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Mathematical Problems in Engineering
VolumeΒ 2011Β (2011), Article IDΒ 810217, 23 pages
http://dx.doi.org/10.1155/2011/810217
Research Article

Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 15 February 2011; Accepted 26 May 2011

Academic Editor: MingΒ Li

Copyright Β© 2011 Liu Zhengrong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation 𝑒𝑑+π‘Ž(1+𝑏𝑒)𝑒𝑒π‘₯+𝑒π‘₯π‘₯π‘₯=0, and PC-like equation π‘£π‘‘π‘‘βˆ’π‘£π‘‘π‘‘π‘₯π‘₯βˆ’(π‘Ž1𝑣+π‘Ž2𝑣2+π‘Ž3𝑣3)π‘₯π‘₯=0, respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interesting that the two equations have similar periodic wave solutions. Our work extend previous some results.

1. Introduction

Many authors have investigated the KdV-like equation 𝑒𝑑+π‘Ž(1+𝑏𝑒)𝑒𝑒π‘₯+𝑒π‘₯π‘₯π‘₯=0,(1.1) and the PC-like equation π‘£π‘‘π‘‘βˆ’π‘£π‘‘π‘‘π‘₯π‘₯βˆ’ξ€·π‘Ž1𝑣+π‘Ž2𝑣2+π‘Ž3𝑣3ξ€Έπ‘₯π‘₯=0.(1.2) For example, Dey [1, 2] studied the exact Himiltonian density and the conservation laws, and gave two kink solutions for (1.1). Zhang et al. [3, 4] gave some solitary wave solutions and singular wave solutions for (1.1) by using two different methods. Yu [5] got an exact kink soliton for (1.1) by using homogeneous balance method. Grimshaw et al. [6] studied the large-amplitude solitons for (1.1). Fan [7, 8] gave some bell-shaped soliton solutions, kink-shaped soliton, and Jacobi periodic solutions for (1.1) by using algebraic method. Tang et al. [9] investigated solitary waves and their bifurcations for (1.1) by employing bifurcation method of dynamical systems. Peng [10] used the modified mapping method to get some solitary wave solutions composed of hyperbolic functions, periodic wave solutions composed of Jacobi elliptic functions, and singular wave solution composed of triangle functions for (1.1). Chow et al. [11] described the interaction between a soliton and a breather for (1.1) by using the Hirota bilinear method. Kaya and Inan [12] studied solitary wave solutions for (1.1) by using Adomian decomposition method. Yomba [13] used Fan's subequation method to construct exact traveling wave solutions composed of hyperbolic functions or Jacobi elliptic functions for (1.1).

Zhang and Ma [14] gave some explicit solitary wave solutions composed of hyperbolic functions by using solving algebraic equations for (1.2). Li and Zhang [15] used bifurcation method of dynamical system to study the bifurcation of traveling wave solutions and construct solitary wave solutions for (1.2). Kaya [16] discussed the exact and numerical solitary wave solutions by using a decomposition method for (1.2). Rafei et al. [17] gave numerical solutions by using He's method for (1.2).

Recently, many authors have presented some useful methods to deal with the problems in equations, for instance [18–30].

In this paper, we use the bifurcation method mentioned above to study the periodic wave solutions for (1.1) and (1.2). Through some special phase orbits, we obtain new expressions of periodic wave solutions which are composed of trigonometric functions sinβ€‰πœ‰ or cosβ€‰πœ‰. These solutions are called trigonometric function periodic wave solutions. We also check the correctness by using the software Mathematica.

In Section 2, we will state our results for (1.1). In Section 3, we will state our results for (1.2). In Sections 4, and 5, we will give derivations for our main results. Some discussions and the orders for testing the correctness of the solutions will be given in Section 6.

2. Trigonometric Function Periodic Wave Solutions for (1.1)

In this section, we state our main results for (1.1). In order to state these results conveniently, we give some preparations. For given constant 𝑐≠0, on π‘Žβˆ’π‘ plane we define some lines and regions as follows.(1)When 𝑐<0, we define lines 𝑙1𝑙:𝑏=0,2π‘Ž:𝑏=βˆ’,𝑙6𝑐3:𝑏=βˆ’3π‘Ž,𝑙16𝑐4:π‘Ž=0,(2.1) and regions 𝐴𝑖(𝑖=1-8), as Figure 1(a).

fig1
Figure 1: The locations of the lines 𝑙𝑖, π‘˜π‘–(𝑖=1,2,3,4) and the regions 𝐴𝑗, 𝐡𝑗(𝑗=1,2,…,8) for given constant 𝑐≠0.
(2)When 𝑐>0, we define lines π‘˜1π‘˜:𝑏=0,2π‘˜:π‘Ž=0,3:𝑏=βˆ’3π‘Ž,π‘˜16𝑐4π‘Ž:𝑏=βˆ’,6𝑐(2.2) and regions 𝐡𝑖(𝑖=1-8), as Figure 1(b).

Using the lines and regions in Figure 1, we narrate our results as follows.

