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

Abstract

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation 𝑢_𝑡+𝑎(1+𝑏𝑢)𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0, and PC-like equation 𝑣_𝑡𝑡−𝑣_{𝑡𝑡𝑥𝑥}−(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 𝑢_𝑡+𝑎(1+𝑏𝑢)𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=0,(1.1) and the PC-like equation 𝑣_𝑡𝑡−𝑣_{𝑡𝑡𝑥𝑥}−𝑎₁𝑣+𝑎₂𝑣²+𝑎₃𝑣³_𝑥𝑥=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 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 𝑐≠0, on 𝑎−𝑏 plane we define some lines and regions as follows.(1)When 𝑐<0, we define lines 𝑙𝑙₁:𝑏=0,₂:𝑏=−𝑎𝑙6𝑐₃,:𝑏=−3𝑎𝑙16𝑐₄,:𝑎=0,(2.1) and regions 𝐴_𝑖(𝑖=1-8), as Figure 1(a).

(a) 𝑐<0

(b) 𝑐>0

Figure 1 

The locations of the lines 𝑙𝑖, 𝑘𝑖(𝑖=1,2,3,4) and the regions 𝐴𝑗, 𝐵𝑗(𝑗=1,2,…,8) for given constant 𝑐≠0.

(2)When 𝑐>0, we define lines 𝑘𝑘₁:𝑏=0,₂𝑘:𝑎=0,₃:𝑏=−3𝑎𝑘16𝑐₄,:𝑏=−𝑎6𝑐,(2.2) and regions 𝐵_𝑖(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 𝜉=𝑥−𝑐𝑡.(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 𝑢^∘1(𝜉)=6𝑐𝑎1+cos√−𝑐𝜉(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 𝑢^∗1(𝜉)=6𝑐𝑎1−cos√−𝑐𝜉(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 𝑢₂(𝜉)=𝛼⁰cos𝑤₀𝜉+𝛽₀𝑝₀cos𝑤₀𝜉+𝑞₀,(2.7) where Δ=3𝑎(3𝑎+16𝑏𝑐),(2.8)𝛼₀=−3𝑎+√Δ𝑎𝑎−√Δ4𝑎²𝑏²𝛽,(2.9)₀=−3𝑎+24𝑏𝑐+√Δ2𝑎𝑏²𝑝,(2.10)₀=𝑎𝑎−√Δ𝑞𝑎𝑏,(2.11)₀=−1𝑎𝑏𝑎+√𝑤Δ,(2.12)₀=3𝑎+16𝑏𝑐+√Δ8𝑏.(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 𝑢^∘2(𝜉)=4𝑐3+2𝑐𝜉²𝑎−9+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 𝑢^∗2(𝜉)=𝑐2sin²√𝑐𝜉/2−3𝑎sin²√𝑐𝜉/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 𝑢₃(𝜉)=𝛼¹cos𝑤₁𝜉+𝛽₁𝑝₁cos𝑤₁𝜉+𝑞₁,(2.16) where 𝛼₁=−3𝑎+√Δ𝑎𝑎+√Δ4𝑎²𝑏²𝛽₁,=3𝑎+24𝑏𝑐−√Δ2𝑎𝑏²𝑝₁,=𝑎𝑎+√Δ𝑞𝑎𝑏₁,=𝑎−√Δ𝑤𝑎𝑏₁,=3𝑎+16𝑏𝑐−√Δ8𝑏.(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 𝑢^∘2(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 𝑢^∗1(𝜉) (see Figure 3).

(a) for 𝑏=0.66

(b) for 𝑏=0.2

(c) for 𝑏=0.00001

Figure 2 

The limiting precess of 𝑢1(𝜉) when 𝑐<0, (𝑎,𝑏)∈𝐴1, and (𝑎,𝑏) tends to the line 𝑙1, where 𝑎=4 and 𝑐=−1.

(a) for 𝑏=−0.66

(b) for 𝑏=−0.2

(c) for 𝑏=−0.00001

Figure 3 

The limiting precess of 𝑢1(𝜉) when 𝑐<0, (𝑎,𝑏)∈𝐴5, and (𝑎,𝑏) tends to the line 𝑙1, where 𝑎=−4 and 𝑐=−1.

