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

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.

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 𝑢_𝑡+𝑢²_𝑥+𝑢²_𝑥𝑥𝑥=0, (1) 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] 𝑢_𝑡+𝑢𝑢_𝑥+𝛼2𝑢𝑢_𝑥𝑥𝑥+4𝑢_𝑥𝑢_𝑥𝑥=0, (2) 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] 𝑢_𝑡+𝑢𝑢_𝑥+𝛼2𝑢𝑢_𝑥𝑥𝑥+4𝑢_𝑥𝑢_𝑥𝑥+𝛽𝑢_𝑥𝑥𝑥=0. (3) 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 that 𝛼≠0. Using the following independent variable transformation: 𝑢(𝑥,𝑡)=𝜙(𝜉),𝜉=𝑥−𝑐𝑡, (4) where 𝑐(𝑐≠0) is the wave speed, and substituting (4) into (3), we obtain −𝑐𝜙^′+𝜙𝜙^′+𝛼2𝜙𝜙^′′′+4𝜙^′𝜙^′′+𝛽𝜙^′′′=0, (5) where “′” is the derivative with respect to 𝜉.

Integrating (5) once with respect to 𝜉, we have 1−𝑐𝜙+2𝜙²+𝛼2𝜙𝜙^′′+𝜙^′2+𝛽𝜙^′′=𝑔, (6) where 𝑔 is the integral constant.

Letting 𝑦=𝑑𝜙/𝑑𝜉, we get the following planar dynamical system: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=()𝜙𝑑𝜉𝑔+𝑐𝜙−1/2²−𝛼𝑦².2𝛼𝜙+𝛽 (7)

The rest of this paper is organized as follows. In Section 2, 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 transformation 𝑑𝜉=(2𝛼𝜙+𝛽)𝑑𝜏, it carries (7) into the Hamiltonian system 𝑑𝜙=𝑑𝜏2𝛼𝜙+𝛽𝑦,𝑑𝑦1𝑑𝜏=𝑔+𝑐𝜙−2𝜙²−𝛼𝑦². (8) System (8) has the following first integral: 𝐻𝜙,𝑦=2𝛼𝜙+𝛽𝑦²−2𝑔𝜙+𝑐𝜙²−13𝜙³=ℎ. (9)

For a fixed ℎ, 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.

Write Δ=−2𝛼(𝛽²+4𝛼𝛽𝑐−8𝑔𝛼²), 𝜙_𝑠=−𝛽/2𝛼. Clearly, when 𝑔>−𝑐²/2, system (8) has two equilibrium points at (𝜙₁,0) and (𝜙₂,0) in 𝜙-axis, where 𝜙_1,2=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=𝜙_𝑠 at (𝜙_𝑠,𝑌_±), where 𝑌_±=±(1/4𝛼²)√Δ. When Δ<0, there is no equilibrium point of system (8) in line 𝜙=𝜙_𝑠.

Let 𝑀(𝜙_𝑒,𝑦_𝑒) be the coefficient matrix of the linearized system of (8) at equilibrium point (𝜙_𝑒,𝑦_𝑒); then we have Trace 𝑀(𝜙_𝑒,0)=0 and 𝐽𝜙₁,0=det𝑀𝜙₁=,02𝑔+𝑐²2𝛼2𝑔+𝑐²+,𝐽𝜙2𝛼𝑐+𝛽₂,0=det𝑀𝜙₂=,02𝑔+𝑐²2𝛼2𝑔+𝑐²−,𝐽𝜙2𝛼𝑐+𝛽_𝑠,𝑌_±=det𝑀𝜙_𝑠,𝑌_±=−4𝛼²𝑌²_±. (10)

For an equilibrium point (𝜙_𝑒,𝑦_𝑒) of a planar integrable system, we know that (𝜙_𝑒,𝑦_𝑒) is a saddle point if 𝐽(𝜙_𝑒,𝑦_𝑒)<0, a center point if 𝐽(𝜙_𝑒,𝑦_𝑒)>0 and Trace 𝑀(𝜙_𝑒,𝑦_𝑒)=0, and a cusp if 𝐽(𝜙_𝑒,𝑦_𝑒)=0 and the Poincaré index of (𝜙_𝑒,𝑦_𝑒) 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=𝜙_𝑠. 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.

(a) 𝑐<−𝛽/2𝛼,𝑔<−𝑐2/2

(b) 𝑐<−𝛽/2𝛼,𝑔=−𝑐2/2

(c) 𝑐<−𝛽/2𝛼,−𝑐2/2<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼2

(d) 𝑐<−𝛽/2𝛼,𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼2

(e) 𝑐<−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼2

(f) 𝑐>−𝛽/2𝛼,𝑔<−𝑐2/2

(g) 𝑐>−𝛽/2𝛼,𝑔=−𝑐2/2

(h) 𝑐>−𝛽/2𝛼,−𝑐2/2<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼2

(i) 𝑐>−𝛽/2𝛼,𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼2

(j) 𝑐>−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼2

Figure 1 

Bifurcation sets and phase portraits of system (7) when 𝛼>0.

