MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation81021710.1155/2011/810217810217Research ArticleTrigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like EquationsZhengrongLiuTianpeiJiangPengQinQinfengXuLiMingDepartment of MathematicsSouth China University of TechnologyGuangzhou 510640Chinascut.edu.cn20111782011201115022011260520112011Copyright © 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.

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 vtt - vttxx - (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, and the PC-like equation vtt-vttxx-(a1v+a2v2+a3v3)xx=0. 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  got an exact kink soliton for (1.1) by using homogeneous balance method. Grimshaw et al.  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.  investigated solitary waves and their bifurcations for (1.1) by employing bifurcation method of dynamical systems. Peng  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.  described the interaction between a soliton and a breather for (1.1) by using the Hirota bilinear method. Kaya and Inan  studied solitary wave solutions for (1.1) by using Adomian decomposition method. Yomba  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  gave some explicit solitary wave solutions composed of hyperbolic functions by using solving algebraic equations for (1.2). Li and Zhang  used bifurcation method of dynamical system to study the bifurcation of traveling wave solutions and construct solitary wave solutions for (1.2). Kaya  discussed the exact and numerical solitary wave solutions by using a decomposition method for (1.2). Rafei et al.  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 .

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 (<xref ref-type="disp-formula" rid="EEq1.1">1.1</xref>)

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 a-b plane we define some lines and regions as follows.

When c<0, we define lines l1:  b=0,l2:  b=-a6c,l3:  b=-3a16c,l4:  a=0, and regions Ai  (i=18), as Figure 1(a).

The locations of the lines li, ki  (i=1,2,3,4) and the regions Aj, Bj  (j=1,2,,8) for given constant c0.

c<0

c>0

When c>0, we define lines k1:  b=0,k2:  a=0,k3:  b=-3a16c,k4:  b=-a6c, and regions Bi  (i=18), 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=x-ct. Then, (1.1) has the following periodic wave solutions.

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), which has the following limit forms.

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)=6ca[1+cos(-cξ)] (see Figure 2).

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)=6ca[1-cos(-cξ)] (see Figure 3).

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.

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, where Δ=3a(3a+16bc),

α0=-(3a+Δ)a(a-Δ)4a2b2,β0=-3a+24bc+Δ2ab2,p0=a(a-Δ)ab,q0=-1ab(a+Δ),w0=3a+16bc+Δ8b. The solution u2(ξ) has the following limit forms.

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)=4c(3+2cξ2)a(-9+2cξ2) (see Figure 4).

When c<0, (a,b)A2 and (a,b) tends to l2, u2(ξ) tends to the trivial solution u(ξ)=0.

When c>0, (a,b)B5 and (a,b) tends to k1, the u2(ξ) tends to the periodic blow-up solution u2*(ξ)=c(2sin2(cξ/2)-3)asin2(cξ/2) (see Figure 5).

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, where α1=(-3a+Δ)a(a+Δ)4a2b2,β1=3a+24bc-Δ2ab2,p1=a(a+Δ)ab,q1=a-Δab,w1=3a+16bc-Δ8b. The solution u3(ξ) has the following limit forms.

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).

When c<0, (a,b)A6 and (a,b) tends to l2, u3(ξ) tends to the trivial solution u(ξ)=0.

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).

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.

for b=0.66

for b=0.2

for b=0.00001

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.

for b=-0.66

for b=-0.2

for b=-0.00001

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.

for b=0.7

for b=0.7499

for b=0.7499999

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.

for b=-1

for b=-0.05

for b=-0.0001

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.

for b=-1.5875

for b=-1.6865

for b=-1.687499

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)=6ca-a(a+6bc)cos(-cξ),u21(ξ)=6ca+a(a+6bc)sin(-cξ),u31(ξ)=6ca-a(a+6bc)sin(-cξ).

(2) When c<0 and (a,b)A2 or when c>0 and (a,b)B5, the functions are u12(ξ)=-α0cos(w0ξ)+β0-p0cos(w0ξ)+q0,u22(ξ)=α0sin(w0ξ)+β0p0sin(w0ξ)+q0,u32(ξ)=-α0sin(w0ξ)+β0-p0sin(w0ξ)+q0.

(3) When c<0 and (a,b)A6, or when c>0 and (a,b)B1, the functions are u13(ξ)=-α1cos(w1ξ)+β1-p1cos(w1ξ)+q1,u23(ξ)=α1sin(w1ξ)+β1p1sin(w1ξ)+q1,u33(ξ)=-α1sin(w1ξ)+β1-p1sin(w1ξ)+q1.

Remark 2.3.

In the given parametric regions, the solutions ui(ξ), ui1(ξ), ui2(ξ), ui3(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 (<xref ref-type="disp-formula" rid="EEq1.2">1.2</xref>)

In this section, we state our main results for (1.2). For given a1 and c  (a1c2), on a2-a3 plane we define some rays and regions as follows.