Proposition 2.1. For arbitrary given constant 𝑐≠0, let πœ‰=π‘₯βˆ’π‘π‘‘.(2.3) Then, (1.1) has the following periodic wave solutions. (1)When c<0 and (π‘Ž,𝑏)∈𝐴1 or 𝐴5, the expression of the periodic wave solution is 𝑒1(πœ‰)=6π‘βˆšπ‘Ž+π‘Žξ‚€βˆš(π‘Ž+6𝑏𝑐)cos,βˆ’π‘πœ‰(2.4) which has the following limit forms.(i)When 𝑐<0, (π‘Ž,𝑏)∈𝐴1 and (π‘Ž,𝑏) tends to the line 𝑙1, 𝑒1(πœ‰) tends to the periodic blow-up solution π‘’βˆ˜1(πœ‰)=6π‘π‘Žξ‚ƒξ‚€βˆš1+cosβˆ’π‘πœ‰ξ‚ξ‚„(2.5) (see Figure 2).(ii)When 𝑐<0, (π‘Ž,𝑏)∈𝐴5 and (π‘Ž,𝑏) tends to the line 𝑙1, 𝑒1(πœ‰) tends to the periodic blow-up solution π‘’βˆ—1(πœ‰)=6π‘π‘Žξ‚ƒξ‚€βˆš1βˆ’cosβˆ’π‘πœ‰ξ‚ξ‚„(2.6) (see Figure 3).(iii)When 𝑐<0, (π‘Ž,𝑏)∈𝐴1 or 𝐴5, and (π‘Ž,𝑏) tends to 𝑙2, 𝑒1(πœ‰) tends to the trivial solution 𝑒(πœ‰)=6𝑐/π‘Ž.(2)When 𝑐<0 and (π‘Ž,𝑏)∈𝐴2, or when 𝑐>0 and (π‘Ž,𝑏)∈𝐡5, the expression of the periodic wave solution is 𝑒2𝛼(πœ‰)=0𝑀cos0πœ‰ξ€Έ+𝛽0𝑝0𝑀cos0πœ‰ξ€Έ+π‘ž0,(2.7) where Ξ”=3π‘Ž(3π‘Ž+16𝑏𝑐),(2.8)𝛼0ξ‚€βˆš=βˆ’3π‘Ž+Ξ”ξ‚ξ‚™π‘Žξ‚€βˆšπ‘Žβˆ’Ξ”ξ‚4π‘Ž2𝑏2,𝛽(2.9)0√=βˆ’3π‘Ž+24𝑏𝑐+Ξ”2π‘Žπ‘2𝑝,(2.10)0=ξ‚™π‘Žξ‚€βˆšπ‘Žβˆ’Ξ”ξ‚π‘žπ‘Žπ‘,(2.11)01=βˆ’ξ‚€βˆšπ‘Žπ‘π‘Ž+Δ𝑀,(2.12)0=ξƒŽβˆš3π‘Ž+16𝑏𝑐+Ξ”.8𝑏(2.13) The solution 𝑒2(πœ‰) has the following limit forms. (i)When 𝑐<0, (π‘Ž,𝑏)∈𝐴2 and (π‘Ž,𝑏) tends to 𝑙3, the 𝑒2(πœ‰) tends to the peak-shaped solitary wave solution π‘’βˆ˜2ξ€·(πœ‰)=4𝑐3+2π‘πœ‰2ξ€Έπ‘Žξ€·βˆ’9+2π‘πœ‰2ξ€Έ(2.14) (see Figure 4).(ii)When 𝑐<0, (π‘Ž,𝑏)∈𝐴2 and (π‘Ž,𝑏) tends to 𝑙2, 𝑒2(πœ‰) tends to the trivial solution 𝑒(πœ‰)=0.(iii)When 𝑐>0, (π‘Ž,𝑏)∈𝐡5 and (π‘Ž,𝑏) tends to π‘˜1, the 𝑒2(πœ‰) tends to the periodic blow-up solution π‘’βˆ—2𝑐(πœ‰)=2sin2ξ‚€βˆšξ‚ξ‚π‘πœ‰/2βˆ’3π‘Žsin2ξ‚€βˆšξ‚π‘πœ‰/2(2.15) (see Figure 5).(3)When 𝑐<0 and (π‘Ž,𝑏)∈𝐴6, or when 𝑐>0 and (π‘Ž,𝑏)∈𝐡1, the expressions of the solution is 𝑒3𝛼(πœ‰)=1𝑀cos1πœ‰ξ€Έ+𝛽1𝑝1𝑀cos1πœ‰ξ€Έ+π‘ž1,(2.16) where 𝛼1=ξ‚€βˆšβˆ’3π‘Ž+Ξ”ξ‚ξ‚™π‘Žξ‚€βˆšπ‘Ž+Δ4π‘Ž2𝑏2,𝛽1=√3π‘Ž+24π‘π‘βˆ’Ξ”2π‘Žπ‘2,𝑝1=ξ‚™π‘Žξ‚€βˆšπ‘Ž+Δ,π‘žπ‘Žπ‘1=βˆšπ‘Žβˆ’Ξ”,π‘€π‘Žπ‘1=ξƒŽβˆš3π‘Ž+16π‘π‘βˆ’Ξ”.8𝑏(2.17) The solution 𝑒3(πœ‰) has the following limit forms.(i)When 𝑐<0, (π‘Ž,𝑏)∈𝐴6 and (π‘Ž,𝑏) tends to 𝑙3, the 𝑒3(πœ‰) tends to the canyon-shaped solitary wave (see Figure 6) solution π‘’βˆ˜2(πœ‰). (ii)When 𝑐<0, (π‘Ž,𝑏)∈𝐴6 and (π‘Ž,𝑏) tends to 𝑙2, 𝑒3(πœ‰) tends to the trivial solution 𝑒(πœ‰)=0. (iii)When 𝑐>0, (π‘Ž,𝑏)∈𝐡1 and (π‘Ž,𝑏) tends to π‘˜1, the 𝑒3(πœ‰) tends to the periodic blow-up wave solution π‘’βˆ—1(πœ‰) (see Figure 3).

fig2
Figure 2: The limiting precess of 𝑒1(πœ‰) when 𝑐<0, (π‘Ž,𝑏)∈𝐴1, and (π‘Ž,𝑏) tends to the line 𝑙1, where π‘Ž=4 and 𝑐=βˆ’1.
fig3
Figure 3: The limiting precess of 𝑒1(πœ‰) when 𝑐<0, (π‘Ž,𝑏)∈𝐴5, and (π‘Ž,𝑏) tends to the line 𝑙1, where π‘Ž=βˆ’4 and 𝑐=βˆ’1.
fig4
Figure 4: The limiting precess of 𝑒2(πœ‰) when 𝑐<0, (π‘Ž,𝑏)∈𝐴2, and (π‘Ž,𝑏) tends to the line 𝑙3, where π‘Ž=4 and 𝑐=βˆ’1.
fig5
Figure 5: The limiting precess of 𝑒2(πœ‰) when 𝑐>0, (π‘Ž,𝑏)∈𝐡5, and (π‘Ž,𝑏) tends to the line π‘˜1, where π‘Ž=βˆ’2 and 𝑐=1.
fig6
Figure 6: The limiting precess of 𝑒3(πœ‰) when 𝑐<0, (π‘Ž,𝑏)∈𝐴6, and (π‘Ž,𝑏) tends to the line 𝑙3, where π‘Ž=βˆ’9 and 𝑐=βˆ’1.

