# The Study of the Solution to a Generalized KdV-mKdV Equation

Authors: Xiumei Lv; Tengwei Shao; Jiacheng Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is employed to investigate a generalized KdV-mKdV equation which possesses high-order nonlinear terms. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions, and the triangle function solutions to the equation are obtained.


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## Body

## 1. Introduction

Many powerful techniques have been established during the past decades for the study of the nonlinear dispersive partial differential equations [1–9]. The inverse scattering method, the Backlund transformation, the Darboux transformation, the Painleve’ analysis, the pseudospectral method, the finite differences method, and the sine-cosine ansatz are used to acquire solitary wave solutions and compactons solutions for some nonlinear equations. Wadati [1] employed the potential function to handle the KdV equation, and the resulting equation was solved iteratively by making use of a formal series for the potential function. Wadati [2] developed the trace method to handle the KP equation. The tanh method, developed by Malfliet and Hereman [3], is heavily used in the literature to deal with nonlinear partial differential equations. Fan and Zhang [4] introduced a useful extended tanh method that combined the standard tanh method with the Riccati equation. The extensive method was effectively used by many researchers to investigate exact solutions for a lot of nonlinear models (see [10, 11]).With the development of the symbolic computation system, the direct methods for constructing travelling wave solutions to differential equations become feasible. With the help of Mathematica, Sirendaoreji and Jiong [12, 13] used the auxiliary equation method to investigate KdV and mKdV equations, Boussinesq equations, sine-Gordon equations, and the nonlinear Klein-Gordon equations, respectively. The Jacobi elliptic function expansion method is confirmed as a powerful technique to solve some nonlinear differential equations (see [14]).Zhang et al. [15] used a sub-ODE technique to investigate the exact solution of the following nonlinear dispersive KdV-mKdV equation:
(1)ut+(α+βup+λu2p)ux+uxxx=0,
where α,β,λ, and p are constants. The bell type solitary wave solution, the kink type solitary wave solution, the algebraic solitary wave solution, and the sinusoidal travelling wave solution of (1) with exponent p>0 were expressed explicitly in [15]. In fact, (1) is turned into a KdV equation if p=1,β≠0,λ=0, and an mKdV equation if p=1,β=0,λ≠0.In this paper, we further develop the work in [15] for the study of (1). By using a mathematical technique different from those in previous works [1–10], we obtain the exact travelling wave solutions including Jacobi elliptic function solutions, degenerated soliton solutions, and triangle function solutions for (1) with exponent p. Many of the solutions obtained are different from those presented in Zhang et al.’s work [15].


## 2. Brief Description of the Approach

To illustrate the basic concept of the auxiliary differential method, we consider that the nonlinear partial differential equation has the form(2)P(u,ut,ux,uxx,uxt,utt,…)=0.
Using the transformation ξ=μ(x-ct)(μ≠0 and c≠0), (2) turns into the following nonlinear ordinary different equation:
(3)Q(u,uξ,uξξ,uξξξ,…)=0.We seek for the solutions of (3) in the form
(4)u(ξ)=∑i=0N‍gizi(ξ),
where gi(i=0,1,2,…,N) are constants which will be determined later. The parameter N is a positive integer and can be determined by balancing the highest-order derivative terms and the highest power nonlinear terms in (3). The highest degree of ∂pu/∂ξp can be calculated by
(5)O[∂pu∂ξp]=N+p,p=0,1,2,…,O[uq∂pu∂ξp]=qN+p,q,p=0,1,2,….We assume thatz(ξ) represents the solutions of the following auxiliary differential equation
(6)(dzdξ)2=c1+c2z2+c32z4,
where ci(i=1,2,3) are real constants.Substituting (4) and (6) into (3) and equating the coefficients of all powers of z(ξ) and zj(ξ)c1+c2z2+(c3/2)z4(j=0,1,2,…) to be zero in the resulting equation, several algebraic equations will be obtained. Then, solving these algebraic equations by the symbolic computation system Matlab and combining (4) and the solutions of the auxiliary equation (6), we can get the exact solutions for (2).


