϕe,ye)=0 and the Poincaré index of (ϕe,ye) is zero.Since both systems (7) and (8) have the same first integral (9), then two systems above have the same topological phase portraits except the line ϕ=ϕs. Therefore we can obtain the bifurcation sets and phase portraits of system (7) from those of system (8).By using the properties of equilibrium points and bifurcation method of dynamical systems, we can show that bifurcation sets and phase portraits of system (7) are drawn in Figures 1 and 2.Bifurcation sets and phase portraits of system (7) when α>0.
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2Bifurcation sets and phase portraits of system (7) when α<0.
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## 3. Exact Travelling Wave Solutions of (3)

Denote thath1,2=-(2c/3)(3g+c2)∓(2/3)(2g+c2)2g+c2 and hs=-β(β2+6αβc-24gα2)/24α3; we present some exact travelling wave solutions of (3) as follows.


### 3.1. Peakon Solutions

From Figure2(d), we see that there are two heteroclinic orbits of system (7) defined by H(ϕ,y)=-β(β2+12αβc+36α2c2)/96α3 connecting with the saddle points (ϕs,±((2αc+β)-6α/8α2)) and ((6αc+β)/4α,0) when α<0, c<-β/2α, and g=-(12α2c2-4αβc-β2)/32α2. Their expressions are
(11)y=±1-6α(ϕ-6αc+β4α),6αc+β4α<ϕ≤ϕs.Substituting (11) into dϕ/dξ=y and integrating it along the heteroclinic orbits yield equation
(12)∫ϕϕs‍dss-((6αc+β)/4α)=1-6α∫0ξ‍ds.Completing (12) and using transformation (4), we can get a peakon solution of (3) as follow:
(13)u(x,t)=6αc+β4α+(ϕs-6αc+β4α)e-ω|x-ct|,
where ω=1/-6α. The profiles of (13) are shown in Figures 3(a) and 3(b).Peakons of (3) for x=1.
(a)
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α
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1
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1From Figure2(k), we see that there are two heteroclinic orbits of system (7) defined by H(ϕ,y)=-β(β2+12αβc+36α2c2)/96α3 connecting with the saddle points (ϕs,∓((2αc+β)-6α/8α2)) and ((6αc+β)/4α,0) when α<0, c>-β/2α, and g=-(12α2c2-4αβc-β2)/32α2. Their expressions are
(14)y=±1-6α(6αc+β4α-ϕ),ϕs≤ϕ<6αc+β4α.Substituting (14) into dϕ/dξ=y and integrating it along the heteroclinic orbits yield equation
(15)∫ϕsϕ‍ds((6αc+β)/4α)-s=1-6α∫0ξ‍ds.Completing (15) and using transformation (4), we can get a peakon solution of (3) which is the same as (13). The profiles are shown in Figures 3(c) and 3(d).


### 3.2. Solitary Wave Solutions

From Figure1(c), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h1 connecting with saddle point (ϕ1,0) and passing point (ϕm,0) when α>0, c<-β/2α, and -c2/2<g<β(4αc+β)/8α2, where ϕm=c-22g+c2. Its expression is
(16)y=±16αϕ1-ϕϕs-ϕ(ϕ-ϕm)(ϕs-ϕ),ϕm≤ϕ<ϕ1.Substituting (16) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(17)∫ϕmϕ‍(ϕs-s)ds(ϕ1-s)(s-ϕm)(ϕs-s)=16α∫0ξ‍ds.Completing (17) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(18)u(x,t)=ϕ1(ϕmcosh2(ωχ)-ϕssinh2(ωχ))-ϕmϕs(ϕmsinh2(ωχ)-ϕscosh2(ωχ))+ϕ1,t=1c(x-6α(χ+2tan-1(2ωtanh(ωχ)))),
where χ is a new parametric variable and ω=(1/2)(ϕ1-ϕm)/(ϕs-ϕ1). The profiles of (18) are shown in Figures 4(a) and 4(b).Solitary waves of (3) for x=1.
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1.7From Figure1(h), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h2 connecting with saddle point (ϕ2,0) and passing point (ϕM,0) when α>0, c>-β/2α, and -c2/2<g<β(4αc+β)/8α2, where ϕM=c+22g+c2. Its expression is
(19)y=±16αϕ-ϕ2ϕ-ϕs(ϕM-ϕ)(ϕ-ϕs),ϕ2<ϕ≤ϕM.Substituting (19) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(20)∫ϕϕM‍(s-ϕs)ds(s-ϕ2)(ϕM-s)(s-ϕs)=16α∫0ξ‍ds.Completing (20) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(21)u(x,t)=ϕ2(ϕMcosh2(ωχ)-ϕssinh2(ωχ))-ϕMϕs(ϕMsinh2(ωχ)-ϕscosh2(ωχ))+ϕ2,t=1c(x-6α(χ-2tan-1(2ωtanh(ωχ)))),
where ω=(1/2)(ϕM-ϕ2)/(ϕ2-ϕs). The profiles of (21) are shown in Figures 4(c) and 4(d).From Figure2(c), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h2 connecting with saddle point (ϕ2,0) and passing point (ϕM,0) when α<0, c<-β/2α, and -c2/2<g<-(12α2c2-4αβc-β2)/32α2, where ϕM=c+22g+c2. Its expression is
(22)y=±1-6αϕ-ϕ2ϕs-ϕ(ϕM-ϕ)(ϕs-ϕ),ϕ2<ϕ≤ϕM.Substituting (22) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(23)∫ϕϕM‍(ϕs-s)ds(s-ϕ2)(ϕM-s)(ϕs-s)=1-6α∫0ξ‍ds.Completing (23) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(24)u(x,t)=ϕ2(ϕMcosh2(ωχ)-ϕssinh2(ωχ))-ϕMϕs(ϕMsinh2(ωχ)-ϕscosh2(ωχ))+ϕ2,t=1c(x--6α(χ+2tanh-1(2ωtanh(ωχ)))),
where ω=(1/2)(ϕM-ϕ2)/(ϕs-ϕ2). The profiles of (24) are shown in Figures 4(e) and 4(f).From Figure2(j), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h1 connecting with saddle point (ϕ1,0) and passing point (ϕm,0) when α<0, c>-β/2α, -c2/2<g<-(12α2c2-4αβc-β2)/32α2, where ϕm=c-22g+c2. Its expression is
(25)y=±1-6αϕ1-ϕϕ-ϕs(ϕ-ϕm)(ϕ-ϕs),ϕm≤ϕ<ϕ1.Substituting (25) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(26)∫ϕmϕ‍(s-ϕs)ds(ϕ1-s)(s-ϕm)(s-ϕs)=1-6α∫0ξ‍ds.Completing (26) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(27)u(x,t)=ϕ1(ϕmcosh2(ωχ)-ϕssinh2(ωχ))-ϕmϕs(ϕmsinh2(ωχ)-ϕscosh2(ωχ))+ϕ1,t=1c(x--6α(χ-2tanh-1(2ωtanh(ωχ)))),
where ω=(1/2)(ϕ1-ϕm)/(ϕ1-ϕs). The profiles of (27) are shown in Figures 4(g) and 4(h).