Remark 2.2. Note that if 𝑒=πœ‘(πœ‰) is a solution of (1.1), then 𝑒=πœ‘(πœ‰+π‘Ÿ) also is solution of (1.1), where π‘Ÿ is a arbitrary constant. According to this fact and the results listed in Proposition 2.1, the following nine functions also are periodic wave solutions of (1.1).
(1) When 𝑐<0 and (π‘Ž,𝑏)∈𝐴1 or 𝐴5, the functions are 𝑒11(πœ‰)=6π‘βˆšπ‘Žβˆ’π‘Žξ‚€βˆš(π‘Ž+6𝑏𝑐)cos,π‘’βˆ’π‘πœ‰12(πœ‰)=6π‘βˆšπ‘Ž+ξ‚€βˆšπ‘Ž(π‘Ž+6𝑏𝑐)sin,π‘’βˆ’π‘πœ‰13(πœ‰)=6π‘βˆšπ‘Žβˆ’ξ‚€βˆšπ‘Ž(π‘Ž+6𝑏𝑐)sin.βˆ’π‘πœ‰(2.18)
(2) When 𝑐<0 and (π‘Ž,𝑏)∈𝐴2 or when 𝑐>0 and (π‘Ž,𝑏)∈𝐡5, the functions are 𝑒21(πœ‰)=βˆ’π›Ό0𝑀cos0πœ‰ξ€Έ+𝛽0βˆ’π‘0𝑀cos0πœ‰ξ€Έ+π‘ž0,𝑒22𝛼(πœ‰)=0𝑀sin0πœ‰ξ€Έ+𝛽0𝑝0𝑀sin0πœ‰ξ€Έ+π‘ž0,𝑒23(πœ‰)=βˆ’π›Ό0𝑀sin0πœ‰ξ€Έ+𝛽0βˆ’π‘0𝑀sin0πœ‰ξ€Έ+π‘ž0.(2.19)
(3) When 𝑐<0 and (π‘Ž,𝑏)∈𝐴6, or when 𝑐>0 and (π‘Ž,𝑏)∈𝐡1, the functions are 𝑒31(πœ‰)=βˆ’π›Ό1𝑀cos1πœ‰ξ€Έ+𝛽1βˆ’π‘1𝑀cos1πœ‰ξ€Έ+π‘ž1,𝑒32𝛼(πœ‰)=1𝑀sin1πœ‰ξ€Έ+𝛽1𝑝1𝑀sin1πœ‰ξ€Έ+π‘ž1,𝑒33(πœ‰)=βˆ’π›Ό1𝑀sin1πœ‰ξ€Έ+𝛽1βˆ’π‘1𝑀sin1πœ‰ξ€Έ+π‘ž1.(2.20)

Remark 2.3. In the given parametric regions, the solutions 𝑒𝑖(πœ‰), 𝑒1𝑖(πœ‰), 𝑒2𝑖(πœ‰), 𝑒3𝑖(πœ‰)(𝑖=1,2,3), and π‘’βˆ˜2(πœ‰) are nonsingular. The solutions π‘’βˆ˜1(πœ‰), π‘’βˆ—1(πœ‰), and π‘’βˆ—2(πœ‰) are singular. The relationships of singular solutions and nonsingular solutions are displayed in the Proposition 2.1.

3. Trigonometric Function Periodic Wave Solutions for (1.2)

In this section, we state our main results for (1.2). For given π‘Ž1 and 𝑐(π‘Ž1≠𝑐2), on π‘Ž2βˆ’π‘Ž3 plane we define some rays and regions as follows.(1)When 𝑐2<π‘Ž1, we define curves Ξ“1:π‘Ž2>0,π‘Ž3Ξ“=0,2:π‘Ž2>0,π‘Ž3=2π‘Ž229ξ€·π‘Ž1βˆ’π‘2ξ€Έ,Ξ“3:π‘Ž2>0,π‘Ž3=π‘Ž224ξ€·π‘Ž1βˆ’π‘2ξ€Έ,Ξ“4:π‘Ž2=0,π‘Ž3Ξ“>0,5:π‘Ž2<0,π‘Ž3=π‘Ž224ξ€·π‘Ž1βˆ’π‘2ξ€Έ,Ξ“6:π‘Ž2<0,π‘Ž3=2π‘Ž229ξ€·π‘Ž1βˆ’π‘2ξ€Έ,Ξ“7:π‘Ž2<0,π‘Ž3Ξ“=0,8:π‘Ž2=0,π‘Ž3<0,(3.1) and region π‘Šπ‘– as the domain surrounded by Γ𝑖 and Γ𝑖+1(𝑖=1-7), π‘Š8 as the domain surrounded by Ξ“8 and Ξ“1 (see Figure 7(a)).

fig7
Figure 7: The locations of the rays Γ𝑖, 𝐿𝑖 and the regions π‘Šπ‘–, Ω𝑖(𝑖=1,2,…,8) for given π‘Ž1 and 𝑐.
(2)When 𝑐2>π‘Ž1, we define curves 𝐿1:π‘Ž2>0,π‘Ž3𝐿=0,2:π‘Ž2=0,π‘Ž3𝐿>0,3:π‘Ž2<0,π‘Ž3𝐿=0,4:π‘Ž2<0,π‘Ž3=2π‘Ž229ξ€·π‘Ž1βˆ’π‘2ξ€Έ,𝐿5:π‘Ž2<0,π‘Ž3=π‘Ž214ξ€·π‘Ž1βˆ’π‘2ξ€Έ,𝐿6:π‘Ž2=0,π‘Ž3𝐿<0,7:π‘Ž2>0,π‘Ž3=π‘Ž224ξ€·π‘Ž1βˆ’π‘2ξ€Έ,𝐿8:π‘Ž2>0,π‘Ž3=2π‘Ž229ξ€·π‘Ž1βˆ’π‘2ξ€Έ,(3.2) and region Ω𝑖 as the domain surrounded by 𝐿𝑖 and 𝐿𝑖+1(𝑖=1-7), Ξ©8 as the domain surrounded by 𝐿8 and 𝐿1 (see Figure 7(b)).

Using the rays and regions above, we state our results as follows.