## 3. Exact Travelling Wave Solutions to (1)

Firstly, settingup=w yields
(7)ux=1pw1/p-1wx,ut=1pw1/p-1wt,uxx=1p(1p-1)w1/p-2(wx)2+1pw1/p-1wxx,uxxx=1p(1p-1)(1p-2)w1/p-3wx3+3p(1p-1)w1/p-2wxwxx+1pw1/p-1wxxx,
which turns (1) into
(8)w2wt+(α+βw+λw2)w2wx+(1p-1)(1p-2)wx3+3(1p-1)wwxwxx+w2wxxx=0.To find the traveling solutions for (1), we use the wave variable ξ=μ(x-ct), where c≠0 and μ≠0. The wave variable ξ transforms (8) into the following ordinary differential equation:
(9)(α+βw+λw2)w2w′+(1p-1)(1p-2)μ2w′3+3μ2(1p-1)ww′w′′-cw2w′+μ2w2w′′′=0.
From (9), we have N=1. Therefore, we choose the ansatz
(10)w(ξ)=g0+g1z,
where z(ξ) may be determined by
(11)(dzdξ)2=c1+c2z2+c32z4,
which possesses several types of solutions listed in Table 1 (see [16]).Table 1
No.
z
(
ξ
)
c
3
c
2
c
1
1
sn
(
ξ
)
,
cd
(
ξ
)
=
cn
(
ξ
)
dn
(
ξ
)
2
r
2
-
(
r
2
+
1
)
1
2
cn
(
ξ
)
-
2
r
2
2
r
2
-
1
1
-
r
2
3
dn
(
ξ
)
-
2
2
-
r
2
r
2
-
1
4
nc
(
ξ
)
=
1
cn
(
ξ
)
2
(
1
-
r
2
)
2
r
2
-
1
-
r
2
5
ns
(
ξ
)
=
1
sn
(
ξ
)
,
dc
(
ξ
)
=
dn
(
ξ
)
cn
(
ξ
)
2
-
(
r
2
+
1
)
r
2
6
nd
(
ξ
)
=
1
dn
(
ξ
)
2
(
r
2
-
1
)
2
-
r
2
-
1
7
cs
(
ξ
)
=
cn
(
ξ
)
sn
(
ξ
)
2
2
-
r
2
1
-
r
2
8
sc
(
ξ
)
=
sn
(
ξ
)
cn
(
ξ
)
2
(
1
-
r
2
)
2
-
r
2
1
9
sd
(
ξ
)
=
sn
(
ξ
)
dn
(
ξ
)
2
r
2
(
r
2
-
1
)
2
r
2
-
1
1
10
ds
(
ξ
)
=
dn
(
ξ
)
sn
(
ξ
)
2
2
r
2
-
1
r
4
-
r
2
11
r
cn
(
ξ
)
±
dn
(
ξ
)
-
1
2
r
2
+
1
2
-
(
1
-
r
2
)
2
4
12
1
sn
(
ξ
)
±
cn
(
ξ
)
sn
(
ξ
)
1
2
-
2
r
2
+
1
2
1
4
13
1
cn
(
ξ
)
±
sn
(
ξ
)
cn
(
ξ
)
1
-
r
2
2
r
2
+
1
2
1
-
r
2
4
14
1
sn
(
ξ
)
±
dn
(
ξ
)
sn
(
ξ
)
1
2
r
2
-
2
2
r
4
4
15
sn
(
ξ
)
±
i
cn
(
ξ
)
,
dn
(
ξ
)
1
-
r
2
sn
(
ξ
)
±
cn
(
ξ
)
r
2
2
r
2
-
2
2
r
2
4
16
r
sn