(a) 𝑐<−𝛽/2𝛼,𝑔<−𝑐2/2

(b) 𝑐<−𝛽/2𝛼,𝑔=−𝑐2/2

(c) 𝑐<−𝛽/2𝛼,−𝑐2/2<𝑔<−(12𝛼2𝑐2−4𝛼𝛽𝑐−𝛽2)/32𝛼2

(d) 𝑐<−𝛽/2𝛼,𝑔=−(12𝛼2𝑐2−4𝛼𝛽𝑐−𝛽2)/32𝛼2

(e) 𝑐<−𝛽/2𝛼,−(12𝛼2𝑐2−4𝛼𝛽𝑐−𝛽2)/32𝛼2<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼2

(f) 𝑐<−𝛽/2𝛼,𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼2

(g) 𝑐<−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼2

(h) 𝑐>−𝛽/2𝛼,𝑔<−𝑐2/2

(i) 𝑐>−𝛽/2𝛼,𝑔=−𝑐2/2

(j) 𝑐>−𝛽/2𝛼,−𝑐2/2<𝑔<−(12𝛼2𝑐2−4𝛼𝛽𝑐−𝛽2)/32𝛼2

(k) 𝑐>−𝛽/2𝛼,𝑔=−(12𝛼2𝑐2−4𝛼𝛽𝑐−𝛽2)/32𝛼2

(l) 𝑐>−𝛽/2𝛼,−(12𝛼2𝑐2−4𝛼𝛽𝑐−𝛽2)/32𝛼2<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼2

(m) 𝑐>−𝛽/2𝛼,𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼2

(n) 𝑐>−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼2

Figure 2 

Bifurcation sets and phase portraits of system (7) when 𝛼<0.

3. Exact Travelling Wave Solutions of (3)

Denote that ℎ_1,2=−(2𝑐/3)(3𝑔+𝑐²)∓(2/3)(2𝑔+𝑐²)2𝑔+𝑐² and ℎ_𝑠=−𝛽(𝛽²+6𝛼𝛽𝑐−24𝑔𝛼²)/24𝛼³; we present some exact travelling wave solutions of (3) as follows.

3.1. Peakon Solutions

From Figure 2(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 (𝜙_𝑠,±((2𝛼𝑐+𝛽)√−6𝛼/8𝛼²)) and ((6𝛼𝑐+𝛽)/4𝛼,0) when 𝛼<0, 𝑐<−𝛽/2𝛼, and 𝑔=−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼². Their expressions are 1𝑦=±√−6𝛼𝜙−6𝛼𝑐+𝛽,4𝛼6𝛼𝑐+𝛽4𝛼<𝜙≤𝜙_𝑠. (11)

Substituting (11) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the heteroclinic orbits yield equation ^𝜙𝑠_𝜙𝑑𝑠=1𝑠−6𝛼𝑐+𝛽/4𝛼√−6𝛼^𝜉0𝑑𝑠. (12)

Completing (12) and using transformation (4), we can get a peakon solution of (3) as follow: 𝑢(𝑥,𝑡)=6𝛼𝑐+𝛽+𝜙4𝛼^𝑠−6𝛼𝑐+𝛽𝑒4𝛼^{−𝜔|𝑥−𝑐𝑡|}, (13) where 𝜔=1/√−6𝛼. The profiles of (13) are shown in Figures 3(a) and 3(b).

(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=1

Figure 3 

Peakons of (3) for 𝑥=1.

From Figure 2(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 (𝜙_𝑠,∓((2𝛼𝑐+𝛽)√−6𝛼/8𝛼²)) and ((6𝛼𝑐+𝛽)/4𝛼,0) when 𝛼<0, 𝑐>−𝛽/2𝛼, and 𝑔=−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼². Their expressions are 1𝑦=±√−6𝛼6𝛼𝑐+𝛽4𝛼−𝜙,𝜙_𝑠≤𝜙<6𝛼𝑐+𝛽.4𝛼 (14)

Substituting (14) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the heteroclinic orbits yield equation ^𝜙_𝜙𝑠𝑑𝑠=16𝛼𝑐+𝛽/4𝛼−𝑠√−6𝛼^𝜉0𝑑𝑠. (15)

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 Figure 1(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 (𝜙_𝑚,0) when 𝛼>0, 𝑐<−𝛽/2𝛼, and −𝑐²/2<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼², where 𝜙_𝑚=𝑐−22𝑔+𝑐². Its expression is 1𝑦=±√𝜙6𝛼¹−𝜙𝜙_𝑠−𝜙𝜙−𝜙_𝑚𝜙_𝑠−𝜙,𝜙_𝑚≤𝜙<𝜙₁. (16)