Proposition 3.1. For given parameter π‘Ž1 and constant 𝑐 satisfying 𝑐2β‰ π‘Ž1, let πœ‰=π‘₯βˆ’π‘π‘‘. Then, (1.2) has the following periodic wave solutions. (1)When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š1 or π‘Š6, the expression of the periodic wave solution is 𝑣1𝑅(πœ‰)=0𝑅1+𝑅2𝑅cos3πœ‰ξ€Έ,(3.3) where 𝑅0𝑐=22βˆ’π‘Ž1ξ€Έ,𝑅1=2π‘Ž23,𝑅2=1318π‘Ž3𝑐2βˆ’π‘Ž1ξ€Έ+4π‘Ž22,𝑅3=ξƒŽπ‘Ž1βˆ’π‘2𝑐2.(3.4) For π‘Ž2β‰ 0, the periodic wave solution 𝑣1(πœ‰) has the following limit forms.(i)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š1 and (π‘Ž2,π‘Ž3) tends to the ray Ξ“1, v1(πœ‰) tends to the periodic blow-up solution π‘£βˆ˜13𝑐(πœ‰)=2βˆ’π‘Ž1ξ€Έπ‘Ž2ξ‚€βˆš1+cosξ‚€ξ‚€π‘Ž1βˆ’π‘2ξ‚πœ‰./|𝑐|(3.5) The limiting process is similar to that in Figure 2.(ii)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š6 and (π‘Ž2,π‘Ž3) tends to the ray Ξ“7, 𝑣1(πœ‰) tends to the periodic blow-up solution π‘£βˆ—13𝑐(πœ‰)=2βˆ’π‘Ž1ξ€Έπ‘Ž2ξ‚€βˆš1βˆ’cosξ‚€ξ‚€π‘Ž1βˆ’π‘2ξ‚πœ‰./|𝑐|(3.6) The limiting process is similar to that in Figure 3.(iii)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š1 and (π‘Ž2,π‘Ž3) tends to the curve Ξ“2, or (π‘Ž2,π‘Ž3)βˆˆπ‘Š6 and (π‘Ž2,π‘Ž3) tends to the curve Ξ“6, 𝑣1(πœ‰) tends to the trivial solution 𝑣(πœ‰)=3(𝑐2βˆ’π‘Ž1)/π‘Ž2.(2)When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š5, or when 𝑐2>π‘Ž1 and (π‘Ž2,π‘Ž3)∈Ω1, the expression of the periodic wave solution is 𝑣2(πœ‰)=𝑆0βˆ’2𝑆1βˆ’π‘†2+𝑆3𝑆cos4πœ‰ξ€Έ,(3.7) where 𝑆0=βˆ’π‘Ž2+βˆšπœ”2π‘Ž3,𝑆1=βˆ’π‘Ž22+4π‘Ž3ξ€·π‘Ž1βˆ’π‘2ξ€Έ+π‘Ž2βˆšπœ”π‘Ž23,𝑆2=23π‘Ž3ξ‚€βˆ’π‘Ž2√+3πœ”ξ‚,𝑆3=23π‘Ž3ξ‚™π‘Ž2ξ‚€π‘Ž2√+3πœ”ξ‚,𝑆4=ξ‚™βˆ’π‘†1π‘Ž32𝑐2,(3.8)π‘Žπœ”=22βˆ’4π‘Ž3ξ€·π‘Ž1βˆ’π‘2ξ€Έ.(3.9) The periodic wave solution 𝑣2(πœ‰) has the following limit forms. (i)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š5, and (π‘Ž2,π‘Ž3) tends to the curve Ξ“6, 𝑣2(πœ‰) tends to the trivial solution 𝑣(πœ‰)=0.(ii)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š5, and (π‘Ž2,π‘Ž3) tends to the curve Ξ“5, the 𝑣2(πœ‰) tends to the canyon-shaped solitary wave solution π‘£βˆ˜22ξ€·π‘Ž(πœ‰)=1βˆ’π‘2ξ€Έξ€Ί12𝑐2βˆ’9𝑐2ξ€·π‘Žβˆ’21βˆ’π‘2ξ€Έπœ‰2ξ€»π‘Ž2ξ€Ί9𝑐2ξ€·π‘Ž+21βˆ’π‘2ξ€Έπœ‰2ξ€».(3.10) The limiting process is similar to that in Figure 6.(iii)When 𝑐2>π‘Ž1, (π‘Ž2,π‘Ž3)∈Ω1, and (π‘Ž2,π‘Ž3) tends to the ray 𝐿1, 𝑣2(πœ‰) tends to the periodic blow-up wave solution π‘£βˆ—2π‘Ž(πœ‰)=1βˆ’π‘22π‘Ž2⎑⎒⎒⎣1+3tan2βŽ›βŽœβŽœβŽξƒŽπ‘2βˆ’π‘Ž14𝑐2πœ‰βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦.(3.11) The limiting process is similar to that in Figure 2.(3)When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š2, or when 𝑐2>π‘Ž1 and (π‘Ž2,π‘Ž3)∈Ω2, the expression of the periodic wave solution is 𝑣3(πœ‰)=𝑇0+2𝑇1βˆ’π‘‡2+𝑇3𝑇cos4πœ‰ξ€Έ,(3.12) where 𝑇0=βˆ’π‘Ž2βˆ’βˆšπœ”2π‘Ž3,𝑇1=βˆ’π‘Ž22+4π‘Ž3ξ€·π‘Ž1βˆ’π‘2ξ€Έβˆ’π‘Ž2βˆšπœ”π‘Ž23,𝑇2=23π‘Ž3ξ‚€π‘Ž2√+3πœ”ξ‚,𝑇3=23π‘Ž3ξ‚™π‘Ž2ξ‚€π‘Ž2βˆšβˆ’3πœ”ξ‚,𝑇4=ξ‚™βˆ’π‘‡1π‘Ž32𝑐2.(3.13) The periodic wave solution 𝑣3(πœ‰) has the following limit forms:(i)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š2, and (π‘Ž2,π‘Ž3) tends to the curve Ξ“2, 𝑣3(πœ‰) tends to the trivial solution 𝑣(πœ‰)=0.(ii)When 𝑐2<π‘Ž1, (π‘Ž2,π‘Ž3)βˆˆπ‘Š2, and (π‘Ž2,π‘Ž3) tends to the curve Ξ“3, the 𝑣3(πœ‰) tends to the peak-shaped solitary wave solution π‘£βˆ˜2(πœ‰). The limiting process is similar to that in Figure 4.(iii)When 𝑐2>π‘Ž1, (π‘Ž2,π‘Ž3)∈Ω2, and (π‘Ž2,π‘Ž3) tends to the ray 𝐿3, the 𝑣3(πœ‰) tends to the periodic blow-up wave solution π‘£βˆ—2(πœ‰). The limiting process is similar to that in Figure 5.