(
ξ
)
±
i
dn
(
ξ
)
,
sn
(
ξ
)
1
±
cn
(
ξ
)
1
2
1
-
2
r
2
2
1
4
17
sn
(
ξ
)
1
±
dn
(
ξ
)
r
2
2
r
2
-
2
2
1
4
18
dn
(
ξ
)
1
±
rsn
(
ξ
)
r
2
-
1
2
r
2
+
1
2
r
2
-
1
4
19
cn
(
ξ
)
1
±
sn
(
ξ
)
1
-
r
2
2
r
2
+
1
2
-
r
2
+
1
4
20
sn
(
ξ
)
dn
(
ξ
)
±
cn
(
ξ
)
(
1
-
r
2
)
2
2
r
2
+
1
2
1
4
21
cn
(
ξ
)
1
-
r
2
±
dn
(
ξ
)
r
4
2
r
2
-
2
2
1
4In Table1, functions sn(ξ)=sn(ξ,r), cn(ξ)=cn(ξ,r), and dn(ξ)=dn(ξ,r), are Jacobian elliptic functions with modulus r(0<r<1), which have the properties sn(-ξ)=-sn(ξ),cn(-ξ)=cn(ξ),dn(-ξ)=dn(ξ), sn2(ξ)+cn2(ξ)=1,  dn2(ξ)+r2sn2(ξ)=1, (sn(ξ))′=cn(ξ)dn(ξ), (cn(ξ))′=-sn(ξ)dn(ξ), and (dn(ξ))′=-r2sn(ξ)cn(ξ). Setting r→0 yields sn(ξ)→sin(ξ),cn(ξ)→cos(ξ), and dn(ξ)→1. When r→1, it derives that sn(ξ)→tanh(ξ), cn(ξ)→sech(ξ), and dn(ξ)→sech(ξ).Substituting (10) and (11) into (9) and letting each coefficient of zjc1+c2z2+(c3/2)z4(0≤j≤4) be zero, we obtain
(12)λg15+μ22(1p-1)(1p-2)g13c3+3(1p-1)μ2g13c3+3μ2g13c3=0,(βg1+2λg0g1)g13+2λg0g14+3(1p-1)μ2g0g12c3+6μ2g0g12c3=0,-cg13+(α+βg0+λg02)g13+2(βg1+2λg0g1)g0g12+λg02g13+μ2(1p-1)(1p-2)g13c2+3(1p-1)μ2g13c2+3c3μ2g02+μ2c2g13=0,-2cg0g12+2(α+βg0+λg02)g12g0+(βg1+2λg0g1)g02g1+3(1p-1)μ2g12g0c2+2μ2g12g0c2=0,-cg02g1+(α+βg0+λg02)g02g1+(1p-1)(1p-2)μ2g13c1+μ2g02g1c2=0.Solving (12) with the help of Matlab, we acquire the following solutions:
(13)p=1,g0=-β2λ,g1=±β2λ-c3c2,μ=±β213λc2,c=α-β26λ,(14)p=12,g0=-2β5λ,g1=±2β5λ-c3c2,μ=±β523λc2,c=α-16β275λ,(15)p=2,μ=±(5c2β2+5ϵβ2c22-2c1c348c1c3λ)1/2,g0=-5β8λ,g1=±μ4-30c3λ,c=α-5β232λ+14μ2c2,(16)g0=-β(2p+1)2λ(p+2),g1=±β(2p+1)2λ(p+2)-c3c2,2c1c3=c22,μ=±βpp+22p+12λ(p+1)c2,c=α-β2(2p+1)λ(p+2)2(p+1),
where the exponent p≠-2,-1, and ϵ=±1.


