Abstract

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation 𝑢𝑡+𝑎(1+𝑏𝑢)𝑢𝑢𝑥+𝑢𝑥𝑥𝑥=0, and PC-like equation 𝑣𝑡𝑡𝑣𝑡𝑡𝑥𝑥(𝑎1𝑣+𝑎2𝑣2+𝑎3𝑣3)𝑥𝑥=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 𝑢𝑡+𝑎(1+𝑏𝑢)𝑢𝑢𝑥+𝑢𝑥𝑥𝑥=0,(1.1) and the PC-like equation 𝑣𝑡𝑡𝑣𝑡𝑡𝑥𝑥𝑎1𝑣+𝑎2𝑣2+𝑎3𝑣3𝑥𝑥=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 [1830].

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 sin 𝜉 or cos 𝜉. 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 𝑐0, on 𝑎𝑏 plane we define some lines and regions as follows.(1)When 𝑐<0, we define lines 𝑙1𝑙:𝑏=0,2𝑎:𝑏=,𝑙6𝑐3:𝑏=3𝑎,𝑙16𝑐4:𝑎=0,(2.1) and regions 𝐴𝑖(𝑖=1-8), as Figure 1(a).

(2)When 𝑐>0, we define lines 𝑘1𝑘:𝑏=0,2𝑘:𝑎=0,3:𝑏=3𝑎,𝑘16𝑐4𝑎:𝑏=,6𝑐(2.2) and regions 𝐵𝑖(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 𝑐0, let 𝜉=𝑥𝑐𝑡.(2.3) Then, (1.1) has the following periodic wave solutions. (1)When c<0 and (𝑎,𝑏)𝐴1 or 𝐴5, the expression of the periodic wave solution is 𝑢1(𝜉)=6𝑐𝑎+𝑎(𝑎+6𝑏𝑐)cos,𝑐𝜉(2.4) which has the following limit forms.(i)When 𝑐<0, (𝑎,𝑏)𝐴1 and (𝑎,𝑏) tends to the line 𝑙1, 𝑢1(𝜉) tends to the periodic blow-up solution 𝑢1(𝜉)=6𝑐𝑎1+cos𝑐𝜉(2.5) (see Figure 2).(ii)When 𝑐<0, (𝑎,𝑏)𝐴5 and (𝑎,𝑏) tends to the line 𝑙1, 𝑢1(𝜉) tends to the periodic blow-up solution 𝑢1(𝜉)=6𝑐𝑎1cos𝑐𝜉(2.6) (see Figure 3).(iii)When 𝑐<0, (𝑎,𝑏)𝐴1 or 𝐴5, and (𝑎,𝑏) tends to 𝑙2, 𝑢1(𝜉) tends to the trivial solution 𝑢(𝜉)=6𝑐/𝑎.(2)When 𝑐<0 and (𝑎,𝑏)𝐴2, or when 𝑐>0 and (𝑎,𝑏)𝐵5, the expression of the periodic wave solution is 𝑢2𝛼(𝜉)=0𝑤cos0𝜉+𝛽0𝑝0𝑤cos0𝜉+𝑞0,(2.7) where Δ=3𝑎(3𝑎+16𝑏𝑐),(2.8)𝛼0=3𝑎+Δ𝑎𝑎Δ4𝑎2𝑏2,𝛽(2.9)0=3𝑎+24𝑏𝑐+Δ2𝑎𝑏2𝑝,(2.10)0=𝑎𝑎Δ𝑞𝑎𝑏,(2.11)01=𝑎𝑏𝑎+Δ𝑤,(2.12)0=3𝑎+16𝑏𝑐+Δ.8𝑏(2.13) The solution 𝑢2(𝜉) has the following limit forms. (i)When 𝑐<0, (𝑎,𝑏)𝐴2 and (𝑎,𝑏) tends to 𝑙3, the 𝑢2(𝜉) tends to the peak-shaped solitary wave solution 𝑢2(𝜉)=4𝑐3+2𝑐𝜉2𝑎9+2𝑐𝜉2(2.14) (see Figure 4).(ii)When 𝑐<0, (𝑎,𝑏)𝐴2 and (𝑎,𝑏) tends to 𝑙2, 𝑢2(𝜉) tends to the trivial solution 𝑢(𝜉)=0.(iii)When 𝑐>0, (𝑎,𝑏)𝐵5 and (𝑎,𝑏) tends to 𝑘1, the 𝑢2(𝜉) tends to the periodic blow-up solution 𝑢2𝑐(𝜉)=2sin2𝑐𝜉/23𝑎sin2𝑐𝜉/2(2.15) (see Figure 5).(3)When 𝑐<0 and (𝑎,𝑏)𝐴6, or when 𝑐>0 and (𝑎,𝑏)𝐵1, the expressions of the solution is 𝑢3𝛼(𝜉)=1𝑤cos1𝜉+𝛽1𝑝1𝑤cos1𝜉+𝑞1,(2.16) where 𝛼1=3𝑎+Δ𝑎𝑎+Δ4𝑎2𝑏2,𝛽1=3𝑎+24𝑏𝑐Δ2𝑎𝑏2,𝑝1=𝑎𝑎+Δ,𝑞𝑎𝑏1=𝑎Δ,𝑤𝑎𝑏1=3𝑎+16𝑏𝑐Δ.8𝑏(2.17) The solution 𝑢3(𝜉) has the following limit forms.(i)When 𝑐<0, (𝑎,𝑏)𝐴6 and (𝑎,𝑏) tends to 𝑙3, the 𝑢3(𝜉) tends to the canyon-shaped solitary wave (see Figure 6) solution 𝑢2(𝜉). (ii)When 𝑐<0, (𝑎,𝑏)𝐴6 and (𝑎,𝑏) tends to 𝑙2, 𝑢3(𝜉) tends to the trivial solution 𝑢(𝜉)=0. (iii)When 𝑐>0, (𝑎,𝑏)𝐵1 and (𝑎,𝑏) tends to 𝑘1, the 𝑢3(𝜉) tends to the periodic blow-up wave solution 𝑢1(𝜉) (see Figure 3).