(a) for 𝑏=0.7

(b) for 𝑏=0.7499

(c) for 𝑏=0.7499999

Figure 4 

The limiting precess of 𝑢2(𝜉) when 𝑐<0, (𝑎,𝑏)∈𝐴2, and (𝑎,𝑏) tends to the line 𝑙3, where 𝑎=4 and 𝑐=−1.

(a) for 𝑏=−1

(b) for 𝑏=−0.05

(c) for 𝑏=−0.0001

Figure 5 

The limiting precess of 𝑢2(𝜉) when 𝑐>0, (𝑎,𝑏)∈𝐵5, and (𝑎,𝑏) tends to the line 𝑘1, where 𝑎=−2 and 𝑐=1.

(a) for 𝑏=−1.5875

(b) for 𝑏=−1.6865

(c) for 𝑏=−1.687499

Figure 6 

The limiting precess of 𝑢3(𝜉) when 𝑐<0, (𝑎,𝑏)∈𝐴6, and (𝑎,𝑏) tends to the line 𝑙3, where 𝑎=−9 and 𝑐=−1.

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)=6𝑐𝑢𝑎−√𝑎(𝑎+6𝑏𝑐)cos√−𝑐𝜉₁₂,(𝜉)=6𝑐𝑢𝑎+√𝑎(𝑎+6𝑏𝑐)sin√−𝑐𝜉₁₃,(𝜉)=6𝑐𝑎−√𝑎(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)=−𝛼⁰cos𝑤₀𝜉+𝛽₀−𝑝₀cos𝑤₀𝜉+𝑞₀𝑢₂₂,(𝜉)=𝛼⁰sin𝑤₀𝜉+𝛽₀𝑝₀sin𝑤₀𝜉+𝑞₀𝑢₂₃,(𝜉)=−𝛼⁰sin𝑤₀𝜉+𝛽₀−𝑝₀sin𝑤₀𝜉+𝑞₀.(2.19)
(3) When 𝑐<0 and (𝑎,𝑏)∈𝐴₆, or when 𝑐>0 and (𝑎,𝑏)∈𝐵₁, the functions are 𝑢³¹(𝜉)=−𝛼¹cos𝑤₁𝜉+𝛽₁−𝑝₁cos𝑤₁𝜉+𝑞₁𝑢₃₂,(𝜉)=𝛼¹sin𝑤₁𝜉+𝛽₁𝑝₁sin𝑤₁𝜉+𝑞₁𝑢₃₃,(𝜉)=−𝛼¹sin𝑤₁𝜉+𝛽₁−𝑝₁sin𝑤₁𝜉+𝑞₁.(2.20)

Remark 2.3. In the given parametric regions, the solutions 𝑢_𝑖(𝜉), 𝑢^1𝑖(𝜉), 𝑢^2𝑖(𝜉), 𝑢^3𝑖(𝜉)(𝑖=1,2,3), and 𝑢^∘2(𝜉) are nonsingular. The solutions 𝑢^∘1(𝜉), 𝑢^∗1(𝜉), and 𝑢^∗2(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(𝑎₁≠𝑐²), on 𝑎₂−𝑎₃ plane we define some rays and regions as follows.(1)When 𝑐²<𝑎₁, we define curves ΓΓ₁:𝑎₂>0,𝑎₃=0,₂:𝑎₂>0,𝑎₃=2𝑎²²Γ₃9𝑎₁−𝑐²,:𝑎₂>0,𝑎₃=𝑎²²Γ₄4𝑎₁−𝑐²,Γ:𝑎₂=0,𝑎₃>0,₅:𝑎₂<0,𝑎₃=𝑎²²Γ₆4𝑎₁−𝑐²,:𝑎₂<0,𝑎₃=2𝑎²²Γ₇9𝑎₁−𝑐²,Γ:𝑎₂<0,𝑎₃=0,₈:𝑎₂=0,𝑎₃<0,(3.1) and region 𝑊_𝑖 as the domain surrounded by Γ_𝑖 and Γ_𝑖+1(𝑖=1-7), 𝑊₈ as the domain surrounded by Γ₈ and Γ₁ (see Figure 7(a)).

(a) 𝑐2<𝑎1

(b) 𝑐2>𝑎1

Figure 7 

The locations of the rays Γ𝑖, 𝐿𝑖 and the regions 𝑊𝑖, Ω𝑖(𝑖=1,2,…,8) for given 𝑎1 and 𝑐.