Substituting (16) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield equation ^𝜙_𝜙𝑚𝜙_𝑠−𝑠𝑑𝑠𝜙₁−𝑠𝑠−𝜙_𝑚𝜙_𝑠=1−𝑠√6𝛼^𝜉0𝑑𝑠. (17)

Completing (17) and using transformation (4), we can get a solitary wave solution of (3) as follows: 𝑢()=𝜙𝑥,𝑡¹𝜙_𝑚cosh²𝜔𝜒−𝜙_𝑠sinh²𝜔𝜒−𝜙_𝑚𝜙_𝑠𝜙_𝑚sinh²𝜔𝜒−𝜙_𝑠cosh²𝜔𝜒+𝜙₁,1𝑡=𝑐𝑥−√6𝛼𝜒+2tan⁻¹2𝜔tanh𝜔𝜒, (18) where 𝜒 is a new parametric variable and 𝜔=(1/2)(𝜙₁−𝜙_𝑚)/(𝜙_𝑠−𝜙₁). The profiles of (18) are shown in Figures 4(a) and 4(b).

(a) 𝛼=1,𝛽=−3,𝑐=1,𝑔=−0.45

(b) 𝛼=1,𝛽=0,𝑐=−2,𝑔=−1

(c) 𝛼=1,𝛽=3,𝑐=−1,𝑔=−0.4

(d) 𝛼=1,𝛽=0,𝑐=2,𝑔=−1

(e) 𝛼=−1,𝛽=3,𝑐=1,𝑔=−0.49

(f) 𝛼=−1,𝛽=0,𝑐=−1,𝑔=−0.4

(g) 𝛼=−1,𝛽=−3,𝑐=−1,𝑔=−0.49

(h) 𝛼=−1,𝛽=0,𝑐=2,𝑔=−1.7

Figure 4 

Solitary waves of (3) for 𝑥=1.

From Figure 1(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 (𝜙_𝑀,0) when 𝛼>0, 𝑐>−𝛽/2𝛼, and −𝑐²/2<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼², where 𝜙_𝑀=𝑐+22𝑔+𝑐². Its expression is 1𝑦=±√6𝛼𝜙−𝜙₂𝜙−𝜙_𝑠𝜙_𝑀−𝜙𝜙−𝜙_𝑠,𝜙₂<𝜙≤𝜙_𝑀. (19)

Substituting (19) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield equation ^𝜙𝑀_𝜙𝑠−𝜙_𝑠𝑑𝑠𝑠−𝜙₂𝜙_𝑀−𝑠𝑠−𝜙_𝑠=1√6𝛼^𝜉0𝑑𝑠. (20)

Completing (20) and using transformation (4), we can get a solitary wave solution of (3) as follows: 𝑢()=𝜙𝑥,𝑡²𝜙_𝑀cosh²𝜔𝜒−𝜙_𝑠sinh²𝜔𝜒−𝜙_𝑀𝜙_𝑠𝜙_𝑀sinh²𝜔𝜒−𝜙_𝑠cosh²𝜔𝜒+𝜙₂,1𝑡=𝑐𝑥−√6𝛼𝜒−2tan⁻¹2𝜔tanh𝜔𝜒, (21) where 𝜔=(1/2)(𝜙_𝑀−𝜙₂)/(𝜙₂−𝜙_𝑠). The profiles of (21) are shown in Figures 4(c) and 4(d).

From Figure 2(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 (𝜙_𝑀,0) when 𝛼<0, 𝑐<−𝛽/2𝛼, and −𝑐²/2<𝑔<−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼², where 𝜙_𝑀=𝑐+22𝑔+𝑐². Its expression is 1𝑦=±√−6𝛼𝜙−𝜙₂𝜙_𝑠−𝜙𝜙_𝑀−𝜙𝜙_𝑠−𝜙,𝜙₂<𝜙≤𝜙_𝑀. (22)

Substituting (22) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield equation ^𝜙𝑀_𝜙𝜙_𝑠−𝑠𝑑𝑠𝑠−𝜙₂𝜙_𝑀−𝑠𝜙_𝑠=1−𝑠√−6𝛼^𝜉0𝑑𝑠. (23)

Completing (23) and using transformation (4), we can get a solitary wave solution of (3) as follows: 𝑢()=𝜙𝑥,𝑡²𝜙_𝑀cosh²𝜔𝜒−𝜙_𝑠sinh²𝜔𝜒−𝜙_𝑀𝜙_𝑠𝜙_𝑀sinh²𝜔𝜒−𝜙_𝑠cosh²𝜔𝜒+𝜙₂,1𝑡=𝑐𝑥−√−6𝛼𝜒+2tanh⁻¹2𝜔tanh𝜔𝜒, (24) where 𝜔=(1/2)(𝜙_𝑀−𝜙₂)/(𝜙_𝑠−𝜙₂). The profiles of (24) are shown in Figures 4(e) and 4(f).