Remark 2.2. Note that if 𝑢=𝜑(𝜉) is a solution of (1.1), then 𝑢=𝜑(𝜉+𝑟) also is solution of (1.1), where 𝑟 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 (𝑎,𝑏)𝐴1 or 𝐴5, the functions are 𝑢11(𝜉)=6𝑐𝑎𝑎(𝑎+6𝑏𝑐)cos,𝑢𝑐𝜉12(𝜉)=6𝑐𝑎+𝑎(𝑎+6𝑏𝑐)sin,𝑢𝑐𝜉13(𝜉)=6𝑐𝑎𝑎(𝑎+6𝑏𝑐)sin.𝑐𝜉(2.18)
(2) When 𝑐<0 and (𝑎,𝑏)𝐴2 or when 𝑐>0 and (𝑎,𝑏)𝐵5, the functions are 𝑢21(𝜉)=𝛼0𝑤cos0𝜉+𝛽0𝑝0𝑤cos0𝜉+𝑞0,𝑢22𝛼(𝜉)=0𝑤sin0𝜉+𝛽0𝑝0𝑤sin0𝜉+𝑞0,𝑢23(𝜉)=𝛼0𝑤sin0𝜉+𝛽0𝑝0𝑤sin0𝜉+𝑞0.(2.19)
(3) When 𝑐<0 and (𝑎,𝑏)𝐴6, or when 𝑐>0 and (𝑎,𝑏)𝐵1, the functions are 𝑢31(𝜉)=𝛼1𝑤cos1𝜉+𝛽1𝑝1𝑤cos1𝜉+𝑞1,𝑢32𝛼(𝜉)=1𝑤sin1𝜉+𝛽1𝑝1𝑤sin1𝜉+𝑞1,𝑢33(𝜉)=𝛼1𝑤sin1𝜉+𝛽1𝑝1𝑤sin1𝜉+𝑞1.(2.20)