### 3.3. Smooth Periodic Wave Solutions

From Figure1(d), we see that there is a periodic orbit of system (7) defined by H(ϕ,y)=β2(3αc+β)/12α3 enclosing the center point ((4αc+β)/2α,0) and passing points (ϕs,0) and ((3αc+β)/α,0) when α>0, c<-β/2α, and g=β(4αc+β)/8α2. Its expression is
(28)y=±16α(ϕ-3αc+βα)(ϕs-ϕ),3αc+βα≤ϕ≤ϕs.Substituting (28) into dϕ/dξ=y and integrating it along the periodic orbit yield equation
(29)∫(3αc+β)/αϕ‍ds(s-((3αc+β)/α))(ϕs-s)=16α∫0ξ‍ds.Completing (29) and using transformation (4), we can get a smooth periodic wave solution of (3) as follows:
(30)u(x,t)=ϕssin2(ω(x-ct))+3αc+βαcos2(ω(x-ct)),
where ω=1/26α. The profiles of (30) are shown in Figures 5(a) and 5(b).Smooth periodic waves of (3) for x=1.
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1From Figure1(i), we see that there is a periodic orbit of system (7) defined by H(ϕ,y)=β2(3αc+β)/12α3 enclosing the center point ((4αc+β)/2α,0) and passing points (ϕs,0) and ((3αc+β)/α,0) when α>0, c>-β/2α, and g=β(4αc+β)/8α2. Its expression is
(31)y=±16α(3αc+βα-ϕ)(ϕ-ϕs),ϕs≤ϕ≤3αc+βα.Substituting (31) into dϕ/dξ=y and integrating it along the periodic orbit yield equation
(32)∫ϕ(3αc+β)/α‍ds(((3αc+β)/α)-s)(s-ϕs)=16α∫0ξ‍ds.Completing (32) and using transformation (4), we can get a smooth periodic wave solution of (3) which is the same as (30). The profiles are shown in Figures 5(c) and 5(d).