Remark 3.2. Similar to the reason in Remark 2.2, the following nine functions also are periodic wave solutions of (1.2).
(1)When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š1 or π‘Š6, the functions are 𝑣11𝑅(πœ‰)=0𝑅1βˆ’π‘…2𝑅cos3πœ‰ξ€Έ,𝑣12(π‘…πœ‰)=0𝑅1+𝑅2𝑅sin3πœ‰ξ€Έ,𝑣13𝑅(πœ‰)=0𝑅1βˆ’π‘…2𝑅sin3πœ‰ξ€Έ.(3.14)(2) When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š5 or when 𝑐2>π‘Ž1 and (π‘Ž2,π‘Ž3)∈Ω1, the functions are 𝑣21(πœ‰)=S0+2𝑆1𝑆2+𝑆3𝑆cos4πœ‰ξ€Έ,𝑣22(πœ‰)=𝑆0βˆ’2𝑆1βˆ’π‘†2+𝑆3𝑆sin4πœ‰ξ€Έ,𝑣23(πœ‰)=𝑆0+2𝑆1𝑆2+𝑆3𝑆sin4πœ‰ξ€Έ.(3.15)(3) When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š2 or when 𝑐2>π‘Ž1 and (π‘Ž2,π‘Ž3)∈Ω2, the functions are 𝑣31(πœ‰)=𝑇0βˆ’2𝑇1𝑇2+𝑇3𝑇cos4πœ‰ξ€Έ,𝑣32(πœ‰)=𝑇0+2𝑇1βˆ’π‘‡2+𝑇3𝑇sin4πœ‰ξ€Έ,𝑣33(πœ‰)=𝑇0βˆ’2𝑇1𝑇2+𝑇3𝑇sin4πœ‰ξ€Έ.(3.16)

Remark 3.3. In the given regions, the solutions 𝑣𝑖(πœ‰), 𝑣1𝑖(πœ‰), 𝑣2𝑖(πœ‰), 𝑣3𝑖(πœ‰)(𝑖=1,2,3), and π‘£βˆ˜2(πœ‰) are nonsingular. The solutions π‘£βˆ˜1(πœ‰), π‘£βˆ—1(πœ‰), and π‘£βˆ—2(πœ‰) are singular. The relationships of nonsingular solutions and singular solutions are displayed in Proposition 3.1.

4. The Derivation on Proposition 2.1

In order to derive the Proposition 2.1, letting 𝑐 be a constant and substituting 𝑒=πœ‘(πœ‰) with πœ‰=π‘₯βˆ’π‘π‘‘ into (1.1), we have βˆ’π‘πœ‘ξ…ž+π‘Žπœ‘πœ‘ξ…ž+π‘Žπ‘πœ‘2πœ‘ξ…ž+πœ‘ξ…žξ…žξ…ž=0.(4.1)

Integrating (4.1) once and letting the integral constant be zero, it follows that π‘Žβˆ’π‘πœ‘+2πœ‘2+π‘Žπ‘3πœ‘3+πœ‘ξ…žξ…ž=0.(4.2)

Letting πœ‘ξ…ž=𝑦, yields the following planar system:πœ‘ξ…ž=𝑦,π‘¦ξ…žπ‘Ž=π‘πœ‘βˆ’2βˆ’π‘Žπ‘3πœ‘3.(4.3)

Obviously, system (4.3) has the first integral 6𝑦2βˆ’6π‘πœ‘2+2π‘Žπœ‘3+π‘Žπ‘πœ‘4=β„Ž.(4.4)

Let πœ‘1=βˆšβˆ’3π‘Žβˆ’Ξ”,πœ‘4π‘Žπ‘2=βˆšβˆ’3π‘Ž+Ξ”,4π‘Žπ‘(4.5) where Ξ” is defined in (2.8). Then, it is easy to see that system (4.3) has three singular points (πœ‘1,0), (0,0) and (πœ‘2,0) when Ξ”>0, two singular points ((βˆ’3/4𝑏),0) and (0,0) when Ξ”=0, unique singular point (0,0) when Ξ”<0.

Let 𝑒𝑖 and 𝑓𝑖(𝑖=1,2,3) be, respectively,𝑒1=βˆšβˆ’π‘Žβˆ’π‘Ž2+6π‘Žπ‘π‘,π‘“π‘Žπ‘1=βˆšβˆ’π‘Ž+π‘Ž2+6π‘Žπ‘π‘,π‘’π‘Žπ‘2=1ξƒ©βˆš4π‘Žπ‘βˆ’π‘Ž+ξ‚™Ξ”βˆ’2π‘Žξ‚€βˆšπ‘Žβˆ’Ξ”ξ‚ξƒͺ,𝑓2=1ξƒ©βˆš4π‘Žπ‘βˆ’π‘Ž+ξ‚™Ξ”+2π‘Žξ‚€βˆšπ‘Žβˆ’Ξ”ξ‚ξƒͺ,𝑒31=βˆ’ξƒ©βˆš4π‘Žπ‘π‘Ž+ξ‚™Ξ”+2π‘Žξ‚€βˆšπ‘Ž+Δξƒͺ,𝑓31=βˆ’ξƒ©βˆš4π‘Žπ‘π‘Ž+ξ‚™Ξ”βˆ’2π‘Žξ‚€βˆšπ‘Ž+Δξƒͺ.(4.6)

Using the qualitative analysis of dynamical systems, we obtain the bifurcation phase portraits of system (4.3) and the locations of 𝑒𝑖 and 𝑓𝑖(𝑖=1,2,3) as Figures 8 and 9.

810217.fig.008
Figure 8: When 𝑐<0, the bifurcation phase portraits of system (4.3) and the locations of 𝑒𝑖 and 𝑓𝑖(𝑖=1,2,3).
810217.fig.009
Figure 9: When 𝑐>0, the bifurcation phase portraits of system (4.3) and the locations of 𝑒𝑖 and 𝑓𝑖(𝑖=1,2,3).