From Figure 2(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 (𝜙_𝑚,0) when 𝛼<0, 𝑐>−𝛽/2𝛼, −𝑐²/2<𝑔<−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼², where 𝜙_𝑚=𝑐−22𝑔+𝑐². Its expression is 1𝑦=±√𝜙−6𝛼¹−𝜙𝜙−𝜙_𝑠𝜙−𝜙_𝑚𝜙−𝜙_𝑠,𝜙_𝑚≤𝜙<𝜙₁. (25)

Substituting (25) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the homoclinic orbit yield equation ^𝜙_𝜙𝑚𝑠−𝜙_𝑠𝑑𝑠𝜙₁−𝑠𝑠−𝜙_𝑚𝑠−𝜙_𝑠=1√−6𝛼^𝜉0𝑑𝑠. (26)

Completing (26) and using transformation (4), we can get a solitary wave solution of (3) as follows: 𝑢()=𝜙𝑥,𝑡¹𝜙_𝑚cosh²𝜔𝜒−𝜙_𝑠sinh²𝜔𝜒−𝜙_𝑚𝜙_𝑠𝜙_𝑚sinh²𝜔𝜒−𝜙_𝑠cosh²𝜔𝜒+𝜙₁,1𝑡=𝑐𝑥−√−6𝛼𝜒−2tanh⁻¹2𝜔tanh𝜔𝜒, (27) where 𝜔=(1/2)(𝜙₁−𝜙_𝑚)/(𝜙₁−𝜙_𝑠). The profiles of (27) are shown in Figures 4(g) and 4(h).

3.3. Smooth Periodic Wave Solutions

From Figure 1(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 (𝜙_𝑠,0) and ((3𝛼𝑐+𝛽)/𝛼,0) when 𝛼>0, 𝑐<−𝛽/2𝛼, and 𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼². Its expression is 1𝑦=±√6𝛼𝜙−3𝛼𝑐+𝛽𝛼𝜙_𝑠−𝜙,3𝛼𝑐+𝛽𝛼≤𝜙≤𝜙_𝑠. (28)

Substituting (28) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield equation ^𝜙_{3𝛼𝑐+𝛽/𝛼}𝑑𝑠𝑠−3𝛼𝑐+𝛽/𝛼𝜙_𝑠=1−𝑠√6𝛼^𝜉0𝑑𝑠. (29)

Completing (29) and using transformation (4), we can get a smooth periodic wave solution of (3) as follows: 𝑢(𝑥,𝑡)=𝜙_𝑠sin²(𝜔(𝑥−𝑐𝑡))+3𝛼𝑐+𝛽𝛼cos²(𝜔(𝑥−𝑐𝑡)), (30) where 𝜔=1/2√6𝛼. The profiles of (30) are shown in Figures 5(a) and 5(b).

(a) 𝛼=1,𝛽=−3,𝑐=1

(b) 𝛼=1.5,𝛽=0,𝑐=−1

(c) 𝛼=1,𝛽=2,𝑐=−0.5

(d) 𝛼=1.5,𝛽=0,𝑐=1

Figure 5 

Smooth periodic waves of (3) for 𝑥=1.

From Figure 1(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 (𝜙_𝑠,0) and ((3𝛼𝑐+𝛽)/𝛼,0) when 𝛼>0, 𝑐>−𝛽/2𝛼, and 𝑔=𝛽(4𝛼𝑐+𝛽)/8𝛼². Its expression is 1𝑦=±√6𝛼3𝛼𝑐+𝛽𝛼−𝜙𝜙−𝜙_𝑠,𝜙_𝑠≤𝜙≤3𝛼𝑐+𝛽𝛼. (31)

Substituting (31) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the periodic orbit yield equation _𝜙^{(3𝛼𝑐+𝛽)/𝛼}𝑑𝑠3𝛼𝑐+𝛽/𝛼−𝑠𝑠−𝜙_𝑠=1√6𝛼^𝜉0𝑑𝑠. (32)

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 given ℎ∈(ℎ₁,ℎ_𝑠), ℎ_𝑠<ℎ₂ (or ℎ∈(ℎ₁,ℎ₂), ℎ_𝑠>ℎ₂) in Figure 1(c), the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(a).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 6 

The level curves defined by 𝐻(𝜙,𝑦)=ℎ.

For given ℎ∈(ℎ_𝑠,ℎ₂), ℎ₁<ℎ_𝑠 in Figure 1(c), ℎ∈(ℎ₁,ℎ₂) in Figure 1(d), and ℎ∈(ℎ_𝑠,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 given ℎ∈(ℎ_𝑠,ℎ₂), ℎ_𝑠>ℎ₁ (or ℎ∈(ℎ₁,ℎ₂), ℎ_𝑠<ℎ₁) in Figure 1(h),the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(c).