### 3.4. Compacton Solutions

For givenh∈(h1,hs), hs<h2 (or h∈(h1,h2), hs>h2) in Figure 1(c), the level curve defined by H(ϕ,y)=h is shown in Figure 6(a).The level curves defined byH(ϕ,y)=h.
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(j)For givenh∈(hs,h2), h1<hs in Figure 1(c), h∈(h1,h2) in Figure 1(d), and h∈(hs,h2) in Figures 1(e) and 1(j), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(b).For givenh∈(hs,h2), hs>h1 (or h∈(h1,h2), hs<h1) in Figure 1(h),the level curve defined by H(ϕ,y)=h is shown in Figure 6(c).For givenh∈(h1,hs), hs<h2 in Figure 1(h), and h∈(h1,h2) in Figure 1(i), h∈(h1,hs) in Figures 1(e) and 1(j), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(d).For givenh∈(hs,h2) in Figures 2(e), 2(f), 2(g), and 2(n), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(e).For givenh∈(h1,hs) in Figures 2(g), 2(l), 2(m), and 2(n), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(f).From Figure6(a), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h when α>0, c<-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(h1,hs), hs<h2 (or h∈(h1,h2), and hs>h2). The open curve passes point (γ1,0) and approaches the line ϕ=ϕs; its expression is
(33)y=±16α(ϕ-γ1)(ϕ-γ2)(ϕ-γ3)ϕs-ϕ,γ1≤ϕ<ϕs,
where γ1, γ2, and γ3(γ3<γ2<γ1<ϕs) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈-2.681985997, γ2≈-2.948887226, γ3≈-3.369126778 when α=0.2, β=1, c=-3, g=-4.48, and h=-8.882, and γ1≈-0.06941350133, γ2≈-0.5430641149, and γ3≈-2.387522384 when α=1, β=0, c=-1, g=-0.25, and h=-0.03.Substituting (33) into dϕ/dξ=y and integrating it along the open curve yield equation
(34)∫γ1ϕ‍ϕs-s(s-γ1)(s-γ2)(s-γ3)ds=16α∫0ξ‍ds.Completing (34) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(35)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(ϕs-γ2)(γ1-γ3)2(ϕs-γ1)6α|x-ct|,
where u~=sn-1((ϕs-γ2)(u(x,t)-γ1)/(ϕs-γ1)(u(x,t)-γ2),k), ϕ~=sin-1((ϕs-γ2)(u(x,t)-γ1)/(ϕs-γ1)(u(x,t)-γ2)), k=(ϕs-γ1)(γ2-γ3)/(ϕs-γ2)(γ1-γ3), and α12=(ϕs-γ1)/(ϕs-γ2); Π(·,·,·) is Legendre’s incomplete elliptic integral of the third kind, and sn-1(·,·) is the inverse function of the Jacobian elliptic function sn(·,·) [15].From Figure6(b), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α>0, c<-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(hs,h2), and h1<hs,(ii)
α>0, c<-β/2α, g=β(4αc+β)/8α2, and h∈(h1,h2),(iii)
α>0, c≠-β/2α, g>β(4αc+β)/8α2, and h∈(hs,h2).The open curve passes point(γ1,0) and approaches the line ϕ=ϕs; its expression is
(36)y=±16α(γ1-ϕ)(ϕ-γ2)(ϕ-γ3)ϕ-ϕs,ϕs<ϕ≤γ1,
where γ1, γ2, and γ3 (γ3<γ2<ϕs<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈0.6863083159, γ2≈0.243331752, and γ3≈-0.1796400679 when α=1,β=-1, c=0.25,g=0, and h=-0.01 and γ1≈5.467670461,γ2≈-1.060802034, γ3≈-1.406868427 when α=1, β=2, c=1, g=2, and h=2.72.Substituting (36) into dϕ/dξ=y and integrating it along the open curve yield equation
(37)∫ϕγ1‍s-ϕs(γ1-s)(s-γ2)(s-γ3)ds=16α∫0ξ‍ds.Completing (37) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(38)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-γ2)(ϕs-γ3)2(γ1-ϕs)6α|x-ct|,
where u~=sn-1((ϕs-γ3)(γ1-u(x,t))/(γ1-ϕs)(u(x,t)-γ3),k), ϕ~=sin-1((ϕs-γ3)(γ1-u(x,t))/(γ1-ϕs)(u(x,t)-γ3)), k=(γ1-ϕs)(γ2-γ3)/(γ1-γ2)(ϕs-γ3), and α12=(ϕs-γ1)/(ϕs-γ3).From Figure6(c), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h when α>0, c>-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(hs,h2), hs>h1 (or h∈(h1,h2), and hs<h1). The open curve passes point (γ3,0) and approaches the line ϕ=ϕs; its expression is
(39)y=±16α(γ1-ϕ)(γ2-ϕ)(γ3-ϕ)ϕ-ϕs,ϕs<ϕ≤γ3,
where γ1, γ2, and γ3 (ϕs<γ3<γ2<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈-1.725343175, γ2≈-1.954509591, and γ3≈-2.320147234 when α=0.2, β=1, c=-2, g=-1.985, and h=-2.608 and γ1≈0.9464101615, γ2=0.3, and γ3≈0.2535898385 when α=2, β=-1, c=0.5, g=-0.1, and h=0.024.Substituting (39) into dϕ/dξ=y and integrating it along the open curve yield equation
(40)∫ϕγ3‍s-ϕs(γ1-s)(γ2-s)(γ3-s)ds=16α∫0ξ‍ds.Completing (40) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follow:
(41)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-γ3)(γ2-ϕs)2(γ3-ϕs)6α|x-ct|,
where u~=sn-1((γ2-ϕs)(γ3-u(x,t))/(γ3-ϕs)(γ2-u(x,t)),k), ϕ~=sin-1((γ2-ϕs)(γ3-u(x,t))/(γ3-ϕs)(γ2-u(x,t))), k=(γ1-γ2)(γ3-ϕs)/(γ1-γ3)(γ2-ϕs), and α12=(γ3-ϕs)/(γ2-ϕs).From Figure6(d), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α>0, c>-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(h1,hs), and hs<h2,(ii)
α>0, c>-β/2α, g=β(4αc+β)/8α2, and h∈(h1,h2),(iii)
α>0, c≠-β/2α, g>β(4αc+β)/8α2, and h∈(h1,hs).The open curve passes point(γ3,0) and approaches the line ϕ=ϕs, its expression is
(42)y=±16α(γ1-ϕ)(γ2-ϕ)(ϕ-γ3)ϕs-ϕ,γ3≤ϕ<ϕs,
where γ1, γ2, and γ3 (γ3<ϕs<γ2<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈2.557900553, γ2≈0.7351513251, and γ3≈-2.393051878 when α=1, β=-1, c=0.3, g=1, and h=-1.5 and γ1≈5.743824982, γ2≈-0.6449004894, γ3≈-8.098924493 when α=0.1, β=1, c=-1, g=7.5, and h=10.Substituting (42) into dϕ/dξ=y and integrating it along the open curve yield equation
(43)∫γ3ϕ‍ϕs-s(γ1-s)(γ2-s)(s-γ3)ds=16α∫0ξ‍ds.Completing (43) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(44)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-ϕs)(γ2-γ3)2(ϕs-γ3)6α|x-ct|,
where u~=sn-1((γ1-ϕs)(u(x,t)-γ3)/(ϕs-γ3)(γ1-u(x,t)),k), ϕ~=sin-1((γ1-ϕs)(u(x,t)-γ3)/(ϕs-γ3)(γ1-u(x,t))), k=(γ1-γ2)(ϕs-γ3)/(γ1-ϕs)(γ2-γ3), and α12=(γ3-ϕs)/(γ1-ϕs).From Figure6(e), we see that there are three open curves of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α<0,c<-β/2α,g>-(12α2c2-4αβc-β2)/32α2, and h∈(hs,h2),(ii)
α<0,c>-β/2α,g>β(4αc+β)/8α2, and h∈(hs,h2).One of them passes point(γ2,0) and approaches the line ϕ=ϕs; its expression is
(45)y=±1-6α(γ1-ϕ)(ϕ-γ2)(ϕ-γ3)ϕs-ϕ,γ2≤ϕ<ϕs,
where γ1, γ2, and γ3(γ3<γ2<ϕs<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈2.064177772,γ2≈-0.3054072893, and γ3≈-4.758770483 when α=-2, β=2, c=-1, and g=1.5 and h=1, γ1≈5.466333036, γ2≈-1.035930847, and γ3≈-1.430402188 when α=-0.5, β=-1, c=1, g=2, and h=2.7.Substituting (45) into dϕ/dξ=y and integrating it along the open curve yield equation
(46)∫γ2ϕ‍ϕs-s(γ1-s)(s-γ2)(s-γ3)ds=1-6α∫0ξ‍ds.Completing (46) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(47)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-γ2)(ϕs-γ3)2(ϕs-γ2)-6α|x-ct|,
where u~=sn-1((ϕs-γ3)(u(x,t)-γ2)/(ϕs-γ2)(u(x,t)-γ3),k), ϕ~=sin-1((ϕs-γ3)(u(x,t)-γ2)/(ϕs-γ2)(u(x,t)-γ3)), k=(ϕs-γ2)(γ1-γ3)/(γ1-γ2)(ϕs-γ3), and α12=(ϕs-γ2)/(ϕs-γ3).From Figure6(f), we see that there are three open curves of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α<0,c<-β/2α,g>β(4αc+β)/8α2, and h∈(h1,hs),(ii)
α<0,c>-β/2α,g>-(12α2c2-4αβc-β2)/32α2, and h∈(h1,hs).One of them passes point(γ2,0) and approaches the line ϕ=ϕs; its expression is
(48)y=±1-6α(γ1-ϕ)(γ2-ϕ)(ϕ-γ3)ϕ-ϕs,ϕs<ϕ≤γ2,
where γ1, γ2, and γ3(γ3<ϕs<γ2<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈0.8657007565, γ2≈-0.091826066674, and γ3≈-3.77387469 when α=-2, β=-1, c=-1, g=0.5, and h=0.1, and γ1≈6.154523009, γ2≈2.662391044, and γ3≈0.1830859473 when α=-0.2, β=1, c=3, g=-3, and h=1.Substituting (48) into dϕ/dξ=y and integrating it along the open curve yield equation
(49)∫ϕγ2‍s-ϕs(γ1-s)(γ2-s)(s-γ3)ds=1-6α∫0ξ‍ds.Completing (49) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(50)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-ϕs)(γ2-γ3)2(γ2-ϕs)-6α|x-ct|,
where u~=sn-1((γ1-ϕs)(γ2-u(x,t))/(γ2-ϕs)(γ1-u(x,t)),k), ϕ~=sin-1((γ1-ϕs)(γ2-u(x,t))/(γ2-ϕs)(γ1-u(x,t))), k=(γ2-ϕs)(γ1-γ3)/(γ1-ϕs)(γ2-γ3), and α12=(γ2-ϕs)/(γ1-ϕs).