### 3.1. The Jacobi Elliptic Function Solutions to (1) in the Case of p=1

In order to make that wave numberμ a real-valued number, we have to choose constants λ and modulus r (c2 depends on r) to satisfy some restrictions. However, in this section, we allow μ to take values in complex number domain. From the expression of g1,g2,μ, and c in (13) and the solutions listed in Table 1, we derive the following Jacobi elliptic function solutions for (1) in the case p=1:
(17)u1.1(x,t)=-β2λ+ϵβr2λ2r2+1×sn[β2-13λ(r2+1)(x-αt+β26λt)],u1.2(x,t)=-β2λ+ϵβr2λ2r2+1×cd[β2-13λ(r2+1)(x-αt+β26λt)],u1.3(x,t)=-β2λ+ϵβr2λ22r2-1×cn[β213λ(2r2-1)(x-αt+β26λt)],u1.4(x,t)=-β2λ+ϵβ2λ22-r2×dn[β213λ(2-r2)(x-αt+β26λt)],u1.5(x,t)=-β2λ+ϵβ2λ2(r2-1)2r2-1×nc[β213λ(2r2-1)(x-αt+β26λt)],u1.6(x,t)=-β2λ+ϵβ2λ2r2+1×ns[β2-13λ(r2+1)(x-αt+β26λt)],u1.7(x,t)=-β2λ+ϵβ2λ2r2+1×dc[β2-13λ(r2+1)(x-αt+β26λt)],u1.8(x,t)=-β2λ+ϵβ2λ2(r2-1)r2-2×nd[β213λ(2-r2)(x-αt+β26λt)],u1.9(x,t)=-β2λ+ϵβ2λ2r2-2×cs[β213λ(2-r2)(x-αt+β26λt)],u1.10(x,t)=-β2λ+ϵβ2λ2(1-r2)r2-2×sc[β213λ(2-r2)(x-αt+β26λt)],u1.11(x,t)=-β2λ+ϵβr2λ2(r2-1)1-2r2×sd[β2-13λ(1-2r2)(x-αt+β26λt)],u1.12(x,t)=-β2λ+ϵβ2λ21-2r2×ds[β2-13λ(1-2r2)(x-αt+β26λt)],u1.13(x,t)=-β2λ+ϵβ2λ11+r2×{rcn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(r2+1)]±nd[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(r2+1)]},u1.14(x,t)=-β2λ+ϵβ2λ12r2-1×{×(x-αt+β2t6λ)[β223λ(1-2r2)])-1)1×(sn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1±(cn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])×(sn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1},u1.15(x,t)=-β2λ+ϵβ2λr2-11+r2×{×(x-αt+β2t6λ)[β223λ(1-2r2)])-11×(cn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1±(sn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])×(cn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1},u1.16(x,t)=-β2λ+ϵβ2λ12-r2×{×(x-αt+β2t6λ)[β223λ(1-2r2)])-11×(sn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1±(dn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])×(sn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1},u1.17(x,t)=-β2λ+ϵβr2λ12-r2×{sn[β223λ(r2-2)(x-αt+β2t6λ)]±icn[β223λ(r2-2)(x-αt+β2t6λ)]},u1.18(x,t)=-β2λ+ϵβr2λ12-r2×((x-αt+β2t6λ)[β223λ(1-2r2)])-1(dn[β223λ(r2-2)(x-αt+β2t6λ)])×(1-r2sn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)]±cn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.19(x,t)=-β2λ+ϵβ2λ12r2-1×{rsn[β223λ(1-2r2)(x-αt+β2t6λ)]±idn[β223λ(1-2r2)(x-αt+β2t6λ)]},u1.20(x,t)=-β2λ+ϵβ2λ12r2-1×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(sn[β213λ(1-2r2)(x-αt+β2t6λ)])×(1±cn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.21(x,t)=-β2λ+ϵβ2λr22-r2×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(sn[β213λ(r2-2)(x-αt+β2t6λ)])×(1±dn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.22(x,t)=-β2λ+ϵβ2λ1-r2r2+1×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(dn[β223λ(r2+1)(x-αt+β2t6λ)])×(1±rsn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.23(x,t)=-β2λ+ϵβ2λr2-11+r2×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(cn[β223λ(r2+1)(x-αt+β2t6λ)])×(1±sn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.24(x,t)=-β2λ+ϵβ(1-r2)2λ-1r2+1×((x-αt+β2t6λ)[β223λ(1-2r2)])-1(sn[β223λ(r2+1)(x-αt+β2t6λ)])×(dn[β223λ(r2+1)×(x-αt+β2t6λ)]±cn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.25(x,t)=-β2λ+ϵβr22λ2-r2×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(cn[β223λ(r2-2)(x-αt+β2t6λ)])×(1-r2±dn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1).Remark 1.
From the properties of Jacobi elliptic functions, we know that when the moduler→0 or r→1, sn, cn, and dn degenerate into triangular or hyperbolic functions, from which we can obtain soliton and triangular function solutions of (1) in the case p=1. In other word, when p=1, we can deduce Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).Remark 2.
Whenα=0, k=(β/2)2/3λ(1-2r2), a0=-β/2λ, and a1=±k-3/2λ, u1.14(x,t) turns into
(18)u1.14(x,t)=a0+a1{nsk[x-(1-2r22k2+λa02+βa0)t]+csk[x-(1-2r22k2+λa02+βa0)t]},
which was obtained by Yomba [17].Remark 3.
Supposingk=(β/6)-1/qνλ, p=(2+3/3)-6qνλ (q and ν are arbitrary constants). When r→1 in u1.17(x,t), we obtain
(19)u1.17.1(x,t)=a0-ϵβ12λq-6qνλ×{±-2qνλ+-6qνλ(tanh-6qνλξ)±isech-6qνλξ},
where a0=-(βp/2λ)-6qνλ+β/2λ,ξ=k(x-(λa02+βa0)t). Solution (19) is in full agreement with that presented in [17] by Yomba.