When c2<a1, we define curves Γ1:  a2>0,a3=0,Γ2:  a2>0,a3=2a229(a1-c2),Γ3:  a2>0,a3=a224(a1-c2),Γ4:  a2=0,a3>0,Γ5:  a2<0,a3=a224(a1-c2),Γ6:  a2<0,a3=2a229(a1-c2),Γ7:  a2<0,a3=0,Γ8:  a2=0,a3<0, and region Wi as the domain surrounded by Γi and Γi+1  (i=17), W8 as the domain surrounded by Γ8 and Γ1 (see Figure 7(a)).

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

c2<a1

c2>a1

When c  2>a1, we define curves L1:  a2>0,a3=0,L2:  a2=0,a3>0,L3:  a2<0,a3=0,L4:  a2<0,a3=2a229(a1-c2),L5:  a2<0,a3=a124(a1-c2),L6:  a2=0,a3<0,L7:  a2>0,a3=a224(a1-c2),L8:  a2>0,a3=2a229(a1-c2), and region Ωi as the domain surrounded by Li and Li+1  (i=17), 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=x-ct. Then, (1.2) has the following periodic wave solutions.

When c  2<a1 and (a2,a3)W1 or W6, the expression of the periodic wave solution is v1(ξ)=R0R1+R2cos(R3ξ), where R0=2(c2-a1),R1=2a23,R2=1318a3(c2-a1)+4a22,R3=a1-c2c2. For a20, the periodic wave solution v1(ξ) has the following limit forms.

When c2<a1, (a2,a3)W1 and (a2,a3) tends to the ray Γ1, v1(ξ) tends to the periodic blow-up solution v1(ξ)=3(c2-a1)a2(1+cos((a1-c2/|c|)ξ)). The limiting process is similar to that in Figure 2.

When c2<a1, (a2,a3)W6 and (a2,a3) tends to the ray Γ7, v1(ξ) tends to the periodic blow-up solution v1*(ξ)=3(c2-a1)a2(1-cos((a1-c2/|c|)ξ)). The limiting process is similar to that in Figure 3.

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  .

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), where S0=-a2+ω2a3,S1=-a22+4a3(a1-c2)+a2ωa32,S2=23a3(-a2+3ω),S3=23a3a2(a2+3ω),S4=-S1a32c2,

ω=a22-4a3(a1-c2). The periodic wave solution v2(ξ) has the following limit forms.

When c2<a1, (a2,a3)W5, and (a2,a3) tends to the curve Γ6, v2(ξ) tends to the trivial solution v(ξ)=0.

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 v2(ξ)=2(a1-c2)[12c2-9c2-2(a1-c2)ξ2]a2[9c2+2(a1-c2)ξ2]. The limiting process is similar to that in Figure 6.

When c2>a1, (a2,a3)Ω1, and (a2,a3) tends to the ray L1, v2(ξ) tends to the periodic blow-up wave solution v2*(ξ)=a1-c22a2[1+3tan2(c2-a14c2ξ)]. The limiting process is similar to that in Figure 2.

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), where T0=-a2-ω2a3,T1=-a22+4a3(a1-c2)-a2ωa32,T2=23a3(a2+3ω),T3=23a3a2(a2-3ω),T4=-T1a32c2. The periodic wave solution v3(ξ) has the following limit forms:

When c2<a1, (a2,a3)W2, and (a2,a3) tends to the curve Γ2, v3(ξ) tends to the trivial solution v(ξ)=0.

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.

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).

When c2<a1 and (a2,a3)W1 or W6, the functions are

v11(ξ)=R0R1-R2cos(R3ξ),v21(ξ)=R0R1+R2sin(R3ξ),v31(ξ)=R0R1-R2sin(R3ξ).

When c2<a1 and (a2,a3)W5 or when c2>a1 and (a2,a3)Ω1, the functions are

v12(ξ)=S0+2S1S2+S3cos(S4ξ),v22(ξ)=S0-2S1-S2+S3sin(S4ξ),v32(ξ)=S0+2S1S2+S3sin(S4ξ).

When c2<a1 and (a2,a3)W2 or when c2>a1 and (a2,a3)Ω2, the functions are v13(ξ)=T0-2T1T2+T3cos(T4ξ),v23(ξ)=T0+2T1-T2+T3sin(T4ξ),v33(ξ)=T0-2T1T2+T3sin(T4ξ).

Remark 3.3.

In the given regions, the solutions vi(ξ), vi1(ξ), vi2(ξ), vi3(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 <xref ref-type="statement" rid="prop2.1">2.1</xref>

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.

Integrating (4.1) once and letting the integral constant be zero, it follows that -cφ+a2φ2+ab3φ3+φ′′=0.

Letting φ=y, yields the following planar system:φ=y,y=cφ-a2-ab3φ3.

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

Let φ1=-3a-Δ4ab,φ2=-3a+Δ4ab, 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=-a-a2+6abcab,f1=-a+a2+6abcab,e2=14ab(-a+Δ-2a(a-Δ)  ),f2=14ab(-a+Δ+2a(a-Δ)  ),e3=-14ab(a+Δ+2a(a+Δ  )  ),f3=-14ab(a+Δ-2a(a+Δ  )  ).

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.

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

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 ui(ξ)  (i=1,2,3) as follows.