Remark 2.3. In the given parametric regions, the solutions 𝑢𝑖(𝜉), 𝑢1𝑖(𝜉), 𝑢2𝑖(𝜉), 𝑢3𝑖(𝜉)(𝑖=1,2,3), and 𝑢2(𝜉) are nonsingular. The solutions 𝑢1(𝜉), 𝑢1(𝜉), and 𝑢2(𝜉) 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 𝑎1 and 𝑐(𝑎1𝑐2), on 𝑎2𝑎3 plane we define some rays and regions as follows.(1)When 𝑐2<𝑎1, we define curves Γ1:𝑎2>0,𝑎3Γ=0,2:𝑎2>0,𝑎3=2𝑎229𝑎1𝑐2,Γ3:𝑎2>0,𝑎3=𝑎224𝑎1𝑐2,Γ4:𝑎2=0,𝑎3Γ>0,5:𝑎2<0,𝑎3=𝑎224𝑎1𝑐2,Γ6:𝑎2<0,𝑎3=2𝑎229𝑎1𝑐2,Γ7:𝑎2<0,𝑎3Γ=0,8:𝑎2=0,𝑎3<0,(3.1) and region 𝑊𝑖 as the domain surrounded by Γ𝑖 and Γ𝑖+1(𝑖=1-7), 𝑊8 as the domain surrounded by Γ8 and Γ1 (see Figure 7(a)).