For given ℎ∈(ℎ₁,ℎ_𝑠), ℎ_𝑠<ℎ₂ in Figure 1(h), and ℎ∈(ℎ₁,ℎ₂) in Figure 1(i), ℎ∈(ℎ₁,ℎ_𝑠) in Figures 1(e) and 1(j), respectively, the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(d).

For given ℎ∈(ℎ_𝑠,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 given ℎ∈(ℎ₁,ℎ_𝑠) 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 Figure 6(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,ℎ_𝑠), ℎ_𝑠<ℎ₂ (or ℎ∈(ℎ₁,ℎ₂), and ℎ_𝑠>ℎ₂). The open curve passes point (𝛾₁,0) and approaches the line 𝜙=𝜙_𝑠; its expression is 1𝑦=±√6𝛼𝜙−𝛾₁𝜙−𝛾₂𝜙−𝛾₃𝜙_𝑠−𝜙,𝛾₁≤𝜙<𝜙_𝑠, (33) where 𝛾₁, 𝛾₂, and 𝛾₃(𝛾₃<𝛾₂<𝛾₁<𝜙_𝑠) 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 ^𝜙_𝛾1𝜙_𝑠−𝑠𝑠−𝛾₁𝑠−𝛾₂𝑠−𝛾₃1𝑑𝑠=√6𝛼^𝜉0𝑑𝑠. (34)

Completing (34) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows: 1𝛼²₁𝑢+𝛼²₁−1Π𝜙,𝛼²₁=,𝑘𝜙_𝑠−𝛾₂𝛾₁−𝛾₃2𝜙_𝑠−𝛾₁√||,6𝛼𝑥−𝑐𝑡 (35) where 𝑢=sn⁻¹((𝜙_𝑠−𝛾₂)(𝑢(𝑥,𝑡)−𝛾₁)/(𝜙_𝑠−𝛾₁)(𝑢(𝑥,𝑡)−𝛾₂),𝑘), 𝜙=sin⁻¹((𝜙_𝑠−𝛾₂)(𝑢(𝑥,𝑡)−𝛾₁)/(𝜙_𝑠−𝛾₁)(𝑢(𝑥,𝑡)−𝛾₂)), 𝑘=(𝜙_𝑠−𝛾₁)(𝛾₂−𝛾₃)/(𝜙_𝑠−𝛾₂)(𝛾₁−𝛾₃), and 𝛼²₁=(𝜙_𝑠−𝛾₁)/(𝜙_𝑠−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 Figure 6(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∈(ℎ_𝑠,ℎ₂), and ℎ₁<ℎ_𝑠,(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∈(ℎ_𝑠,ℎ₂).

The open curve passes point (𝛾₁,0) and approaches the line 𝜙=𝜙_𝑠; its expression is 1𝑦=±√6𝛼𝛾₁−𝜙𝜙−𝛾₂𝜙−𝛾₃𝜙−𝜙_𝑠,𝜙_𝑠<𝜙≤𝛾₁, (36) where 𝛾₁, 𝛾₂, and 𝛾₃ (𝛾₃<𝛾₂<𝜙_𝑠<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 ^𝛾1_𝜙𝑠−𝜙_𝑠𝛾₁−𝑠𝑠−𝛾₂𝑠−𝛾₃1𝑑𝑠=√6𝛼^𝜉0𝑑𝑠. (37)

Completing (37) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows: 1𝛼²₁𝑢+𝛼²₁−1Π𝜙,𝛼²₁=,𝑘𝛾₁−𝛾₂𝜙_𝑠−𝛾₃2𝛾₁−𝜙_𝑠√||,6𝛼𝑥−𝑐𝑡 (38) where 𝑢=sn⁻¹((𝜙_𝑠−𝛾₃)(𝛾₁−𝑢(𝑥,𝑡))/(𝛾₁−𝜙_𝑠)(𝑢(𝑥,𝑡)−𝛾₃),𝑘), 𝜙=sin⁻¹((𝜙_𝑠−𝛾₃)(𝛾₁−𝑢(𝑥,𝑡))/(𝛾₁−𝜙_𝑠)(𝑢(𝑥,𝑡)−𝛾₃)), 𝑘=(𝛾₁−𝜙_𝑠)(𝛾₂−𝛾₃)/(𝛾₁−𝛾₂)(𝜙_𝑠−𝛾₃), and 𝛼²₁=(𝜙_𝑠−𝛾₁)/(𝜙_𝑠−𝛾₃).