(2)When 𝑐²>𝑎₁, we define curves 𝐿𝐿₁:𝑎₂>0,𝑎₃=0,₂𝐿:𝑎₂=0,𝑎₃>0,₃𝐿:𝑎₂<0,𝑎₃=0,₄:𝑎₂<0,𝑎₃=2𝑎²²𝐿₅9𝑎₁−𝑐²,:𝑎₂<0,𝑎₃=𝑎²¹𝐿₆4𝑎₁−𝑐²,𝐿:𝑎₂=0,𝑎₃<0,₇:𝑎₂>0,𝑎₃=𝑎²²𝐿₈4𝑎₁−𝑐²,:𝑎₂>0,𝑎₃=2𝑎²²9𝑎₁−𝑐²,(3.2) and region Ω_𝑖 as the domain surrounded by 𝐿_𝑖 and 𝐿_𝑖+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 𝑐²≠𝑎₁, let 𝜉=𝑥−𝑐𝑡. 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 𝑣₁(𝜉)=𝑅⁰𝑅₁+𝑅₂cos𝑅₃𝜉,(3.3) where 𝑅𝑅₀₁=2𝑐²−𝑎₁,=2𝑎²𝑅₂3,=13𝑅₃18𝑎₃𝑐²−𝑎₁+4𝑎²²,=𝑎₁−𝑐²𝑐².(3.4) For 𝑎₂≠0, 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 𝑣^∘1(𝜉)=3𝑐²−𝑎₁𝑎₂1+cos√𝑎₁−𝑐²/|𝑐|𝜉.(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 𝑣^∗1(𝜉)=3𝑐²−𝑎₁𝑎₂1−cos√𝑎₁−𝑐²/|𝑐|𝜉.(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(𝑐²−𝑎₁)/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)=𝑆₀−2𝑆¹−𝑆₂+𝑆₃cos𝑆₄𝜉,(3.7) where 𝑆₀=−𝑎²+√𝜔2𝑎₃𝑆₁,=−𝑎²²+4𝑎₃𝑎₁−𝑐²+𝑎₂√𝜔𝑎²³𝑆₂,=23𝑎₃−𝑎²𝑆₃+3√𝜔,=23𝑎₃𝑆₄𝑎₂𝑎²+3√𝜔,=−𝑆¹𝑎₃2𝑐²,(3.8)𝜔=𝑎²²−4𝑎₃𝑎₁−𝑐².(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 𝑣^∘2(𝜉)=2𝑎¹−𝑐²12𝑐²−9𝑐²−2𝑎₁−𝑐²𝜉²𝑎₂9𝑐²+2𝑎₁−𝑐²𝜉².(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 𝑣^∗2(𝜉)=𝑎¹−𝑐²2𝑎₂⎡⎢⎢⎣1+3tan₂⎛⎜⎜⎝𝑐²−𝑎₁4𝑐²𝜉⎞⎟⎟⎠⎤⎥⎥⎦.(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+2𝑇¹−𝑇₂+𝑇₃cos𝑇₄𝜉,(3.12) where 𝑇₀=−𝑎²−√𝜔2𝑎₃𝑇₁,=−𝑎²²+4𝑎₃𝑎₁−𝑐²−𝑎₂√𝜔𝑎²³𝑇₂,=23𝑎₃𝑎²𝑇₃+3√𝜔,=23𝑎₃𝑇₄𝑎₂𝑎²−3√𝜔,=−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 𝑣^∘2(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 𝑣^∗2(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 𝑣¹¹(𝜉)=𝑅⁰𝑣₁₂𝑅₁−𝑅₂cos𝑅₃𝜉,(𝜉)=𝑅⁰𝑣₁₃𝑅₁+𝑅₂sin𝑅₃𝜉,(𝜉)=𝑅⁰𝑅₁−𝑅₂sin𝑅₃𝜉.(3.14)(2) When 𝑐²<𝑎₁ and (𝑎₂,𝑎₃)∈𝑊₅ or when 𝑐²>𝑎₁ and (𝑎₂,𝑎₃)∈Ω₁, the functions are 𝑣²¹(𝜉)=S₀+2𝑆¹𝑣₂₂𝑆₂+𝑆₃cos𝑆₄𝜉,(𝜉)=𝑆₀−2𝑆¹𝑣₂₃−𝑆₂+𝑆₃sin𝑆₄𝜉,(𝜉)=𝑆₀+2𝑆¹𝑆₂+𝑆₃sin𝑆₄𝜉.(3.15)(3) When 𝑐²<𝑎₁ and (𝑎₂,𝑎₃)∈𝑊₂ or when 𝑐²>𝑎₁ and (𝑎₂,𝑎₃)∈Ω₂, the functions are 𝑣³¹(𝜉)=𝑇₀−2𝑇¹𝑣₃₂𝑇₂+𝑇₃cos𝑇₄𝜉,(𝜉)=𝑇₀+2𝑇¹𝑣₃₃−𝑇₂+𝑇₃sin𝑇₄𝜉,(𝜉)=𝑇₀−2𝑇¹𝑇₂+𝑇₃sin𝑇₄𝜉.(3.16)