### 3.5. Periodic Cusp Wave Solutions

For givenh=hs in Figure 1(e), the level curve defined by H(ϕ,y)=h is shown in Figure 6(g).For givenh=hs in Figure 1(j), the level curve defined by H(ϕ,y)=h is shown in Figure 6(h).For givenh=hs in Figure 2(e), the level curve defined by H(ϕ,y)=h is shown in Figure 6(i).For givenh=hs in Figure 2(l), the level curve defined by H(ϕ,y)=h is shown in Figure 6(j).From Figures6(g) and 6(h), we see that there are two heteroclinic orbits of system (7) defined by H(ϕ,y)=hs connecting with the saddle points (ϕs,Y±) and passing points (ϕm,0), and (ϕM,0), respectively, when α>0, c≠-β/2α, and g>β(4αc+β)/8α2, where ϕM,m=((β+6αc)±δ)/4α and δ=-3β2-12αβc+36α2c2+96α2g. Their expressions are, respectively,
(51)y=±16α(ϕ-ϕm)(ϕM-ϕ),ϕm≤ϕ≤ϕs,(52)y=±16α(ϕ-ϕm)(ϕM-ϕ),ϕs≤ϕ≤ϕM.Substituting (51) into the dϕ/dξ=y and integrating it along the heteroclinic orbit, yields equation
(53)∫ϕϕs‍ds(s-ϕm)(ϕM-s)=16α∫0ξ‍ds.Completing (53) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(54)u(x,t)=ϕmcos2(Ω-ω|x-ct-2nT|)+ϕMsin2(Ω-ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…, ω=1/26α, Ω=tan-1((ϕs-ϕm)/(ϕM-ϕs)), and T=2|Ω|.Substituting (52) into dϕ/dξ=y and integrating it along the heteroclinic orbit yield equation
(55)∫ϕsϕ‍ds(s-ϕm)(ϕM-s)=16α∫0ξ‍ds.Completing (55) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(56)u(x,t)=ϕmsin2(Ω+ω|x-ct-2nT|)+ϕMcos2(Ω+ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…, ω=1/26α, Ω=tan-1((ϕM-ϕs)/(ϕs-ϕm)), and T=2|Ω|.From Figure6(i), we see that there are a heteroclinic orbit and an open curve of system (7) defined by H(ϕ,y)=hs when α<0,c<-β/2α, and -(12α2c2-4αβc-β2)/32α2<g<β(4αc+β)/8α2. The heteroclinic orbit connecting with the saddle points (ϕs,Y±) and passing point (ϕM,0); its expression is
(57)y=±1-6α(ϕ-ϕM)(ϕ-ϕm),ϕM≤ϕ≤ϕs,
where ϕm,M=((β+6αc)±δ)/4α and δ=-3β2-12αβc+36α2c2+96α2g.Substituting (57) into dϕ/dξ=y and integrating it along the heteroclinic orbit yield equation
(58)∫ϕϕs‍ds(s-ϕM)(s-ϕm)=1-6α∫0ξ‍ds.Completing (58) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(59)u(x,t)=ϕMcosh2(Ω-ω|x-ct-2nT|)-ϕmsinh2(Ω-ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…,ω=1/2-6α, Ω=tanh-1((ϕs-ϕM)/(ϕs-ϕm)), and T=2|Ω|.From Figure6(j), we see that there are a heteroclinic orbit and an open curve of system (7) defined by H(ϕ,y)=hs when α<0,c>-β/2α, and -(12α2c2-4αβc-β2)/32α2<g<β(4αc+β)/8α2. The heteroclinic orbit connecting with the saddle points (ϕs,Y±) and passing point (ϕm,0); its expression is
(60)y=±1-6α(ϕm-ϕ)(ϕM-ϕ),ϕs≤ϕ≤ϕm,
where ϕm,M=((β+6αc)±δ)/4α and δ=-3β2-12αβc+36α2c2+96α2g.Substituting (60) into dϕ/dξ=y and integrating it along the heteroclinic orbit yield equation
(61)∫ϕsϕ‍ds(ϕm-s)(ϕM-s)=1-6α∫0ξ‍ds.Completing (61) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(62)u(x,t)=ϕmcosh2(Ω+ω|x-ct-2nT|)-ϕMsinh2(Ω+ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…, ω=1/2-6α, Ω=tanh-1((ϕm-ϕs)/(ϕM-ϕs)), and T=2|Ω|.


## 3.1. Peakon Solutions

From Figure2(d), we see that there are two heteroclinic orbits of system (7) defined by H(ϕ,y)=-β(β2+12αβc+36α2c2)/96α3 connecting with the saddle points (ϕs,±((2αc+β)-6α/8α2)) and ((6αc+β)/4α,0) when α<0, c<-β/2α, and g=-(12α2c2-4αβc-β2)/32α2. Their expressions are
(11)y=±1-6α(ϕ-6αc+β4α),6αc+β4α<ϕ≤ϕs.Substituting (11) into dϕ/dξ=y and integrating it along the heteroclinic orbits yield equation
(12)∫ϕϕs‍dss-((6αc+β)/4α)=1-6α∫0ξ‍ds.Completing (12) and using transformation (4), we can get a peakon solution of (3) as follow:
(13)u(x,t)=6αc+β4α+(ϕs-6αc+β4α)e-ω|x-ct|,
where ω=1/-6α. The profiles of (13) are shown in Figures 3(a) and 3(b).Peakons of (3) for x=1.
(a)
α
=
-
1
,
β
=
3
,
c
=
1
(b)
α
=
-
1
,
β
=
0
,
c
=
-
1
(c)
α
=
-
1
,
β
=
-
3
,
c
=
-
1
(d)
α
=
-
1
,
β
=
0
,
c
=
1From Figure2(k), we see that there are two heteroclinic orbits of system (7) defined by H(ϕ,y)=-β(β2+12αβc+36α2c2)/96α3 connecting with the saddle points (ϕs,∓((2αc+β)-6α/8α2)) and ((6αc+β)/4α,0) when α<0, c>-β/2α, and g=-(12α2c2-4αβc-β2)/32α2. Their expressions are
(14)y=±1-6α(6αc+β4α-ϕ),ϕs≤ϕ<6αc+β4α.Substituting (14) into dϕ/dξ=y and integrating it along the heteroclinic orbits yield equation
(15)∫ϕsϕ‍ds((6αc+β)/4α)-s=1-6α∫0ξ‍ds.Completing (15) and using transformation (4), we can get a peakon solution of (3) which is the same as (13). The profiles are shown in Figures 3(c) and 3(d).