From Figure 6(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∈(ℎ_𝑠,ℎ₂), ℎ_𝑠>ℎ₁ (or ℎ∈(ℎ₁,ℎ₂), and ℎ_𝑠<ℎ₁). The open curve passes point (𝛾₃,0) and approaches the line 𝜙=𝜙_𝑠; its expression is 1𝑦=±√6𝛼𝛾₁−𝜙𝛾₂−𝜙𝛾₃−𝜙𝜙−𝜙_𝑠,𝜙_𝑠<𝜙≤𝛾₃, (39) where 𝛾₁, 𝛾₂, and 𝛾₃ (𝜙_𝑠<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 ^𝛾3_𝜙𝑠−𝜙_𝑠𝛾₁−𝑠𝛾₂−𝑠𝛾₃1−𝑠𝑑𝑠=√6𝛼^𝜉0𝑑𝑠. (40)

Completing (40) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follow: 1𝛼²₁𝑢+𝛼²₁−1Π𝜙,𝛼²₁=,𝑘𝛾₁−𝛾₃𝛾₂−𝜙_𝑠2𝛾₃−𝜙_𝑠√||,6𝛼𝑥−𝑐𝑡 (41) where 𝑢=sn⁻¹((𝛾₂−𝜙_𝑠)(𝛾₃−𝑢(𝑥,𝑡))/(𝛾₃−𝜙_𝑠)(𝛾₂−𝑢(𝑥,𝑡)),𝑘), 𝜙=sin⁻¹((𝛾₂−𝜙_𝑠)(𝛾₃−𝑢(𝑥,𝑡))/(𝛾₃−𝜙_𝑠)(𝛾₂−𝑢(𝑥,𝑡))), 𝑘=(𝛾₁−𝛾₂)(𝛾₃−𝜙_𝑠)/(𝛾₁−𝛾₃)(𝛾₂−𝜙_𝑠), and 𝛼²₁=(𝛾₃−𝜙_𝑠)/(𝛾₂−𝜙_𝑠).

From Figure 6(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,ℎ_𝑠), and ℎ_𝑠<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,ℎ_𝑠).

The open curve passes point (𝛾₃,0) and approaches the line 𝜙=𝜙_𝑠, its expression is 1𝑦=±√6𝛼𝛾₁−𝜙𝛾₂−𝜙𝜙−𝛾₃𝜙_𝑠−𝜙,𝛾₃≤𝜙<𝜙_𝑠, (42) where 𝛾₁, 𝛾₂, and 𝛾₃ (𝛾₃<𝜙_𝑠<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 ^𝜙_𝛾3𝜙_𝑠−𝑠𝛾₁−𝑠𝛾₂−𝑠𝑠−𝛾₃1𝑑𝑠=√6𝛼^𝜉0𝑑𝑠. (43)

Completing (43) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows: 1𝛼²₁𝑢+𝛼²₁−1Π𝜙,𝛼²₁=,𝑘𝛾₁−𝜙_𝑠𝛾₂−𝛾₃2𝜙_𝑠−𝛾₃√||,6𝛼𝑥−𝑐𝑡 (44) where 𝑢=sn⁻¹((𝛾₁−𝜙_𝑠)(𝑢(𝑥,𝑡)−𝛾₃)/(𝜙_𝑠−𝛾₃)(𝛾₁−𝑢(𝑥,𝑡)),𝑘), 𝜙=sin⁻¹((𝛾₁−𝜙_𝑠)(𝑢(𝑥,𝑡)−𝛾₃)/(𝜙_𝑠−𝛾₃)(𝛾₁−𝑢(𝑥,𝑡))), 𝑘=(𝛾₁−𝛾₂)(𝜙_𝑠−𝛾₃)/(𝛾₁−𝜙_𝑠)(𝛾₂−𝛾₃), and 𝛼²₁=(𝛾₃−𝜙_𝑠)/(𝛾₁−𝜙_𝑠).

From Figure 6(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∈(ℎ_𝑠,ℎ₂),(ii)𝛼<0,𝑐>−𝛽/2𝛼,𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼², and ℎ∈(ℎ_𝑠,ℎ₂).

One of them passes point (𝛾₂,0) and approaches the line 𝜙=𝜙_𝑠; its expression is 1𝑦=±√−6𝛼𝛾₁−𝜙𝜙−𝛾₂𝜙−𝛾₃𝜙_𝑠−𝜙,𝛾₂≤𝜙<𝜙_𝑠, (45) where 𝛾₁, 𝛾₂, and 𝛾₃(𝛾₃<𝛾₂<𝜙_𝑠<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 ^𝜙_𝛾2𝜙_𝑠−𝑠𝛾₁−𝑠𝑠−𝛾₂𝑠−𝛾₃1𝑑𝑠=√−6𝛼^𝜉0𝑑𝑠. (46)

Completing (46) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows: 1𝛼²₁𝑢+𝛼²₁−1Π𝜙,𝛼²₁=,𝑘𝛾₁−𝛾₂𝜙_𝑠−𝛾₃2𝜙_𝑠−𝛾₂√||,−6𝛼𝑥−𝑐𝑡 (47) where 𝑢=sn⁻¹((𝜙_𝑠−𝛾₃)(𝑢(𝑥,𝑡)−𝛾₂)/(𝜙_𝑠−𝛾₂)(𝑢(𝑥,𝑡)−𝛾₃),𝑘), 𝜙=sin⁻¹((𝜙_𝑠−𝛾₃)(𝑢(𝑥,𝑡)−𝛾₂)/(𝜙_𝑠−𝛾₂)(𝑢(𝑥,𝑡)−𝛾₃)), 𝑘=(𝜙_𝑠−𝛾₂)(𝛾₁−𝛾₃)/(𝛾₁−𝛾₂)(𝜙_𝑠−𝛾₃), and 𝛼²₁=(𝜙_𝑠−𝛾₂)/(𝜙_𝑠−𝛾₃).