### 3.2. The Jacobi Elliptic Function Solutions to (1) in the Case of p=1/2

Similarly as in Section3.1, we allow μ to take values in complex number field. Making use of (14) and the solutions listed in Table 1, we obtain the following Jacobi elliptic function solutions for (1) in the case p=1/2:
(20)u2.1(x,t)={-2β5λ+2ϵβr5λ2r2+1×sn[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.2(x,t)={-2β5λ+2ϵβr5λ2r2+1×cd[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.3(x,t)={[2β275λ(2r2-1)(x-αt+16β275λt)]-2β5λ+2ϵβr5λ22r2-1×cn[2β275λ(2r2-1)(x-αt+16β275λt)]}2,u2.4(x,t)={-2β5λ+2ϵβ5λ22-r2×dn[β523λ(2-r2)(x-αt+16β275λt)]}2,u2.5(x,t)={-2β5λ+2ϵβ5λ2r2-22r2-1×nc[2β275λ(2r2-1)(x-αt+16β275λt)]}2,u2.6(x,t)={-2β5λ+2ϵβ5λ2r2+1×ns[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.7(x,t)={-2β5λ+2ϵβ5λ2r2+1×dc[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.8(x,t)={[β523λ(2-r2)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ2-2r22-r2×nd[β523λ(2-r2)(x-αt+16β275λt)]2-2r22-r2}2,u2.9(x,t)={-2β5λ+2ϵβ5λ-22-r2×cs[β523λ(2-r2)(x-αt+16β275λt)]}2,u2.10(x,t)={[β523λ(2-r2)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ2r2-22-r2×sc[β523λ(2-r2)(x-αt+16β275λt)]2r2-22-r2}2,u2.11(x,t)={[β523λ(2r2-1)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ2r2(1-r2)2r2-1×sd[β523λ(2r2-1)(x-αt+16β275λt)]2r2(1-r2)2r2-1}2,u2.12(x,t)={[2β275λ(2r2-1)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ21-2r2×ds[2β275λ(2r2-1)(x-αt+16β275λt)]}2,u2.13(x,t)={-2β5λ+2ϵβ5λ1r2+1×(rcn[β543λ(r2+1)(x-αt+16β275λt)]±dn[β543λ(r2+1)(x-αt+16β275λt)])}2,u2.14(x,t)={×(x-αt+16β275λt)43λ(-2r2+1)])-1)×(x-αt+16β275λt)])-1)-2β5λ+2ϵβ5λ-11-2r2×(×(x-αt+16β275λt)43λ(-2r2+1)])-11×(sn[β543λ(-2r2+1)×(x-αt+16β275λt)43λ(-2r2+1)])-1±(cn[β543λ(-2r2+1)×(x-αt+16β275λt)43λ(-2r2+1)])×(sn[β543λ(-2r2+1)×(x-αt+16β275λt)43λ(-2r2+1)])-1)}2,u2.15(x,t)={×(x-αt+16β275λt)])-1)-2β5λ+2ϵβ5λ-1-r2r2+1×(×(x-αt+16β275λt)43λ(r2+1)])-11×(cn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1±(sn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])×(cn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)}2,u2.16(x,t)={×(x-αt+16β275λt)43λ(r2-2)])-1)-2β5λ+2ϵβ5λ-1r