(2)When 𝑐2>𝑎1, we define curves 𝐿1:𝑎2>0,𝑎3𝐿=0,2:𝑎2=0,𝑎3𝐿>0,3:𝑎2<0,𝑎3𝐿=0,4:𝑎2<0,𝑎3=2𝑎229𝑎1𝑐2,𝐿5:𝑎2<0,𝑎3=𝑎214𝑎1𝑐2,𝐿6:𝑎2=0,𝑎3𝐿<0,7:𝑎2>0,𝑎3=𝑎224𝑎1𝑐2,𝐿8:𝑎2>0,𝑎3=2𝑎229𝑎1𝑐2,(3.2) and region Ω𝑖 as the domain surrounded by 𝐿𝑖 and 𝐿𝑖+1(𝑖=1-7), Ω8 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 𝑎1 and constant 𝑐 satisfying 𝑐2𝑎1, let 𝜉=𝑥𝑐𝑡. Then, (1.2) has the following periodic wave solutions. (1)When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊1 or 𝑊6, the expression of the periodic wave solution is 𝑣1𝑅(𝜉)=0𝑅1+𝑅2𝑅cos3𝜉,(3.3) where 𝑅0𝑐=22𝑎1,𝑅1=2𝑎23,𝑅2=1318𝑎3𝑐2𝑎1+4𝑎22,𝑅3=𝑎1𝑐2𝑐2.(3.4) For 𝑎20, the periodic wave solution 𝑣1(𝜉) has the following limit forms.(i)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊1 and (𝑎2,𝑎3) tends to the ray Γ1, v1(𝜉) tends to the periodic blow-up solution 𝑣13𝑐(𝜉)=2𝑎1𝑎21+cos𝑎1𝑐2𝜉./|𝑐|(3.5) The limiting process is similar to that in Figure 2.(ii)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊6 and (𝑎2,𝑎3) tends to the ray Γ7, 𝑣1(𝜉) tends to the periodic blow-up solution 𝑣13𝑐(𝜉)=2𝑎1𝑎21cos𝑎1𝑐2𝜉./|𝑐|(3.6) The limiting process is similar to that in Figure 3.(iii)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊1 and (𝑎2,𝑎3) tends to the curve Γ2, or (𝑎2,𝑎3)𝑊6 and (𝑎2,𝑎3) tends to the curve Γ6, 𝑣1(𝜉) tends to the trivial solution 𝑣(𝜉)=3(𝑐2𝑎1)/𝑎2.(2)When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊5, or when 𝑐2>𝑎1 and (𝑎2,𝑎3)Ω1, the expression of the periodic wave solution is 𝑣2(𝜉)=𝑆02𝑆1𝑆2+𝑆3𝑆cos4𝜉,(3.7) where 𝑆0=𝑎2+𝜔2𝑎3,𝑆1=𝑎22+4𝑎3𝑎1𝑐2+𝑎2𝜔𝑎23,𝑆2=23𝑎3𝑎2+3𝜔,𝑆3=23𝑎3𝑎2𝑎2+3𝜔,𝑆4=𝑆1𝑎32𝑐2,(3.8)𝑎𝜔=224𝑎3𝑎1𝑐2.(3.9) The periodic wave solution 𝑣2(𝜉) has the following limit forms. (i)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊5, and (𝑎2,𝑎3) tends to the curve Γ6, 𝑣2(𝜉) tends to the trivial solution 𝑣(𝜉)=0.(ii)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊5, and (𝑎2,𝑎3) tends to the curve Γ5, the 𝑣2(𝜉) tends to the canyon-shaped solitary wave solution 𝑣22𝑎(𝜉)=1𝑐212𝑐29𝑐2𝑎21𝑐2𝜉2𝑎29𝑐2𝑎+21𝑐2𝜉2.(3.10) The limiting process is similar to that in Figure 6.(iii)When 𝑐2>𝑎1, (𝑎2,𝑎3)Ω1, and (𝑎2,𝑎3) tends to the ray 𝐿1, 𝑣2(𝜉) tends to the periodic blow-up wave solution 𝑣2𝑎(𝜉)=1𝑐22𝑎21+3tan2𝑐2𝑎14𝑐2𝜉.(3.11) The limiting process is similar to that in Figure 2.(3)When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊2, or when 𝑐2>𝑎1 and (𝑎2,𝑎3)Ω2, the expression of the periodic wave solution is 𝑣3(𝜉)=𝑇0+2𝑇1𝑇2+𝑇3𝑇cos4𝜉,(3.12) where 𝑇0=𝑎2𝜔2𝑎3,𝑇1=𝑎22+4𝑎3𝑎1𝑐2𝑎2𝜔𝑎23,𝑇2=23𝑎3𝑎2+3𝜔,𝑇3=23𝑎3𝑎2𝑎23𝜔,𝑇4=𝑇1𝑎32𝑐2.(3.13) The periodic wave solution 𝑣3(𝜉) has the following limit forms:(i)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊2, and (𝑎2,𝑎3) tends to the curve Γ2, 𝑣3(𝜉) tends to the trivial solution 𝑣(𝜉)=0.(ii)When 𝑐2<𝑎1, (𝑎2,𝑎3)𝑊2, and (𝑎2,𝑎3) tends to the curve Γ3, the 𝑣3(𝜉) tends to the peak-shaped solitary wave solution 𝑣2(𝜉). The limiting process is similar to that in Figure 4.(iii)When 𝑐2>𝑎1, (𝑎2,𝑎3)Ω2, and (𝑎2,𝑎3) tends to the ray 𝐿3, the 𝑣3(𝜉) tends to the periodic blow-up wave solution 𝑣2(𝜉). 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<𝑎1 and (𝑎2,𝑎3)𝑊1 or 𝑊6, the functions are 𝑣11𝑅(𝜉)=0𝑅1𝑅2𝑅cos3𝜉,𝑣12(𝑅𝜉)=0𝑅1+𝑅2𝑅sin3𝜉,𝑣13𝑅(𝜉)=0𝑅1𝑅2𝑅sin3𝜉.(3.14)(2) When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊5 or when 𝑐2>𝑎1 and (𝑎2,𝑎3)Ω1, the functions are 𝑣21(𝜉)=S0+2𝑆1𝑆2+𝑆3𝑆cos4𝜉,𝑣22(𝜉)=𝑆02𝑆1𝑆2+𝑆3𝑆sin4𝜉,𝑣23(𝜉)=𝑆0+2𝑆1𝑆2+𝑆3𝑆sin4𝜉.(3.15)(3) When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊2 or when 𝑐2>𝑎1 and (𝑎2,𝑎3)Ω2, the functions are 𝑣31(𝜉)=𝑇02𝑇1𝑇2+𝑇3𝑇cos4𝜉,𝑣32(𝜉)=𝑇0+2𝑇1𝑇2+𝑇3𝑇sin4𝜉,𝑣33(𝜉)=𝑇02𝑇1𝑇2+𝑇3𝑇sin4𝜉.(3.16)

Remark 3.3. In the given regions, the solutions 𝑣𝑖(𝜉), 𝑣1𝑖(𝜉), 𝑣2𝑖(𝜉), 𝑣3𝑖(𝜉)(𝑖=1,2,3), and 𝑣2(𝜉) are nonsingular. The solutions 𝑣1(𝜉), 𝑣1(𝜉), and 𝑣2(𝜉) 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 𝑢=𝜑(𝜉) with 𝜉=𝑥𝑐𝑡 into (1.1), we have 𝑐𝜑+𝑎𝜑𝜑+𝑎𝑏𝜑2𝜑+𝜑=0.(4.1)