It is easy to test that the closed orbit passing (𝑒𝑖,0) passes (𝑓𝑖,0)(𝑖=1,2,3). Thus, using the phase portraits in Figures 8 and 9, we derive 𝑒𝑖(πœ‰)(𝑖=1,2,3) as follows.

(1)When 𝑐<0 and (π‘Ž,𝑏)∈𝐴1 or 𝐴5, the closed orbit passing the points (𝑒1,0) and (𝑓1,0) has expression 𝑦=Β±π‘Žπ‘6πœ‘ξ”βˆ’π‘’1𝑓1+𝑒1+𝑓1ξ€Έπœ‘βˆ’πœ‘2,where𝑒1β‰€πœ‘β‰€π‘“1.(4.7) Substituting (4.7) into dπœ‘/𝑦=dπœ‰, we have dπœ‘ξ”βˆ’π‘’1𝑓1+𝑒1+𝑓1ξ€Έπœ‘βˆ’πœ‘2=ξ‚™π‘Žπ‘6dπœ‰.(4.8) Integrating (4.8) along the closed orbit and noting that 𝑒=πœ‘(πœ‰), we obtain the solution 𝑒1(πœ‰) as (2.4).(2) When 𝑐<0 and (π‘Ž,𝑏)∈𝐴2 or when 𝑐>0 and (π‘Ž,𝑏)∈𝐡5, the closed orbit passing the points (𝑒2,0) and (𝑓2,0) has expression 𝑦=Β±π‘Žπ‘6ξ€·πœ‘βˆ’πœ‘1ξ€Έξ”βˆ’π‘’2𝑓2+𝑒2+𝑓2ξ€Έπœ‘βˆ’πœ‘2,where𝑒2β‰€πœ‘β‰€π‘“2.(4.9) Substituting (4.9) into dπœ‘/𝑦=dπœ‰, we get dπœ‘ξ€·πœ‘βˆ’πœ‘1ξ€Έξ”βˆ’π‘’2𝑓2+𝑒2+𝑓2ξ€Έπœ‘βˆ’πœ‘2=ξ‚™π‘Žπ‘6dπœ‰.(4.10) Along the closed orbit integrating (4.10) and noting that 𝑒=πœ‘(πœ‰), we get the solution 𝑒2(πœ‰) as (2.7).(3) When 𝑐<0 and (π‘Ž,𝑏)∈𝐴6 or when 𝑐>0 and (π‘Ž,𝑏)∈𝐡1, the closed orbit passing the points (𝑒3,0) and (𝑓3,0) has expression 𝑦=Β±π‘Žπ‘6ξ€·πœ‘2ξ€Έξ”βˆ’πœ‘βˆ’π‘’3𝑓3+𝑒3+𝑓3ξ€Έπœ‘βˆ’πœ‘2,where𝑒3β‰€πœ‘β‰€π‘“3.(4.11) Substituting (4.11) into dπœ‘/𝑦=dπœ‰, it follows that dπœ‘ξ€·πœ‘2ξ€Έξ”βˆ’πœ‘βˆ’π‘’3𝑓3+𝑒3+𝑓3ξ€Έπœ‘βˆ’πœ‘2=ξ‚™π‘Žπ‘6dπœ‰.(4.12) Similarly, along the closed orbit integrating (4.12), we obtain 𝑒3(πœ‰) as (2.16). From the expressions of these solutions, we get their limit forms. This completes the derivation on Proposition 2.1.

5. The Derivation on Proposition 3.1

In this section, we give derivation on Proposition 3.1. Let 𝑣=πœ“(πœ‰) with πœ‰=π‘₯βˆ’π‘π‘‘, where 𝑐 is a constant. Thus, (1.2) becomes 𝑐2πœ“ξ…žξ…žβˆ’π‘2πœ“ξ…žξ…žξ…žξ…žβˆ’ξ€·π‘Ž1πœ“+π‘Ž2πœ“2+π‘Ž3πœ“3ξ€Έξ…žξ…ž=0.(5.1)

Integrating (5.1) twice and letting integral constant be zero, we get 𝑐2ξ€·πœ“βˆ’πœ“ξ…žξ…žξ€Έ=π‘Ž1πœ“+π‘Ž2πœ“2+π‘Ž3πœ“3.(5.2)

Letting πœ“ξ…ž=𝑦, we have the planar system πœ“ξ…ž=𝑦,𝑐2π‘¦ξ…ž=𝑐2βˆ’π‘Ž1ξ€Έπœ“βˆ’π‘Ž2πœ“2βˆ’π‘Ž3πœ“3.(5.3)

It is easy to see that system (5.3) has the first integral 𝑐2𝑦2+πœ“2ξ‚΅π‘Ž32πœ“2+2π‘Ž23πœ“+π‘Ž1βˆ’π‘2ξ‚Ά=β„Ž,(5.4) and three singular points (0,0), (πœ“1,0), and (πœ“2,0), where πœ“1=βˆ’π‘Ž2βˆ’βˆšπœ”2π‘Ž3,πœ“2=βˆ’π‘Ž2+βˆšπœ”2π‘Ž3(5.5) and πœ” is defined in (3.9).

Let π‘šπ‘– and 𝑛𝑖(𝑖=1,2,3) be, respectively,π‘š1=βˆ’2π‘Ž2βˆ’ξ”2ξ€·π‘Ž22βˆ’9π‘Ž1π‘Ž3+9π‘Ž3𝑐2ξ€Έ3π‘Ž3,𝑛1=βˆ’2π‘Ž2+2ξ€·π‘Ž22βˆ’9π‘Ž1π‘Ž3+9π‘Ž3𝑐2ξ€Έ3π‘Ž3,π‘š2=βˆ’π‘Ž2βˆšβˆ’3ξ‚™πœ”βˆ’2π‘Ž2ξ‚€π‘Ž2√+3πœ”ξ‚6π‘Ž3,𝑛2=βˆ’π‘Ž2βˆšβˆ’3ξ‚™πœ”+2π‘Ž2ξ‚€π‘Ž2√+3πœ”ξ‚6π‘Ž3,π‘š3=βˆ’π‘Ž2√+3ξ‚™πœ”βˆ’2π‘Ž2ξ‚€π‘Ž2βˆšβˆ’3πœ”ξ‚6π‘Ž3,𝑛3=βˆ’π‘Ž2√+3ξ‚™πœ”+2π‘Ž2ξ‚€π‘Ž2βˆšβˆ’3πœ”ξ‚6π‘Ž3.(5.6)

Similarly, using the qualitative analysis of dynamical systems, we get the bifurcation phase portraits of system (5.3) and the locations of π‘šπ‘– and 𝑛𝑖(𝑖=1,2,3) as Figures 10 and 11.