Remark 3.3. In the given regions, the solutions 𝑣_𝑖(𝜉), 𝑣^1𝑖(𝜉), 𝑣^2𝑖(𝜉), 𝑣^3𝑖(𝜉)(𝑖=1,2,3), and 𝑣^∘2(𝜉) are nonsingular. The solutions 𝑣^∘1(𝜉), 𝑣^∗1(𝜉), and 𝑣^∗2(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=𝑥−𝑐𝑡 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 −𝑐𝜑+𝑎2𝜑²+𝑎𝑏3𝜑³+𝜑^=0.(4.2)

Letting 𝜑^=𝑦, yields the following planar system:𝜑^=𝑦,𝑦^=𝑐𝜑−𝑎2−𝑎3𝑏𝜑³.(4.3)

Obviously, system (4.3) has the first integral 6𝑦²−6𝑐𝜑²+2𝑎𝜑³+𝑎𝑏𝜑⁴=ℎ.(4.4)

Let 𝜑₁=−3𝑎−√Δ𝜑4𝑎𝑏₂,=−3𝑎+√Δ4𝑎𝑏,(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 𝑒_𝑖 and 𝑓_𝑖(𝑖=1,2,3) be, respectively,𝑒₁=−𝑎−√𝑎²+6𝑎𝑏𝑐𝑓𝑎𝑏₁,=−𝑎+√𝑎²+6𝑎𝑏𝑐𝑒𝑎𝑏₂,=14𝑎𝑏−𝑎+√𝑓₂Δ−2𝑎𝑎−√Δ,=14𝑎𝑏−𝑎+√𝑒₃Δ+2𝑎𝑎−√Δ,=−14𝑎𝑏𝑎+√𝑓₃Δ+2𝑎𝑎+√Δ,=−14𝑎𝑏𝑎+√Δ−2𝑎𝑎+√Δ.(4.6)

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

Figure 8 

When 𝑐<0, the bifurcation phase portraits of system (4.3) and the locations of 𝑒𝑖 and 𝑓𝑖(𝑖=1,2,3).

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 (𝑒_𝑖,0) passes (𝑓_𝑖,0)(𝑖=1,2,3). Thus, using the phase portraits in Figures 8 and 9, we derive 𝑢_𝑖(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=±𝑎𝑏6𝜑−𝑒₁𝑓₁+𝑒₁+𝑓₁𝜑−𝜑²,where𝑒₁≤𝜑≤𝑓₁.(4.7) Substituting (4.7) into d𝜑/𝑦=d𝜉, we have d𝜑−𝑒₁𝑓₁+𝑒₁+𝑓₁𝜑−𝜑²=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=±𝑎𝑏6𝜑−𝜑¹−𝑒₂𝑓₂+𝑒₂+𝑓₂𝜑−𝜑²,where𝑒₂≤𝜑≤𝑓₂.(4.9) Substituting (4.9) into d𝜑/𝑦=d𝜉, we get d𝜑𝜑−𝜑₁−𝑒₂𝑓₂+𝑒₂+𝑓₂𝜑−𝜑²=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=±𝑎𝑏6𝜑²−𝜑−𝑒₃𝑓₃+𝑒₃+𝑓₃𝜑−𝜑²,where𝑒₃≤𝜑≤𝑓₃.(4.11) Substituting (4.11) into d𝜑/𝑦=d𝜉, it follows that d𝜑𝜑₂−𝜑−𝑒₃𝑓₃+𝑒₃+𝑓₃𝜑−𝜑²=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=𝑥−𝑐𝑡, where 𝑐 is a constant. Thus, (1.2) becomes 𝑐²𝜓^−𝑐²𝜓^−𝑎₁𝜓+𝑎₂𝜓²+𝑎₃𝜓³^=0.(5.1)