Integrating (4.1) once and letting the integral constant be zero, it follows that 𝑎𝑐𝜑+2𝜑2+𝑎𝑏3𝜑3+𝜑=0.(4.2)

Letting 𝜑=𝑦, yields the following planar system:𝜑=𝑦,𝑦𝑎=𝑐𝜑2𝑎𝑏3𝜑3.(4.3)

Obviously, system (4.3) has the first integral 6𝑦26𝑐𝜑2+2𝑎𝜑3+𝑎𝑏𝜑4=.(4.4)

Let 𝜑1=3𝑎Δ,𝜑4𝑎𝑏2=3𝑎+Δ,4𝑎𝑏(4.5) where Δ is defined in (2.8). Then, it is easy to see that system (4.3) has three singular points (𝜑1,0), (0,0) and (𝜑2,0) when Δ>0, two singular points ((3/4𝑏),0) and (0,0) when Δ=0, unique singular point (0,0) when Δ<0.

Let 𝑒𝑖 and 𝑓𝑖(𝑖=1,2,3) be, respectively,𝑒1=𝑎𝑎2+6𝑎𝑏𝑐,𝑓𝑎𝑏1=𝑎+𝑎2+6𝑎𝑏𝑐,𝑒𝑎𝑏2=14𝑎𝑏𝑎+Δ2𝑎𝑎Δ,𝑓2=14𝑎𝑏𝑎+Δ+2𝑎𝑎Δ,𝑒31=4𝑎𝑏𝑎+Δ+2𝑎𝑎+Δ,𝑓31=4𝑎𝑏𝑎+Δ2𝑎𝑎+Δ.(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.

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 𝑢𝑖(𝜉)(𝑖=1,2,3) as follows.

(1)When 𝑐<0 and (𝑎,𝑏)𝐴1 or 𝐴5, the closed orbit passing the points (𝑒1,0) and (𝑓1,0) has expression 𝑦=±𝑎𝑏6𝜑𝑒1𝑓1+𝑒1+𝑓1𝜑𝜑2,where𝑒1𝜑𝑓1.(4.7) Substituting (4.7) into d𝜑/𝑦=d𝜉, we have d𝜑𝑒1𝑓1+𝑒1+𝑓1𝜑𝜑2=𝑎𝑏6d𝜉.(4.8) Integrating (4.8) along the closed orbit and noting that 𝑢=𝜑(𝜉), we obtain the solution 𝑢1(𝜉) as (2.4).(2) When 𝑐<0 and (𝑎,𝑏)𝐴2 or when 𝑐>0 and (𝑎,𝑏)𝐵5, the closed orbit passing the points (𝑒2,0) and (𝑓2,0) has expression 𝑦=±𝑎𝑏6𝜑𝜑1𝑒2𝑓2+𝑒2+𝑓2𝜑𝜑2,where𝑒2𝜑𝑓2.(4.9) Substituting (4.9) into d𝜑/𝑦=d𝜉, we get d𝜑𝜑𝜑1𝑒2𝑓2+𝑒2+𝑓2𝜑𝜑2=𝑎𝑏6d𝜉.(4.10) Along the closed orbit integrating (4.10) and noting that 𝑢=𝜑(𝜉), we get the solution 𝑢2(𝜉) as (2.7).(3) When 𝑐<0 and (𝑎,𝑏)𝐴6 or when 𝑐>0 and (𝑎,𝑏)𝐵1, the closed orbit passing the points (𝑒3,0) and (𝑓3,0) has expression 𝑦=±𝑎𝑏6𝜑2𝜑𝑒3𝑓3+𝑒3+𝑓3𝜑𝜑2,where𝑒3𝜑𝑓3.(4.11) Substituting (4.11) into d𝜑/𝑦=d𝜉, it follows that d𝜑𝜑2𝜑𝑒3𝑓3+𝑒3+𝑓3𝜑𝜑2=𝑎𝑏6d𝜉.(4.12) Similarly, along the closed orbit integrating (4.12), we obtain 𝑢3(𝜉) 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=𝜓(𝜉) with 𝜉=𝑥𝑐𝑡, where 𝑐 is a constant. Thus, (1.2) becomes 𝑐2𝜓𝑐2𝜓𝑎1𝜓+𝑎2𝜓2+𝑎3𝜓3=0.(5.1)