810217.fig.0010
Figure 10: When 𝑐2<π‘Ž1, the bifurcation phase portraits of system (5.3) and the locations of π‘šπ‘– and 𝑛𝑖(𝑖=1,2,3).
810217.fig.0011
Figure 11: When 𝑐2>π‘Ž1, the bifurcation phase portraits of system (5.3) and the locations of π‘šπ‘– and 𝑛𝑖(𝑖=1,2,3).

It is easy to test that the closed orbit passing (π‘šπ‘–,0) passes (𝑛𝑖,0)(𝑖=1,2,3). Thus, using the phase portraits in Figures 10 and 11, we derive 𝑣𝑖(πœ‰)(𝑖=1,2,3) as follows.(1)When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š1 or π‘Š6, the closed orbit passing the points (π‘š1,0) and (𝑛1,0) has expression 𝑦=Β±π‘Ž32𝑐2πœ“ξ”βˆ’π‘š1𝑛1+ξ€·π‘š1+𝑛1ξ€Έπœ“βˆ’πœ“2,whereπ‘š1β‰€πœ“β‰€π‘›1.(5.7) Substituting (5.7) into dπœ“/𝑦=dπœ‰, we have dπœ“πœ“ξ”βˆ’π‘š1𝑛1+ξ€·π‘š1+𝑛1ξ€Έπœ“βˆ’πœ“2=ξ‚™π‘Ž32𝑐2dπœ‰.(5.8) Integrating (5.8) along the closed orbit and noting that 𝑣=πœ“(πœ‰), we get the solution 𝑣1(πœ‰) as (3.3). (2) When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š5, or when 𝑐2>π‘Ž1 and (π‘Ž2,π‘Ž3)∈Ω1, the closed orbit passing the points (π‘š2,0) and (𝑛2,0) has expression 𝑦=Β±π‘Ž32𝑐2ξ€·πœ“2ξ€Έξ”βˆ’πœ“βˆ’π‘š2𝑛2+ξ€·π‘š2+𝑛2ξ€Έπœ“βˆ’πœ“2,whereπ‘š2β‰€πœ“β‰€π‘›2.(5.9) From dπœ“/𝑦=dπœ‰ and (5.9), it follows that dπœ“ξ€·πœ“2ξ€Έξ”βˆ’πœ“βˆ’π‘š2𝑛2+ξ€·π‘š2+𝑛2ξ€Έπœ“βˆ’πœ“2=ξ‚™π‘Ž32𝑐2dπœ‰.(5.10) Integrating (5.10) along the closed orbit, we get 𝑣2(πœ‰) as (3.7).(3) When 𝑐2<π‘Ž1 and (π‘Ž2,π‘Ž3)βˆˆπ‘Š2, or when 𝑐2>π‘Ž1 and (π‘Ž2,π‘Ž3)∈Ω2, the closed orbit passing the points (π‘š3,0) and (𝑛3,0) has expression 𝑦=Β±π‘Ž32𝑐2ξ€·πœ“βˆ’πœ“1ξ€Έξ”βˆ’π‘š3𝑛3+ξ€·π‘š3+𝑛3ξ€Έπœ“βˆ’πœ“2,whereπ‘š3β‰€πœ“β‰€π‘›3.(5.11) Substituting (5.11) into dπœ“/𝑦=dπœ‰, we have dπœ“ξ€·πœ“βˆ’πœ“1ξ€Έξ”βˆ’π‘š3𝑛3+ξ€·π‘š3+𝑛3ξ€Έπœ“βˆ’πœ“2=ξ‚™π‘Ž32𝑐2dπœ‰.(5.12) Integrating (5.12) along the closed orbit, we obtain 𝑣3(πœ‰) as (3.12). From the expressions of these solutions, we get their limiting properties. This completes the derivation on Proposition 3.1.

6. Discussions and Testing Orders

In this paper, Using the special closed orbits, we have obtained trigonometric function periodic wave solutions for (1.1) and (1.2), respectively. Their limit forms have been given. From these expressions, an interesting phenomena has been seen, that is, (1.1) and (1.2) have similar periodic wave solutions. Our work has extended previous results on periodic wave solutions.

Now, we point out that the trigonometric function periodic wave solutions can be obtained from the limits of the elliplic function periodic wave solution. For given real number πœ‡, let πœ‡1=1βŽ›βŽœβŽœβŽ4ξ‚€βˆš12π‘Žπ‘βˆ’4π‘Ž(2+π‘πœ‡)+1+i3ξ‚π‘ŽπΉ02πΉξ‚€βˆš+2ii+3ξ‚πΉβŽžβŽŸβŽŸβŽ ,πœ‡2=1βŽ›βŽœβŽœβŽ4ξ‚€βˆš12π‘Žπ‘βˆ’4π‘Ž(2+π‘πœ‡)+1βˆ’i3ξ‚π‘ŽπΉ02πΉξ‚€βˆšβˆ’2ii+3ξ‚πΉβŽžβŽŸβŽŸβŽ ,πœ‡3=1ξ‚΅6π‘Žπ‘βˆ’2π‘Ž(2+π‘πœ‡)βˆ’4π‘ŽπΉ02𝐹,+2𝐹(6.1) where 𝐹01=ξ€·8βˆ’6π‘πœ‡+15𝑏2πœ‡2+10𝑏3πœ‡3ξ€Έ,𝐹02=ξ€·ξ€·βˆ’9𝑏𝑐+π‘Žβˆ’2+π‘πœ‡+𝑏2πœ‡2,𝐹03=ξ‚™π‘Ž3ξ‚€8𝐹302ξ€·+π‘Žβˆ’54𝑏𝑐(βˆ’1+π‘πœ‡)+π‘ŽπΉ01ξ€Έ2,𝐹=54π‘Ž2𝑏𝑐(βˆ’1+π‘πœ‡)βˆ’π‘Ž3𝐹01+𝐹03ξ€Έ1/3.(6.2)