From Figure 6(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,ℎ_𝑠),(ii)𝛼<0,𝑐>−𝛽/2𝛼,𝑔>−(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼², and ℎ∈(ℎ₁,ℎ_𝑠).

One of them passes point (𝛾₂,0) and approaches the line 𝜙=𝜙_𝑠; its expression is 1𝑦=±√−6𝛼𝛾₁−𝜙𝛾₂−𝜙𝜙−𝛾₃𝜙−𝜙_𝑠,𝜙_𝑠<𝜙≤𝛾₂, (48) where 𝛾₁, 𝛾₂, and 𝛾₃(𝛾₃<𝜙_𝑠<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 ^𝛾2_𝜙𝑠−𝜙_𝑠𝛾₁−𝑠𝛾₂−𝑠𝑠−𝛾₃1𝑑𝑠=√−6𝛼^𝜉0𝑑𝑠. (49)

Completing (49) and using transformation (4), we can get the implicit representation of a compacton solution of (3) as follows: 1𝛼²₁𝑢+𝛼²₁−1Π𝜙,𝛼²₁=,𝑘𝛾₁−𝜙_𝑠𝛾₂−𝛾₃2𝛾₂−𝜙_𝑠√||,−6𝛼𝑥−𝑐𝑡 (50) where 𝑢=sn⁻¹((𝛾₁−𝜙_𝑠)(𝛾₂−𝑢(𝑥,𝑡))/(𝛾₂−𝜙_𝑠)(𝛾₁−𝑢(𝑥,𝑡)),𝑘), 𝜙=sin⁻¹((𝛾₁−𝜙_𝑠)(𝛾₂−𝑢(𝑥,𝑡))/(𝛾₂−𝜙_𝑠)(𝛾₁−𝑢(𝑥,𝑡))), 𝑘=(𝛾₂−𝜙_𝑠)(𝛾₁−𝛾₃)/(𝛾₁−𝜙_𝑠)(𝛾₂−𝛾₃), and 𝛼²₁=(𝛾₂−𝜙_𝑠)/(𝛾₁−𝜙_𝑠).

3.5. Periodic Cusp Wave Solutions

For given ℎ=ℎ_𝑠 in Figure 1(e), the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(g).

For given ℎ=ℎ_𝑠 in Figure 1(j), the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(h).

For given ℎ=ℎ_𝑠 in Figure 2(e), the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(i).

For given ℎ=ℎ_𝑠 in Figure 2(l), the level curve defined by 𝐻(𝜙,𝑦)=ℎ is shown in Figure 6(j).

From Figures 6(g) and 6(h), we see that there are two heteroclinic orbits of system (7) defined by 𝐻(𝜙,𝑦)=ℎ_𝑠 connecting with the saddle points (𝜙_𝑠,𝑌_±) and passing points (𝜙_𝑚,0), and (𝜙_𝑀,0), respectively, when 𝛼>0, 𝑐≠−𝛽/2𝛼, and 𝑔>𝛽(4𝛼𝑐+𝛽)/8𝛼², where 𝜙_𝑀,𝑚=((𝛽+6𝛼𝑐)±√𝛿)/4𝛼 and 𝛿=−3𝛽²−12𝛼𝛽𝑐+36𝛼²𝑐²+96𝛼²𝑔. Their expressions are, respectively, 1𝑦=±√6𝛼𝜙−𝜙_𝑚𝜙_𝑀−𝜙,𝜙_𝑚≤𝜙≤𝜙_𝑠,1𝑦=±√6𝛼𝜙−𝜙_𝑚𝜙_𝑀−𝜙,𝜙_𝑠≤𝜙≤𝜙_𝑀. (51)(52)

Substituting (51) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the heteroclinic orbit, yields equation ^𝜙𝑠_𝜙𝑑𝑠𝑠−𝜙_𝑚𝜙_𝑀=1−𝑠√6𝛼^𝜉0𝑑𝑠. (53)

Completing (53) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows: 𝑢(𝑥,𝑡)=𝜙_𝑚cos²(Ω−𝜔|𝑥−𝑐𝑡−2𝑛𝑇|)+𝜙_𝑀sin²(|,)()𝑇],Ω−𝜔𝑥−𝑐𝑡−2𝑛𝑇|)𝑥−𝑐𝑡∈[(2𝑛−1𝑇,2𝑛+1 (54) where 𝑛=0,±1,±2,…, 𝜔=1/2√6𝛼, Ω=tan⁻¹((𝜙_𝑠−𝜙_𝑚)/(𝜙_𝑀−𝜙_𝑠)), and 𝑇=2|Ω|.