Integrating (5.1) twice and letting integral constant be zero, we get 𝑐2𝜓𝜓=𝑎1𝜓+𝑎2𝜓2+𝑎3𝜓3.(5.2)

Letting 𝜓=𝑦, we have the planar system 𝜓=𝑦,𝑐2𝑦=𝑐2𝑎1𝜓𝑎2𝜓2𝑎3𝜓3.(5.3)

It is easy to see that system (5.3) has the first integral 𝑐2𝑦2+𝜓2𝑎32𝜓2+2𝑎23𝜓+𝑎1𝑐2=,(5.4) and three singular points (0,0), (𝜓1,0), and (𝜓2,0), where 𝜓1=𝑎2𝜔2𝑎3,𝜓2=𝑎2+𝜔2𝑎3(5.5) and 𝜔 is defined in (3.9).

Let 𝑚𝑖 and 𝑛𝑖(𝑖=1,2,3) be, respectively,𝑚1=2𝑎22𝑎229𝑎1𝑎3+9𝑎3𝑐23𝑎3,𝑛1=2𝑎2+2𝑎229𝑎1𝑎3+9𝑎3𝑐23𝑎3,𝑚2=𝑎23𝜔2𝑎2𝑎2+3𝜔6𝑎3,𝑛2=𝑎23𝜔+2𝑎2𝑎2+3𝜔6𝑎3,𝑚3=𝑎2+3𝜔2𝑎2𝑎23𝜔6𝑎3,𝑛3=𝑎2+3𝜔+2𝑎2𝑎23𝜔6𝑎3.(5.6)

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

It is easy to test that the closed orbit passing (𝑚𝑖,0) passes (𝑛𝑖,0)(𝑖=1,2,3). Thus, using the phase portraits in Figures 10 and 11, we derive 𝑣𝑖(𝜉)(𝑖=1,2,3) as follows.(1)When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊1 or 𝑊6, the closed orbit passing the points (𝑚1,0) and (𝑛1,0) has expression 𝑦=±𝑎32𝑐2𝜓𝑚1𝑛1+𝑚1+𝑛1𝜓𝜓2,where𝑚1𝜓𝑛1.(5.7) Substituting (5.7) into d𝜓/𝑦=d𝜉, we have d𝜓𝜓𝑚1𝑛1+𝑚1+𝑛1𝜓𝜓2=𝑎32𝑐2d𝜉.(5.8) Integrating (5.8) along the closed orbit and noting that 𝑣=𝜓(𝜉), we get the solution 𝑣1(𝜉) as (3.3). (2) When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊5, or when 𝑐2>𝑎1 and (𝑎2,𝑎3)Ω1, the closed orbit passing the points (𝑚2,0) and (𝑛2,0) has expression 𝑦=±𝑎32𝑐2𝜓2𝜓𝑚2𝑛2+𝑚2+𝑛2𝜓𝜓2,where𝑚2𝜓𝑛2.(5.9) From d𝜓/𝑦=d𝜉 and (5.9), it follows that d𝜓𝜓2𝜓𝑚2𝑛2+𝑚2+𝑛2𝜓𝜓2=𝑎32𝑐2d𝜉.(5.10) Integrating (5.10) along the closed orbit, we get 𝑣2(𝜉) as (3.7).(3) When 𝑐2<𝑎1 and (𝑎2,𝑎3)𝑊2, or when 𝑐2>𝑎1 and (𝑎2,𝑎3)Ω2, the closed orbit passing the points (𝑚3,0) and (𝑛3,0) has expression 𝑦=±𝑎32𝑐2𝜓𝜓1𝑚3𝑛3+𝑚3+𝑛3𝜓𝜓2,where𝑚3𝜓𝑛3.(5.11) Substituting (5.11) into d𝜓/𝑦=d𝜉, we have d𝜓𝜓𝜓1𝑚3𝑛3+𝑚3+𝑛3𝜓𝜓2=𝑎32𝑐2d𝜉.(5.12) Integrating (5.12) along the closed orbit, we obtain 𝑣3(𝜉) 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 𝜇, let 𝜇1=1412𝑎𝑏4𝑎(2+𝑏𝜇)+1+i3𝑎𝐹02𝐹+2ii+3𝐹,𝜇2=1412𝑎𝑏4𝑎(2+𝑏𝜇)+1i3𝑎𝐹02𝐹2ii+3𝐹,𝜇3=16𝑎𝑏2𝑎(2+𝑏𝜇)4𝑎𝐹02𝐹,+2𝐹(6.1) where 𝐹01=86𝑏𝜇+15𝑏2𝜇2+10𝑏3𝜇3,𝐹02=9𝑏𝑐+𝑎2+𝑏𝜇+𝑏2𝜇2,𝐹03=𝑎38𝐹302+𝑎54𝑏𝑐(1+𝑏𝜇)+𝑎𝐹012,𝐹=54𝑎2𝑏𝑐(1+𝑏𝜇)𝑎3𝐹01+𝐹031/3.(6.2)