Assume that 𝑐<0, (π‘Ž,𝑏)∈(𝐴1), and πœ‘1<πœ‡<𝑒1. It is easy to check that πœ‡π‘–(𝑖=1,2,3) are real and satisfy πœ‡<𝑒1<πœ‘2<𝑓1<πœ‡1<πœ‘3<πœ‡2<0<πœ‡3<πœ‘4.(6.3)

There are two closed orbits 𝑙1πœ‡ and 𝑙2πœ‡ (see Figure 12). The closed orbit 𝑙1πœ‡ passes the points (πœ‡,0) and (πœ‡1,0). The closed orbit 𝑙2πœ‡ passes the points (πœ‡2,0) and (πœ‡3,0).

810217.fig.0012
Figure 12: The locations of 𝑙1πœ‡ and 𝑙2πœ‡ when 𝑐<0 and (π‘Ž,𝑏)∈𝐴1.

On πœ‘βˆ’π‘¦ plane, the expression of 𝑙1πœ‡ is 𝑦2=π‘Žπ‘6ξ€·πœ‡3πœ‡βˆ’πœ‘ξ€Έξ€·2πœ‡βˆ’πœ‘ξ€Έξ€·1ξ€Έβˆ’πœ‘(πœ‘βˆ’πœ‡),whereπœ‡β‰€πœ‘β‰€πœ‡1.(6.4)

Substituting (6.4) into dπœ‘/𝑦=dπœ‰ and integrating it along 𝑙1πœ‡, we have 𝑔snβˆ’1ξ‚™(sin𝑧,π‘˜)=π‘Žπ‘6||πœ‰||,(6.5) where 2𝑔=ξ”ξ€·πœ‡3βˆ’πœ‡1πœ‡ξ€Έξ€·2ξ€Έ,ξ„Άξ„΅ξ„΅βŽ·βˆ’πœ‡π‘˜=ξ€·πœ‡3βˆ’πœ‡2πœ‡ξ€Έξ€·1ξ€Έβˆ’πœ‡ξ€·πœ‡3βˆ’πœ‡1πœ‡ξ€Έξ€·2ξ€Έ,ξ„Άξ„΅ξ„΅βŽ·βˆ’πœ‡sin𝑧=ξ€·πœ‡3βˆ’πœ‡1ξ€Έ(πœ‘βˆ’πœ‡)ξ€·πœ‡1πœ‡βˆ’πœ‡ξ€Έξ€·3ξ€Έ.βˆ’πœ‘(6.6)

Solving (6.5) for πœ‘ and noting that 𝑒=πœ‘(πœ‰), we obtain an elliptic function periodic wave solution π‘’πœ‡ξ€·πœ‡(πœ‰)=3βˆ’πœ‡1ξ€Έ+πœ‡3ξ€·πœ‡1ξ€Έβˆ’πœ‡sn2(πœ‚πœ‰,π‘˜)πœ‡3βˆ’πœ‡1+ξ€·πœ‡1ξ€Έβˆ’πœ‡sn2,(πœ‚πœ‰,π‘˜)(6.7) where ξƒŽπœ‚=ξ€·πœ‡π‘Žπ‘3βˆ’πœ‡1πœ‡ξ€Έξ€·2ξ€Έβˆ’πœ‡.24(6.8)

Letting πœ‡β†’π‘’1βˆ’0, it follows that πœ‡1→𝑓1, πœ‡2β†’0, πœ‡3β†’0, π‘˜β†’0, βˆšπœ‚β†’(π‘Žπ‘π‘’1𝑓1)/24 and sn2(πœ‚πœ‰,π‘˜)β†’sn2(√(π‘Žπ‘π‘’1𝑓1/24)πœ‰,0)=sin2(√(π‘Žπ‘π‘’1𝑓1/24)πœ‰).

Therefore, in (6.7) letting πœ‡β†’π‘’1βˆ’0, we obtain the trigonometric function periodic wave solution 𝑒𝑒(πœ‰)=1𝑓1𝑓1+𝑒1βˆ’π‘“1ξ€Έsin2ξ‚΅ξ”ξ€·π‘Žπ‘π‘’1𝑓1ξ€Έπœ‰ξ‚Ά=/24βˆ’6π‘βˆšβˆ’π‘Ž+βˆšπ‘Ž(π‘Ž+6𝑏𝑐)βˆ’2π‘Ž(π‘Ž+6𝑏𝑐)sin2βˆšξ‚€ξ‚€ξ‚πœ‰ξ‚=|𝑐|/26π‘βˆšπ‘Žβˆ’π‘Žξ‚€βˆš(π‘Ž+6𝑏𝑐)cos|𝑐|πœ‰=𝑒11(πœ‰).(6.9)

Via Remark 2.2 and 𝑒11(πœ‰), further we get 𝑒12(πœ‰), 𝑒13(πœ‰) and 𝑒1(πœ‰). Similarly, we can derive others trigonometric function periodic wave solutions.

We also have tested the correctness of each solution by using the software Mathematica. Here, we list two testing orders. Others testing orders are similar.(1)The orders for testing 𝑒1(πœ‰)u=6c√a+aξ‚ƒβˆš(a+6bc)cosξ‚„βˆ’c(xβˆ’ct)(6.10) Simplify [D[u,t]+a(1+bu)D[u,x]u+D[u,{x,3}]]. (2) The orders for testing 𝑣1(πœ‰)R0ξ€·=2βˆ’a1+c2ξ€Έ,R1=2a23,R2=ξƒŽ2a3ξ€·βˆ’a1+c2ξ€Έ+4a229,R3=ξƒŽa1βˆ’c2c2,Rv=0R1+R2ξ€ΊRcos3ξ€»,D[],[](xβˆ’ct)vtt=v,{t,2}vttxx=Dvtt,{x,2}(6.11) Simplify [vttβˆ’vttxxβˆ’D[a1v+a2v2+a3v3,{x,2}]].

Acknowledgment

Research is supported by the National Natural Science Foundation of China (no. 10871073) and the Research Expences of Central Universities for students.

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