## 3.2. Solitary Wave Solutions

From Figure1(c), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h1 connecting with saddle point (ϕ1,0) and passing point (ϕm,0) when α>0, c<-β/2α, and -c2/2<g<β(4αc+β)/8α2, where ϕm=c-22g+c2. Its expression is
(16)y=±16αϕ1-ϕϕs-ϕ(ϕ-ϕm)(ϕs-ϕ),ϕm≤ϕ<ϕ1.Substituting (16) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(17)∫ϕmϕ‍(ϕs-s)ds(ϕ1-s)(s-ϕm)(ϕs-s)=16α∫0ξ‍ds.Completing (17) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(18)u(x,t)=ϕ1(ϕmcosh2(ωχ)-ϕssinh2(ωχ))-ϕmϕs(ϕmsinh2(ωχ)-ϕscosh2(ωχ))+ϕ1,t=1c(x-6α(χ+2tan-1(2ωtanh(ωχ)))),
where χ is a new parametric variable and ω=(1/2)(ϕ1-ϕm)/(ϕs-ϕ1). The profiles of (18) are shown in Figures 4(a) and 4(b).Solitary waves of (3) for x=1.
(a)
α
=
1
,
β
=
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3
,
c
=
1
,
g
=
-
0.45
(b)
α
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1
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β
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0
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c
=
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2
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g
=
-
1
(c)
α
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1
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β
=
3
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c
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1
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g
=
-
0.4
(d)
α
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1
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β
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0
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c
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2
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g
=
-
1
(e)
α
=
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1
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β
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3
,
c
=
1
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g
=
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0.49
(f)
α
=
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1
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β
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0
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c
=
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1
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g
=
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0.4
(g)
α
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1
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β
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3
,
c
=
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1
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g
=
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0.49
(h)
α
=
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1
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β
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0
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c
=
2
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g
=
-
1.7From Figure1(h), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h2 connecting with saddle point (ϕ2,0) and passing point (ϕM,0) when α>0, c>-β/2α, and -c2/2<g<β(4αc+β)/8α2, where ϕM=c+22g+c2. Its expression is
(19)y=±16αϕ-ϕ2ϕ-ϕs(ϕM-ϕ)(ϕ-ϕs),ϕ2<ϕ≤ϕM.Substituting (19) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(20)∫ϕϕM‍(s-ϕs)ds(s-ϕ2)(ϕM-s)(s-ϕs)=16α∫0ξ‍ds.Completing (20) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(21)u(x,t)=ϕ2(ϕMcosh2(ωχ)-ϕssinh2(ωχ))-ϕMϕs(ϕMsinh2(ωχ)-ϕscosh2(ωχ))+ϕ2,t=1c(x-6α(χ-2tan-1(2ωtanh(ωχ)))),
where ω=(1/2)(ϕM-ϕ2)/(ϕ2-ϕs). The profiles of (21) are shown in Figures 4(c) and 4(d).From Figure2(c), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h2 connecting with saddle point (ϕ2,0) and passing point (ϕM,0) when α<0, c<-β/2α, and -c2/2<g<-(12α2c2-4αβc-β2)/32α2, where ϕM=c+22g+c2. Its expression is
(22)y=±1-6αϕ-ϕ2ϕs-ϕ(ϕM-ϕ)(ϕs-ϕ),ϕ2<ϕ≤ϕM.Substituting (22) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(23)∫ϕϕM‍(ϕs-s)ds(s-ϕ2)(ϕM-s)(ϕs-s)=1-6α∫0ξ‍ds.Completing (23) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(24)u(x,t)=ϕ2(ϕMcosh2(ωχ)-ϕssinh2(ωχ))-ϕMϕs(ϕMsinh2(ωχ)-ϕscosh2(ωχ))+ϕ2,t=1c(x--6α(χ+2tanh-1(2ωtanh(ωχ)))),
where ω=(1/2)(ϕM-ϕ2)/(ϕs-ϕ2). The profiles of (24) are shown in Figures 4(e) and 4(f).From Figure2(j), we see that there is one homoclinic orbit of system (7) defined by H(ϕ,y)=h1 connecting with saddle point (ϕ1,0) and passing point (ϕm,0) when α<0, c>-β/2α, -c2/2<g<-(12α2c2-4αβc-β2)/32α2, where ϕm=c-22g+c2. Its expression is
(25)y=±1-6αϕ1-ϕϕ-ϕs(ϕ-ϕm)(ϕ-ϕs),ϕm≤ϕ<ϕ1.Substituting (25) into dϕ/dξ=y and integrating it along the homoclinic orbit yield equation
(26)∫ϕmϕ‍(s-ϕs)ds(ϕ1-s)(s-ϕm)(s-ϕs)=1-6α∫0ξ‍ds.Completing (26) and using transformation (4), we can get a solitary wave solution of (3) as follows:
(27)u(x,t)=ϕ1(ϕmcosh2(ωχ)-ϕssinh2(ωχ))-ϕmϕs(ϕmsinh2(ωχ)-ϕscosh2(ωχ))+ϕ1,t=1c(x--6α(χ-2tanh-1(2ωtanh(ωχ)))),
where ω=(1/2)(ϕ1-ϕm)/(ϕ1-ϕs). The profiles of (27) are shown in Figures 4(g) and 4(h).


## 3.3. Smooth Periodic Wave Solutions

From Figure1(d), we see that there is a periodic orbit of system (7) defined by H(ϕ,y)=β2(3αc+β)/12α3 enclosing the center point ((4αc+β)/2α,0) and passing points (ϕs,0) and ((3αc+β)/α,0) when α>0, c<-β/2α, and g=β(4αc+β)/8α2. Its expression is
(28)y=±16α(ϕ-3αc+βα)(ϕs-ϕ),3αc+βα≤ϕ≤ϕs.Substituting (28) into dϕ/dξ=y and integrating it along the periodic orbit yield equation
(29)∫(3αc+β)/αϕ‍ds(s-((3αc+β)/α))(ϕs-s)=16α∫0ξ‍ds.Completing (29) and using transformation (4), we can get a smooth periodic wave solution of (3) as follows:
(30)u(x,t)=ϕssin2(ω(x-ct))+3αc+βαcos2(ω(x-ct)),
where ω=1/26α. The profiles of (30) are shown in Figures 5(a) and 5(b).Smooth periodic waves of (3) for x=1.
(a)
α
=
1
,
β
=
-
3
,
c
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α
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1From Figure1(i), we see that there is a periodic orbit of system (7) defined by H(ϕ,y)=β2(3αc+β)/12α3 enclosing the center point ((4αc+β)/2α,0) and passing points (ϕs,0) and ((3αc+β)/α,0) when α>0, c>-β/2α, and g=β(4αc+β)/8α2. Its expression is
(31)y=±16α(3αc+βα-ϕ)(ϕ-ϕs),ϕs≤ϕ≤3αc+βα.Substituting (31) into dϕ/dξ=y and integrating it along the periodic orbit yield equation
(32)∫ϕ(3αc+β)/α‍ds(((3αc+β)/α)-s)(s-ϕs)=16α∫0ξ‍ds.Completing (32) and using transformation (4), we can get a smooth periodic wave solution of (3) which is the same as (30). The profiles are shown in Figures 5(c) and 5(d).