Integrating (5.1) twice and letting integral constant be zero, we get 𝑐²𝜓−𝜓^=𝑎₁𝜓+𝑎₂𝜓²+𝑎₃𝜓³.(5.2)

Letting 𝜓^=𝑦, we have the planar system 𝜓^=𝑦,𝑐²𝑦^=𝑐²−𝑎₁𝜓−𝑎₂𝜓²−𝑎₃𝜓³.(5.3)

It is easy to see that system (5.3) has the first integral 𝑐²𝑦²+𝜓²𝑎2³𝜓²+2𝑎²3𝜓+𝑎¹−𝑐²=ℎ,(5.4) and three singular points (0,0), (𝜓₁,0), and (𝜓₂,0), where 𝜓₁=−𝑎²−√𝜔2𝑎₃𝜓₂,=−𝑎²+√𝜔2𝑎₃(5.5) and 𝜔 is defined in (3.9).

Let 𝑚_𝑖 and 𝑛_𝑖(𝑖=1,2,3) be, respectively,𝑚₁=−2𝑎²−2𝑎²²−9𝑎₁𝑎₃+9𝑎₃𝑐²3𝑎₃𝑛₁,=−2𝑎²+2𝑎²²−9𝑎₁𝑎₃+9𝑎₃𝑐²3𝑎₃𝑚₂,=−𝑎²−3√𝜔−2𝑎₂𝑎²+3√𝜔6𝑎₃𝑛₂,=−𝑎²−3√𝜔+2𝑎₂𝑎²+3√𝜔6𝑎₃𝑚₃,=−𝑎²+3√𝜔−2𝑎₂𝑎²−3√𝜔6𝑎₃𝑛₃,=−𝑎²+3√𝜔+2𝑎₂𝑎²−3√𝜔6𝑎₃.(5.6)

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

Figure 10 

When 𝑐2<𝑎1, the bifurcation phase portraits of system (5.3) and the locations of 𝑚𝑖 and 𝑛𝑖(𝑖=1,2,3).

Figure 11 

When 𝑐2>𝑎1, the bifurcation phase portraits of system (5.3) and the locations of 𝑚𝑖 and 𝑛𝑖(𝑖=1,2,3).

It is easy to test that the closed orbit passing (𝑚_𝑖,0) passes (𝑛_𝑖,0)(𝑖=1,2,3). Thus, using the phase portraits in Figures 10 and 11, we derive 𝑣_𝑖(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=±𝑎₃2𝑐²𝜓−𝑚₁𝑛₁+𝑚₁+𝑛₁𝜓−𝜓²,where𝑚₁≤𝜓≤𝑛₁.(5.7) Substituting (5.7) into d𝜓/𝑦=d𝜉, we have d𝜓𝜓−𝑚₁𝑛₁+𝑚₁+𝑛₁𝜓−𝜓²=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=±𝑎₃2𝑐²𝜓₂−𝜓−𝑚₂𝑛₂+𝑚₂+𝑛₂𝜓−𝜓²,where𝑚₂≤𝜓≤𝑛₂.(5.9) From d𝜓/𝑦=d𝜉 and (5.9), it follows that d𝜓𝜓₂−𝜓−𝑚₂𝑛₂+𝑚₂+𝑛₂𝜓−𝜓²=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=±𝑎₃2𝑐²𝜓−𝜓₁−𝑚₃𝑛₃+𝑚₃+𝑛₃𝜓−𝜓²,where𝑚₃≤𝜓≤𝑛₃.(5.11) Substituting (5.11) into d𝜓/𝑦=d𝜉, we have d𝜓𝜓−𝜓₁−𝑚₃𝑛₃+𝑚₃+𝑛₃𝜓−𝜓²=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=112𝑎𝑏⎛⎜⎜⎝−4𝑎(2+𝑏𝜇)+41+i√3𝑎𝐹₀₂𝐹+2ii+√𝜇₂3𝐹⎞⎟⎟⎠,=112𝑎𝑏⎛⎜⎜⎝−4𝑎(2+𝑏𝜇)+41−i√3𝑎𝐹₀₂𝐹−2ii+√𝜇₃3𝐹⎞⎟⎟⎠,=16𝑎𝑏−2𝑎(2+𝑏𝜇)−4𝑎𝐹𝐹⁰²+2𝐹,(6.1) where 𝐹𝐹₀₁₀₂=8−6𝑏𝜇+15𝑏²𝜇²+10𝑏³𝜇³,𝐹=−9𝑏𝑐+𝑎−2+𝑏𝜇+𝑏²𝜇²₀₃,𝐹==54𝑎𝑎³₂8𝐹³⁰²+𝑎−54𝑏𝑐(−1+𝑏𝜇)+𝑎𝐹₀₁²,𝑏𝑐(−1+𝑏𝜇)−𝑎³𝐹₀₁+𝐹₀₃^1/3.(6.2)