2-2×(×(x-αt+16β275λt)])-11×(sn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])-1±(dn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])×(sn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])-1)}2,u2.17(x,t)={β543λ(r2-2)-2β5λ+2ϵβr5λ-1r2-2×(sn[β543λ(r2-2)×(x-αt+16β275λt)[β223λ(1-2r2)]±icn[β543λ(r2-2)×(x-αt+16β275λt)β543λ(r2-2)])}2,u2.18(x,t)={4β275λ(r2-2)×(x-αt+16β275λt)4β275λ(r2-2)])-1-2β5λ+2ϵβr5λ-1r2-2×(×(x-αt+16β275λt)4β275λ(r2-2)])-1(dn[β543λ(r2-2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1-r2sn[4β275λ(r2-2)×(x-αt+16β275λt)4β275λ(r2-2)]±cn[4β275λ(r2-2)×(x-αt+16β275λt)4β275λ(r2-2)])-1)}2,u2.19(x,t)={×(x-αt+16β275λt)])-1×(x-αt+16β275λt)43λ(1-2r2)])-1)-2β5λ+2ϵβ5λ2r2-1×(×(x-αt+16β275λt)43λ(1-2r2)])-1(sn[β543λ(1-2r2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±cn[β543λ(1-2r2)×(x-αt+16β275λt)43λ(1-2r2)])-1)}2,u2.20(x,t)={β543λ(r2-2)-2β5λ+2ϵβ5λ12r2-1×(rsn[β543λ(1-2r2)×(x-αt+16β275λt)[β223λ(1-2r2)]±idn[β543λ(1-2r2)×(x-αt+16β275λt)β543λ(r2-2)])}2,u2.21(x,t)={×(x-αt+16β275λt)])-1×(x-αt+16β275λt)43λ(2-r2)])-1)-2β5λ+2ϵβr5λ12-r2×(×(x-αt+16β275λt)43λ(2-r2)])-1(sn[β543λ(2-r2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±dn[β543λ(2-r2)×(x-αt+16β275λt)43λ(2-r2)])-1)}2,u2.22(x,t)={-2β5λ+2ϵβ5λ1-r2r2+1×(×(x-αt+16β275λt)43λ(r2+1)])-1(dn[β543λ(r2+1)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±rsn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)}2,u2.23(x,t)={-2β5λ+2ϵβ5λr2-1r2+1×((cn[β543λ(r2+1)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±sn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)r2-1r2+1}2,u2.24(x,t)={×(x-αt+16β275λt)43λ(r2+1)])-1)-2β5λ+2ϵβ(1-r2)5λ-1r2+1×(×(x-αt+16β275λt)43λ(r2+1)])-1(sn[β543λ(r2+1)×(x-αt+16β275λt)[β223λ(1-2r2)])×(dn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)]±cn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)}2,u2.25(x,t)={×(x-αt+16β275λt)43λ(r2-2)])-1)-2β5λ+2ϵβr25λ12-r2×((cn[β543λ(r2-2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1-r2±dn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])-1)}2.Whenp=1/2, we can also obtain Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).