## 3.4. Compacton Solutions

For givenh∈(h1,hs), hs<h2 (or h∈(h1,h2), hs>h2) in Figure 1(c), the level curve defined by H(ϕ,y)=h is shown in Figure 6(a).The level curves defined byH(ϕ,y)=h.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)For givenh∈(hs,h2), h1<hs in Figure 1(c), h∈(h1,h2) in Figure 1(d), and h∈(hs,h2) in Figures 1(e) and 1(j), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(b).For givenh∈(hs,h2), hs>h1 (or h∈(h1,h2), hs<h1) in Figure 1(h),the level curve defined by H(ϕ,y)=h is shown in Figure 6(c).For givenh∈(h1,hs), hs<h2 in Figure 1(h), and h∈(h1,h2) in Figure 1(i), h∈(h1,hs) in Figures 1(e) and 1(j), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(d).For givenh∈(hs,h2) in Figures 2(e), 2(f), 2(g), and 2(n), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(e).For givenh∈(h1,hs) in Figures 2(g), 2(l), 2(m), and 2(n), respectively, the level curve defined by H(ϕ,y)=h is shown in Figure 6(f).From Figure6(a), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h when α>0, c<-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(h1,hs), hs<h2 (or h∈(h1,h2), and hs>h2). The open curve passes point (γ1,0) and approaches the line ϕ=ϕs; its expression is
(33)y=±16α(ϕ-γ1)(ϕ-γ2)(ϕ-γ3)ϕs-ϕ,γ1≤ϕ<ϕs,
where γ1, γ2, and γ3(γ3<γ2<γ1<ϕs) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈-2.681985997, γ2≈-2.948887226, γ3≈-3.369126778 when α=0.2, β=1, c=-3, g=-4.48, and h=-8.882, and γ1≈-0.06941350133, γ2≈-0.5430641149, and γ3≈-2.387522384 when α=1, β=0, c=-1, g=-0.25, and h=-0.03.Substituting (33) into dϕ/dξ=y and integrating it along the open curve yield equation
(34)∫γ1ϕ‍ϕs-s(s-γ1)(s-γ2)(s-γ3)ds=16α∫0ξ‍ds.Completing (34) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(35)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(ϕs-γ2)(γ1-γ3)2(ϕs-γ1)6α|x-ct|,
where u~=sn-1((ϕs-γ2)(u(x,t)-γ1)/(ϕs-γ1)(u(x,t)-γ2),k), ϕ~=sin-1((ϕs-γ2)(u(x,t)-γ1)/(ϕs-γ1)(u(x,t)-γ2)), k=(ϕs-γ1)(γ2-γ3)/(ϕs-γ2)(γ1-γ3), and α12=(ϕs-γ1)/(ϕs-γ2); Π(·,·,·) is Legendre’s incomplete elliptic integral of the third kind, and sn-1(·,·) is the inverse function of the Jacobian elliptic function sn(·,·) [15].From Figure6(b), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α>0, c<-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(hs,h2), and h1<hs,(ii)
α>0, c<-β/2α, g=β(4αc+β)/8α2, and h∈(h1,h2),(iii)
α>0, c≠-β/2α, g>β(4αc+β)/8α2, and h∈(hs,h2).The open curve passes point(γ1,0) and approaches the line ϕ=ϕs; its expression is
(36)y=±16α(γ1-ϕ)(ϕ-γ2)(ϕ-γ3)ϕ-ϕs,ϕs<ϕ≤γ1,
where γ1, γ2, and γ3 (γ3<γ2<ϕs<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈0.6863083159, γ2≈0.243331752, and γ3≈-0.1796400679 when α=1,β=-1, c=0.25,g=0, and h=-0.01 and γ1≈5.467670461,γ2≈-1.060802034, γ3≈-1.406868427 when α=1, β=2, c=1, g=2, and h=2.72.Substituting (36) into dϕ/dξ=y and integrating it along the open curve yield equation
(37)∫ϕγ1‍s-ϕs(γ1-s)(s-γ2)(s-γ3)ds=16α∫0ξ‍ds.Completing (37) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(38)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-γ2)(ϕs-γ3)2(γ1-ϕs)6α|x-ct|,
where u~=sn-1((ϕs-γ3)(γ1-u(x,t))/(γ1-ϕs)(u(x,t)-γ3),k), ϕ~=sin-1((ϕs-γ3)(γ1-u(x,t))/(γ1-ϕs)(u(x,t)-γ3)), k=(γ1-ϕs)(γ2-γ3)/(γ1-γ2)(ϕs-γ3), and α12=(ϕs-γ1)/(ϕs-γ3).From Figure6(c), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h when α>0, c>-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(hs,h2), hs>h1 (or h∈(h1,h2), and hs<h1). The open curve passes point (γ3,0) and approaches the line ϕ=ϕs; its expression is
(39)y=±16α(γ1-ϕ)(γ2-ϕ)(γ3-ϕ)ϕ-ϕs,ϕs<ϕ≤γ3,
where γ1, γ2, and γ3 (ϕs<γ3<γ2<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈-1.725343175, γ2≈-1.954509591, and γ3≈-2.320147234 when α=0.2, β=1, c=-2, g=-1.985, and h=-2.608 and γ1≈0.9464101615, γ2=0.3, and γ3≈0.2535898385 when α=2, β=-1, c=0.5, g=-0.1, and h=0.024.Substituting (39) into dϕ/dξ=y and integrating it along the open curve yield equation
(40)∫ϕγ3‍s-ϕs(γ1-s)(γ2-s)(γ3-s)ds=16α∫0ξ‍ds.Completing (40) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follow:
(41)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-γ3)(γ2-ϕs)2(γ3-ϕs)6α|x-ct|,
where u~=sn-1((γ2-ϕs)(γ3-u(x,t))/(γ3-ϕs)(γ2-u(x,t)),k), ϕ~=sin-1((γ2-ϕs)(γ3-u(x,t))/(γ3-ϕs)(γ2-u(x,t))), k=(γ1-γ2)(γ3-ϕs)/(γ1-γ3)(γ2-ϕs), and α12=(γ3-ϕs)/(γ2-ϕs).From Figure6(d), we see that there are a periodic orbit and an open curve of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α>0, c>-β/2α, -c2/2<g<β(4αc+β)/8α2, h∈(h1,hs), and hs<h2,(ii)
α>0, c>-β/2α, g=β(4αc+β)/8α2, and h∈(h1,h2),(iii)
α>0, c≠-β/2α, g>β(4αc+β)/8α2, and h∈(h1,hs).The open curve passes point(γ3,0) and approaches the line ϕ=ϕs, its expression is
(42)y=±16α(γ1-ϕ)(γ2-ϕ)(ϕ-γ3)ϕs-ϕ,γ3≤ϕ<ϕs,
where γ1, γ2, and γ3 (γ3<ϕs<γ2<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈2.557900553, γ2≈0.7351513251, and γ3≈-2.393051878 when α=1, β=-1, c=0.3, g=1, and h=-1.5 and γ1≈5.743824982, γ2≈-0.6449004894, γ3≈-8.098924493 when α=0.1, β=1, c=-1, g=7.5, and h=10.Substituting (42) into dϕ/dξ=y and integrating it along the open curve yield equation
(43)∫γ3ϕ‍ϕs-s(γ1-s)(γ2-s)(s-γ3)ds=16α∫0ξ‍ds.Completing (43) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(44)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-ϕs)(γ2-γ3)2(ϕs-γ3)6α|x-ct|,
where u~=sn-1((γ1-ϕs)(u(x,t)-γ3)/(ϕs-γ3)(γ1-u(x,t)),k), ϕ~=sin-1((γ1-ϕs)(u(x,t)-γ3)/(ϕs-γ3)(γ1-u(x,t))), k=(γ1-γ2)(ϕs-γ3)/(γ1-ϕs)(γ2-γ3), and α12=(γ3-ϕs)/(γ1-ϕs).From Figure6(e), we see that there are three open curves of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α<0,c<-β/2α,g>-(12α2c2-4αβc-β2)/32α2, and h∈(hs,h2),(ii)
α<0,c>-β/2α,g>β(4αc+β)/8α2, and h∈(hs,h2).One of them passes point(γ2,0) and approaches the line ϕ=ϕs; its expression is
(45)y=±1-6α(γ1-ϕ)(ϕ-γ2)(ϕ-γ3)ϕs-ϕ,γ2≤ϕ<ϕs,
where γ1, γ2, and γ3(γ3<γ2<ϕs<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈2.064177772,γ2≈-0.3054072893, and γ3≈-4.758770483 when α=-2, β=2, c=-1, and g=1.5 and h=1, γ1≈5.466333036, γ2≈-1.035930847, and γ3≈-1.430402188 when α=-0.5, β=-1, c=1, g=2, and h=2.7.Substituting (45) into dϕ/dξ=y and integrating it along the open curve yield equation
(46)∫γ2ϕ‍ϕs-s(γ1-s)(s-γ2)(s-γ3)ds=1-6α∫0ξ‍ds.Completing (46) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(47)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-γ2)(ϕs-γ3)2(ϕs-γ2)-6α|x-ct|,
where u~=sn-1((ϕs-γ3)(u(x,t)-γ2)/(ϕs-γ2)(u(x,t)-γ3),k), ϕ~=sin-1((ϕs-γ3)(u(x,t)-γ2)/(ϕs-γ2)(u(x,t)-γ3)), k=(ϕs-γ2)(γ1-γ3)/(γ1-γ2)(ϕs-γ3), and α12=(ϕs-γ2)/(ϕs-γ3).From Figure6(f), we see that there are three open curves of system (7) defined by H(ϕ,y)=h if and only if one of the following conditions holds:(i)
α<0,c<-β/2α,g>β(4αc+β)/8α2, and h∈(h1,hs),(ii)
α<0,c>-β/2α,g>-(12α2c2-4αβc-β2)/32α2, and h∈(h1,hs).One of them passes point(γ2,0) and approaches the line ϕ=ϕs; its expression is
(48)y=±1-6α(γ1-ϕ)(γ2-ϕ)(ϕ-γ3)ϕ-ϕs,ϕs<ϕ≤γ2,
where γ1, γ2, and γ3(γ3<ϕs<γ2<γ1) are three real roots of ψ3-3cψ2-6gψ-3h=0. For example, γ1≈0.8657007565, γ2≈-0.091826066674, and γ3≈-3.77387469 when α=-2, β=-1, c=-1, g=0.5, and h=0.1, and γ1≈6.154523009, γ2≈2.662391044, and γ3≈0.1830859473 when α=-0.2, β=1, c=3, g=-3, and h=1.Substituting (48) into dϕ/dξ=y and integrating it along the open curve yield equation
(49)∫ϕγ2‍s-ϕs(γ1-s)(γ2-s)(s-γ3)ds=1-6α∫0ξ‍ds.Completing (49) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows:
(50)1α12(u~+(α12-1)Π(ϕ~,α12,k))=(γ1-ϕs)(γ2-γ3)2(γ2-ϕs)-6α|x-ct|,
where u~=sn-1((γ1-ϕs)(γ2-u(x,t))/(γ2-ϕs)(γ1-u(x,t)),k), ϕ~=sin-1((γ1-ϕs)(γ2-u(x,t))/(γ2-ϕs)(γ1-u(x,t))), k=(γ2-ϕs)(γ1-γ3)/(γ1-ϕs)(γ2-γ3), and α12=(γ2-ϕs)/(γ1-ϕs).