Assume that 𝑐<0, (𝑎,𝑏)∈(𝐴₁), and 𝜑₁<𝜇<𝑒₁. It is easy to check that 𝜇_𝑖(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 𝑙^1𝜇 and 𝑙^2𝜇 (see Figure 12). The closed orbit 𝑙^1𝜇 passes the points (𝜇,0) and (𝜇₁,0). The closed orbit 𝑙^2𝜇 passes the points (𝜇₂,0) and (𝜇₃,0).

Figure 12 

The locations of 𝑙1𝜇 and 𝑙2𝜇 when 𝑐<0 and (𝑎,𝑏)∈𝐴1.

On 𝜑−𝑦 plane, the expression of 𝑙^1𝜇 is 𝑦²=𝑎𝑏6𝜇³−𝜑𝜇₂−𝜑𝜇₁−𝜑(𝜑−𝜇),where𝜇≤𝜑≤𝜇₁.(6.4)

Substituting (6.4) into d𝜑/𝑦=d𝜉 and integrating it along 𝑙^1𝜇, we have 𝑔sn⁻¹(sin𝑧,𝑘)=𝑎𝑏6||𝜉||,(6.5) where 𝑔=2𝑘=⎷𝜇₃−𝜇₁𝜇₂−𝜇,𝜇₃−𝜇₂𝜇₁−𝜇sin𝑧=⎷𝜇₃−𝜇₁𝜇₂−𝜇,𝜇₃−𝜇₁(𝜑−𝜇)𝜇₁−𝜇𝜇₃−𝜑.(6.6)

Solving (6.5) for 𝜑 and noting that 𝑢=𝜑(𝜉), we obtain an elliptic function periodic wave solution 𝑢(𝜉)=𝜇𝜇³−𝜇₁+𝜇₃𝜇₁−𝜇sn²(𝜂𝜉,𝑘)𝜇₃−𝜇₁+𝜇₁−𝜇sn²(𝜂𝜉,𝑘),(6.7) where 𝜂=𝑎𝑏𝜇₃−𝜇₁𝜇₂−𝜇24.(6.8)

Letting 𝜇→𝑒₁−0, it follows that 𝜇₁→𝑓₁, 𝜇₂→0, 𝜇₃→0, 𝑘→0, 𝜂→√(𝑎𝑏𝑒₁𝑓₁)/24 and sn²(𝜂𝜉,𝑘)→sn²(√(𝑎𝑏𝑒₁𝑓₁/24)𝜉,0)=sin²(√(𝑎𝑏𝑒₁𝑓₁/24)𝜉).

Therefore, in (6.7) letting 𝜇→𝑒₁−0, we obtain the trigonometric function periodic wave solution 𝑢(𝜉)=𝑒¹𝑓₁𝑓₁+𝑒₁−𝑓₁sin²𝑎𝑏𝑒₁=−6𝑐𝑓₁/24𝜉=−𝑎+√𝑎(𝑎+66𝑐𝑏𝑐)−2√𝑎(𝑎+6𝑏𝑐)sin²√|𝑐|/2𝜉𝑎−√𝑎(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)cos√−c(x−ct)(6.10) Simplify [D[u,t]+a(1+bu)D[u,x]u+D[u,{x,3}]]. (2) The orders for testing 𝑣₁(𝜉)RR₀₁=2−a₁+c²,=2a²R₂3,=2a₃−a₁+c²+4a²²R₃9,=a₁−c²c²v=,R₀vttvxtxt==DDR[₁[vv+t,t,{R{t,x₂2,c}2o]}s],R₃(x−ct),(6.11) Simplify [vtt−vttxx−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.