Substituting (52) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating it along the heteroclinic orbit yield equation ^𝜙_𝜙𝑠𝑑𝑠𝑠−𝜙_𝑚𝜙_𝑀=1−𝑠√6𝛼^𝜉0𝑑𝑠. (55)

Completing (55) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows: 𝑢(𝑥,𝑡)=𝜙_𝑚sin²(Ω+𝜔|𝑥−𝑐𝑡−2𝑛𝑇|)+𝜙_𝑀cos²(|,)()𝑇],Ω+𝜔𝑥−𝑐𝑡−2𝑛𝑇|)𝑥−𝑐𝑡∈[(2𝑛−1𝑇,2𝑛+1 (56) where 𝑛=0,±1,±2,…, 𝜔=1/2√6𝛼, Ω=tan⁻¹((𝜙_𝑀−𝜙_𝑠)/(𝜙_𝑠−𝜙_𝑚)), and 𝑇=2|Ω|.

From Figure 6(i), we see that there are a heteroclinic orbit and an open curve of system (7) defined by 𝐻(𝜙,𝑦)=ℎ_𝑠 when 𝛼<0,𝑐<−𝛽/2𝛼, and −(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼². The heteroclinic orbit connecting with the saddle points (𝜙_𝑠,𝑌_±) and passing point (𝜙_𝑀,0); its expression is 1𝑦=±√−6𝛼𝜙−𝜙_𝑀𝜙−𝜙_𝑚,𝜙_𝑀≤𝜙≤𝜙_𝑠, (57) where 𝜙_𝑚,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 ^𝜙𝑠_𝜙𝑑𝑠𝑠−𝜙_𝑀𝑠−𝜙_𝑚=1√−6𝛼^𝜉0𝑑𝑠. (58)

Completing (58) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows: 𝑢(𝑥,𝑡)=𝜙_𝑀cosh²(Ω−𝜔|𝑥−𝑐𝑡−2𝑛𝑇|)−𝜙_𝑚sinh²(|,)()𝑇],Ω−𝜔𝑥−𝑐𝑡−2𝑛𝑇|)𝑥−𝑐𝑡∈[(2𝑛−1𝑇,2𝑛+1 (59) where 𝑛=0,±1,±2,…,𝜔=1/2√−6𝛼, Ω=tanh⁻¹((𝜙_𝑠−𝜙_𝑀)/(𝜙_𝑠−𝜙_𝑚)), and 𝑇=2|Ω|.

From Figure 6(j), we see that there are a heteroclinic orbit and an open curve of system (7) defined by 𝐻(𝜙,𝑦)=ℎ_𝑠 when 𝛼<0,𝑐>−𝛽/2𝛼, and −(12𝛼²𝑐²−4𝛼𝛽𝑐−𝛽²)/32𝛼²<𝑔<𝛽(4𝛼𝑐+𝛽)/8𝛼². The heteroclinic orbit connecting with the saddle points (𝜙_𝑠,𝑌_±) and passing point (𝜙_𝑚,0); its expression is 1𝑦=±√−6𝛼𝜙_𝑚−𝜙𝜙_𝑀−𝜙,𝜙_𝑠≤𝜙≤𝜙_𝑚, (60) where 𝜙_𝑚,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 ^𝜙_𝜙𝑠𝑑𝑠𝜙_𝑚−𝑠𝜙_𝑀=1−𝑠√−6𝛼^𝜉0𝑑𝑠. (61)

Completing (61) and using transformation (4), we can get a periodic cusp wave solution of (3) as follows: 𝑢(𝑥,𝑡)=𝜙_𝑚cosh²(Ω+𝜔|𝑥−𝑐𝑡−2𝑛𝑇|)−𝜙_𝑀sinh²(|,)()𝑇],Ω+𝜔𝑥−𝑐𝑡−2𝑛𝑇|)𝑥−𝑐𝑡∈[(2𝑛−1𝑇,2𝑛+1 (62) where 𝑛=0,±1,±2,…, 𝜔=1/2√−6𝛼, Ω=tanh⁻¹((𝜙_𝑚−𝜙_𝑠)/(𝜙_𝑀−𝜙_𝑠)), 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. Choose 𝛼, 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).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 7 

The numerical simulations of integral curves of (3).

Example 2. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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. Choose 𝛼, 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).

Figure 7 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].

Acknowledgments

This work is supported by the Natural Science Foundation of Yunnan Province, China (no. 2013FZ117), and the National Natural Science Foundation of China (no. 11161020).