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,  where  e1φf1. Substituting (4.7) into dφ/y=dξ, we have dφ-e1f1+(e1+f1)φ-φ2=ab6dξ. Integrating (4.8) along the closed orbit and noting that u=φ(ξ), we obtain the solution u1(ξ) as (2.4).

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,where  e2φf2. Substituting (4.9) into dφ/y=dξ, we get dφ(φ-φ1)-e2f2+(e2+f2)φ-φ2=ab6dξ. Along the closed orbit integrating (4.10) and noting that u=φ(ξ), we get the solution u2(ξ) as (2.7).

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,where  e3φf3. Substituting (4.11) into dφ/y=dξ, it follows that dφ(φ2-φ)-e3f3+(e3+f3)φ-φ2=ab6dξ. 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 <xref ref-type="statement" rid="prop3.1">3.1</xref>

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.

Integrating (5.1) twice and letting integral constant be zero, we get c2(ψ-ψ′′)=a1ψ+a2ψ2+a3ψ3.

Letting ψ=y, we have the planar system ψ=y,c2y=(c2-a1)ψ-a2ψ2-a3ψ3.

It is easy to see that system (5.3) has the first integral c2y2+ψ2(a32ψ2+2a23ψ+a1-c2)=h, and three singular points (0,0), (ψ1,0), and (ψ2,0), where ψ1=-a2-ω2a3,ψ2=-a2+ω2a3 and ω is defined in (3.9).

Let mi and ni  (i=1,2,3) be, respectively,m1=-2a2-2(a22-9a1a3+9a3c2)3a3,n1=-2a2+2(a22-9a1a3+9a3c2)3a3,m2=-a2-3ω-2a2(a2+3ω)6a3,n2=-a2-3ω+2a2(a2+3ω)6a3,m3=-a2+3ω-2a2(a2-3ω)6a3,n3=-a2+3ω+2a2(a2-3ω)6a3.

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.

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

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 (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.

When c  2<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,where  m1ψn1. Substituting (5.7) into dψ/y=dξ, we have dψψ-m1n1+(m1+n1)ψ-ψ2=a32c2dξ. Integrating (5.8) along the closed orbit and noting that v=ψ(ξ), we get the solution v1(ξ) as (3.3).

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,where  m2ψn2. From dψ/y=dξ and (5.9), it follows that dψ(ψ2-ψ)-m2n2+(m2+n2)ψ-ψ2=a32c2dξ. Integrating (5.10) along the closed orbit, we get v2(ξ) as (3.7).

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,where  m3ψn3. Substituting (5.11) into dψ/y=dξ, we have dψ(ψ-ψ1)-m3n3+(m3+n3)ψ-ψ2=a32c2dξ. 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+i3)aF02F+2i(i+3)F  ),μ2=112ab(-4a(2+bμ)+4(1-i3)aF02F-2i(i+3)F  ),μ3=16ab(-2a(2+bμ)-4aF02F+2F  ), where F01=(8-6bμ+15b2μ2+10b3μ3),F02=(-9bc+a(-2+bμ+b2μ2)),F03=a3(8F023+a(-54bc(-1+bμ)+aF01)2),F=(54a2bc(-1+bμ)-a3F01+F03)1/3.

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.

There are two closed orbits lμ1 and lμ2 (see Figure 12). The closed orbit lμ1 passes the points (μ,0) and (μ1,0). The closed orbit lμ2 passes the points (μ2,0) and (μ3,0).

The locations of lμ1 and lμ2 when c<0 and (a,b)A1.

On φ-y plane, the expression of lμ1 is y2=ab6(μ3-φ)(μ2-φ)(μ1-φ)(φ-μ),where  μφμ1.

Substituting (6.4) into dφ/y=dξ and integrating it along lμ1, we have gsn-1(sinz,k)=ab6|ξ|, where g=2(μ3-μ1)(μ2-μ),k=(μ3-μ2)(μ1-μ)(μ3-μ1)(μ2-μ),sinz=(μ3-μ1)(φ-μ)(μ1-μ)(μ3-φ).

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), where η=ab(μ3-μ1)(μ2-μ)24.

Letting μe1-0, 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 μe1-0, we obtain the trigonometric function periodic wave solution u(ξ)=e1f1f1+(e1-f1)sin2((abe1f1/24)ξ)=-6c-a+a(a+6bc)-2a(a+6bc)sin2((|c|/2)ξ)=6ca-a(a+6bc)cos(|c|ξ)=u11(ξ).

Via Remark 2.2 and u11(ξ), further we get u21(ξ), u31(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.

The orders for testing u1(ξ)  u=6ca+a(a+6bc)cos[-c(x-ct)] Simplify [D[u,t]+a(1+bu)D[u,x]u+D[u,{x,3}]].

The orders for testing v1(ξ)R0=2(-a1+c2),R1=2a23,R2=2a3(-a1+c2)+4a229,R3=a1-c2c2,v=R0R1+R2cos[R3(x-ct)],vtt=D[v,{t,2}],vttxx=D[vtt,{x,2}] 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.

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