Assume that 𝑐<0, (𝑎,𝑏)(𝐴1), and 𝜑1<𝜇<𝑒1. It is easy to check that 𝜇𝑖(𝑖=1,2,3) are real and satisfy 𝜇<𝑒1<𝜑2<𝑓1<𝜇1<𝜑3<𝜇2<0<𝜇3<𝜑4.(6.3)

There are two closed orbits 𝑙1𝜇 and 𝑙2𝜇 (see Figure 12). The closed orbit 𝑙1𝜇 passes the points (𝜇,0) and (𝜇1,0). The closed orbit 𝑙2𝜇 passes the points (𝜇2,0) and (𝜇3,0).

On 𝜑𝑦 plane, the expression of 𝑙1𝜇 is 𝑦2=𝑎𝑏6𝜇3𝜇𝜑2𝜇𝜑1𝜑(𝜑𝜇),where𝜇𝜑𝜇1.(6.4)

Substituting (6.4) into d𝜑/𝑦=d𝜉 and integrating it along 𝑙1𝜇, we have 𝑔sn1(sin𝑧,𝑘)=𝑎𝑏6||𝜉||,(6.5) where 2𝑔=𝜇3𝜇1𝜇2,𝜇𝑘=𝜇3𝜇2𝜇1𝜇𝜇3𝜇1𝜇2,𝜇sin𝑧=𝜇3𝜇1(𝜑𝜇)𝜇1𝜇𝜇3.𝜑(6.6)

Solving (6.5) for 𝜑 and noting that 𝑢=𝜑(𝜉), we obtain an elliptic function periodic wave solution 𝑢𝜇𝜇(𝜉)=3𝜇1+𝜇3𝜇1𝜇sn2(𝜂𝜉,𝑘)𝜇3𝜇1+𝜇1𝜇sn2,(𝜂𝜉,𝑘)(6.7) where 𝜂=𝜇𝑎𝑏3𝜇1𝜇2𝜇.24(6.8)

Letting 𝜇𝑒10, it follows that 𝜇1𝑓1, 𝜇20, 𝜇30, 𝑘0, 𝜂(𝑎𝑏𝑒1𝑓1)/24 and sn2(𝜂𝜉,𝑘)sn2((𝑎𝑏𝑒1𝑓1/24)𝜉,0)=sin2((𝑎𝑏𝑒1𝑓1/24)𝜉).

Therefore, in (6.7) letting 𝜇𝑒10, we obtain the trigonometric function periodic wave solution 𝑒𝑢(𝜉)=1𝑓1𝑓1+𝑒1𝑓1sin2𝑎𝑏𝑒1𝑓1𝜉=/246𝑐𝑎+𝑎(𝑎+6𝑏𝑐)2𝑎(𝑎+6𝑏𝑐)sin2𝜉=|𝑐|/26𝑐𝑎𝑎(𝑎+6𝑏𝑐)cos|𝑐|𝜉=𝑢11(𝜉).(6.9)

Via Remark 2.2 and 𝑢11(𝜉), further we get 𝑢12(𝜉), 𝑢13(𝜉) and 𝑢1(𝜉). 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 𝑢1(𝜉)u=6ca+a(a+6bc)cosc(xct)(6.10) Simplify [D[u,t]+a(1+bu)D[u,x]u+D[u,{x,3}]]. (2) The orders for testing 𝑣1(𝜉)R0=2a1+c2,R1=2a23,R2=2a3a1+c2+4a229,R3=a1c2c2,Rv=0R1+R2Rcos3,D[],[](xct)vtt=v,{t,2}vttxx=Dvtt,{x,2}(6.11) Simplify [vttvttxxD[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.