## 3.5. Periodic Cusp Wave Solutions

For givenh=hs in Figure 1(e), the level curve defined by H(ϕ,y)=h is shown in Figure 6(g).For givenh=hs in Figure 1(j), the level curve defined by H(ϕ,y)=h is shown in Figure 6(h).For givenh=hs in Figure 2(e), the level curve defined by H(ϕ,y)=h is shown in Figure 6(i).For givenh=hs in Figure 2(l), the level curve defined by H(ϕ,y)=h is shown in Figure 6(j).From Figures6(g) and 6(h), we see that there are two heteroclinic orbits of system (7) defined by H(ϕ,y)=hs connecting with the saddle points (ϕs,Y±) and passing points (ϕm,0), and (ϕM,0), respectively, when α>0, c≠-β/2α, and g>β(4αc+β)/8α2, where ϕM,m=((β+6αc)±δ)/4α and δ=-3β2-12αβc+36α2c2+96α2g. Their expressions are, respectively,
(51)y=±16α(ϕ-ϕm)(ϕM-ϕ),ϕm≤ϕ≤ϕs,(52)y=±16α(ϕ-ϕm)(ϕM-ϕ),ϕs≤ϕ≤ϕM.Substituting (51) into the dϕ/dξ=y and integrating it along the heteroclinic orbit, yields equation
(53)∫ϕϕs‍ds(s-ϕm)(ϕM-s)=16α∫0ξ‍ds.Completing (53) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(54)u(x,t)=ϕmcos2(Ω-ω|x-ct-2nT|)+ϕMsin2(Ω-ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…, ω=1/26α, Ω=tan-1((ϕs-ϕm)/(ϕM-ϕs)), and T=2|Ω|.Substituting (52) into dϕ/dξ=y and integrating it along the heteroclinic orbit yield equation
(55)∫ϕsϕ‍ds(s-ϕm)(ϕM-s)=16α∫0ξ‍ds.Completing (55) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(56)u(x,t)=ϕmsin2(Ω+ω|x-ct-2nT|)+ϕMcos2(Ω+ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…, ω=1/26α, Ω=tan-1((ϕM-ϕs)/(ϕs-ϕm)), and T=2|Ω|.From Figure6(i), we see that there are a heteroclinic orbit and an open curve of system (7) defined by H(ϕ,y)=hs when α<0,c<-β/2α, and -(12α2c2-4αβc-β2)/32α2<g<β(4αc+β)/8α2. The heteroclinic orbit connecting with the saddle points (ϕs,Y±) and passing point (ϕM,0); its expression is
(57)y=±1-6α(ϕ-ϕM)(ϕ-ϕm),ϕM≤ϕ≤ϕs,
where ϕm,M=((β+6αc)±δ)/4α and δ=-3β2-12αβc+36α2c2+96α2g.Substituting (57) into dϕ/dξ=y and integrating it along the heteroclinic orbit yield equation
(58)∫ϕϕs‍ds(s-ϕM)(s-ϕm)=1-6α∫0ξ‍ds.Completing (58) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(59)u(x,t)=ϕMcosh2(Ω-ω|x-ct-2nT|)-ϕmsinh2(Ω-ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…,ω=1/2-6α, Ω=tanh-1((ϕs-ϕM)/(ϕs-ϕm)), and T=2|Ω|.From Figure6(j), we see that there are a heteroclinic orbit and an open curve of system (7) defined by H(ϕ,y)=hs when α<0,c>-β/2α, and -(12α2c2-4αβc-β2)/32α2<g<β(4αc+β)/8α2. The heteroclinic orbit connecting with the saddle points (ϕs,Y±) and passing point (ϕm,0); its expression is
(60)y=±1-6α(ϕm-ϕ)(ϕM-ϕ),ϕs≤ϕ≤ϕm,
where ϕm,M=((β+6αc)±δ)/4α and δ=-3β2-12αβc+36α2c2+96α2g.Substituting (60) into dϕ/dξ=y and integrating it along the heteroclinic orbit yield equation
(61)∫ϕsϕ‍ds(ϕm-s)(ϕM-s)=1-6α∫0ξ‍ds.Completing (61) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows:
(62)u(x,t)=ϕmcosh2(Ω+ω|x-ct-2nT|)-ϕMsinh2(Ω+ω|x-ct-2nT|),x-ct∈[(2n-1)T,(2n+1)T],
where n=0,±1,±2,…, ω=1/2-6α, Ω=tanh-1((ϕm-ϕs)/(ϕM-ϕs)), and T=2|Ω|.


## 4. Numerical Simulations

In this section, we simulate the planar diagrams of the compactons and the periodic cusp waves of (3). From the derivations of (6) and (7), it is seen that if ϕ=ϕ(ξ) and y=y(ξ) are the parameter expressions of an orbit of system (7) and they satisfy ϕ(0)=ϕ0 and y(0)=y0, then the diagram of u=ϕ(ξ) and the integral curve of (6) with initial conditions ϕ(0)=ϕ0 and ϕ′(0)=y0 are the same. Therefore we can use the simulations of integral curves of (6) to test the validity of exact travelling wave solutions (35), (38), (41), (44), (47), (50), (54), (56), (59), and (62).Example 1.
Chooseα, β, c, g, and h satisfying the parametric conditions of Figure 6(a); that is α=0.2, β=1, c=-3, g=-4.48, and h=-8.88, and let ϕ(0)=-2.6 in (33), then it follows that ϕ′(0)=y(0)≈0.3651483603 or ϕ′(0)=y(0)≈-0.3651483603. Taking ϕ(0)=-2.6, and y(0)=0.3651483603 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (33) as Figure 7(a).The numerical simulations of integral curves of (3).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)Example 2.
Chooseα, β, c, g, and h satisfying the parametric conditions of Figure 6(b); that is α=0.2, β=0, c=-1, g=-0.25, and h=0.2, and let ϕ(0)=0.001 in (36); then it follows that ϕ′(0)=y(0)≈22.33265544 or ϕ′(0)=y(0)≈-22.33265544. Taking ϕ(0)=0.001, and y(0)=22.33265544 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (36) as Figure 7(b).Example 3.
Chooseα, β, c, g, and h satisfying the parametric conditions of Figure 6(c); that is α=0.2, β=1, c=-2, g=-1.985, h=-2.608 and let ϕ(0)=-2.4 in (39); then it follows that ϕ′(0)=y(0)≈0.4472135955 or ϕ′(0)=y(0)≈-0.4472135955. Taking ϕ(0)=-2.4 and y(0)=-0.4472135955 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (39) as Figure 7(c).Example 4.
Chooseα, β, c, g, and h satisfying the parametric conditions of Figure 6(d); that is α=0.2, β=0, c=2, g=0, and h=-5, and let ϕ(0)=-1.2 in (42); then it follows that ϕ′(0)=y(0)≈1.793506807 or ϕ′(0)=y(0)≈-1.793506807. Taking ϕ(0)=-1.2 and y(0)=1.793506807 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (42) as Figure 7(d).Example 5.
Chooseα, β, c, g, and h satisfying the parametric conditions of Figure 6(e); that is α=-0.2, β=-1, c=-3, g=-4.43, h=-8.58, and let ϕ(0)=-2.9 in (45); then it follows that ϕ′(0)=y(0)≈0.2922613022 or ϕ′(0)=y(0)≈-0.2922613022. Taking ϕ(0)=-2.9 and y(0)=0.2922613022 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (45) as Figure 7(e).Example 6.
Chooseα, β, c, g, and h satisfying the parametric conditions of Figure 6(f); that is α=-0.2, β=-1, c=-2, g=-1.92, and h=-2.36, and let ϕ(0)=-2.1 in (48); then it follows that ϕ′(0)=y(0)≈0.4257346591 or ϕ′(0)=y(0)≈-0.4257346591. Taking ϕ(0)=-2.1 and y(0)=0.4257346591 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the open curve (48) as Figure 7(f).Example 7.
Chooseα, β, c, and g satisfying the parametric conditions of Figure 6(g); that is α=0.2, β=1, c=-3, and g=-4.1. Let ϕ(0)=-4 in (51); then it follows that ϕ′(0)=y(0)≈1.172603942 or ϕ′(0)=y(0)≈-1.172603942. Taking ϕ(0)=-4 and y(0)=1.172603942 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (51) as Figure 7(g). Let ϕ(0)=-2 in (52), then it follows that ϕ′(0)=y(0)≈0.7337120246 or ϕ′(0)=y(0)≈-0.7337120246. Taking ϕ(0)=-2,y(0)=0.7337120246 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (52) as Figure 7(h).Example 8.
Chooseα, β, c, and g satisfying the parametric conditions of Figure 6(h); that is α=0.2, β=-1, c=3, and g=-3. Let ϕ(0)=2 in (51), then it follows that ϕ′(0)=y(0)≈2.457912394 or ϕ′(0)=y(0)≈-2.457912394. Taking ϕ(0)=2 and y(0)=2.457912394 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (51) as Figure 7(i). Let ϕ(0)=5 in (52); then it follows that ϕ′(0)=y(0)≈2.188957133 or ϕ′(0)=y(0)≈-2.188957133. Taking ϕ(0)=5,y(0)=2.188957133 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (52) as Figure 7(j).Example 9.
Chooseα, β, c, and g satisfying the parametric conditions of Figure 6(i); that is α=-0.2, β=-1, c=-3, and g=-4.43 and let ϕ(0)=-2.7 in (57), then it follows that ϕ′(0)=y(0)≈0.2397915762 or ϕ′(0)=y(0)≈-0.2397915762. Taking ϕ(0)=-2.7 and y(0)=0.2397915762 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (57) as Figure 7(k).Example 10.
Chooseα, β, c, and g satisfying the parametric conditions of Figure 6(j); that is α=-1, β=2, c=2, and g=-1.6 and let ϕ(0)=-1.1 in (60); then it follows that ϕ′(0)=y(0)≈0.2272296966 or ϕ′(0)=y(0)≈-0.2272296966. Taking ϕ(0)=1.1 and y(0)=0.2272296966 as initial values of (6), using the Maple, we get the numerical simulation of the integral curve which corresponds to the heteroclinic orbit (60) as Figure 7(l).Figure7 implies that our theoretic results of the compacton and the periodic cusp wave solutions are identical with the numerical simulations.


## 5. Conclusion

In this paper, we studied the bifurcations of travelling wave solutions of (3) using the approach of dynamical system and obtained some exact peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions. Also, the compactons and the periodic cusp waves are simulated by approach of numerical simulation. The results of this paper have enriched results of references [9–11].



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