### 3.3. The Jacobi Elliptic Function Solutions to (1) in the Case ofp=2

From the expressions of (15), we get the following jacobi elliptic function solutions to (1):
(21)u3.1(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμr2-15λ×sn[μ(x-αt+5β232λt+t4(r2+1)μ2)]}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.2(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμr2-15λ×cd[μ(x-αt+5β232λt+t4(r2+1)μ2)]}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.3(x,t)={-t4(2r2-1)μ25β232λ)-5β8λ+ϵμr215λ×cn[μ(x-αt+5β232λt-t4(2r2-1)μ2)]}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.4(x,t)={-t4(2-r2)μ25β232λ)]-5β8λ+ϵμ215λ×dn[μ(x-αt+5β232λt-t4(2-r2)μ2)]15λ}1/2,μ=±β4(5(2-r2)+5ϵr26(1-r2)λ)1/2,u3.5(x,t)={-t4(2r2-1)μ25β232λ)]-5β8λ+ϵμ2-15(1-r2)λ×nc[μ(x-αt+5β232λt-t4(2r2-1)μ2)]}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.6(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμ2-15λ×ns[μ(x-αt+5β232λt+t4(r2+1)μ2)]-15λ}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.7(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμ2-15λ×dc[μ(x-αt+5β232λt+t4(r2+1)μ2)]-15λ}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.8(x,t)={-5β8λ+ϵμ2-15(r2-1)λ×nd[μ(x-αt+5β232λt-t4(2-r2)μ2)]-15(r2-1)λ}1/2,μ=±β4(5(2-r2)+5ϵr26(1-r2)λ)1/2,u3.9(x,t)={-5β8λ+ϵμ2-15λ×cs[μ(x-αt+5β232λt-t4(2-r2)μ2)]}1/2,μ=±β4(5(2-r2)+5ϵr26λ(1-r2))1/2,u3.10(x,t)={-5β8λ+ϵμ2-15(1-r2)λ×sc[μ(x-αt+5β232λt-t4(2-r2)μ2)]-15(1-r2)λ}1/2,μ=±β4(5(2-r2)+5ϵr26λ(1-r2))1/2,u3.11(x,t)={-t4(2r2-1)μ25β232λ)]-5β8λ+ϵμr215(1-r2)λ×sd[μ(x-αt+5β232λt-t4(2r2-1)μ2)]15(1-r2)λ}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.12(x,t)={-5β8λ+ϵμ2-15λ×ds[μ(x-αt+5β232λt-t4(2r2-1)μ2)]}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.13(x,t)={-t8(r2+1)μ25β232λ)]-5β8λ+ϵμ415λ×(rcn[μ(x-αt+5β232λt-t8(r2+1)μ2)]±dn[μ(x-αt+5β232λt-t8(r2+1)μ2)])15λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.14(x,t)={-t8(1-2r2)μ25β232λ)])-1)-5β8λ+ϵμ4-15λ×(-t8(1-2r2)μ25β232λ)])-11×(sn[μ(x-αt+5β232λt-t8(1-2r2)5β232λ)])-1±(cn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])×(sn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])-1)-15λ}1/2,μ=±β2(5(1-2r2)+10ϵrr2-13λ)1/2,u3.15(x,t)={-5β8λ+ϵμ4-15(1-r2)λ×(-t8(r2+1)μ25β232λ)])-11×(cn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1±(dn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])×(cn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)-15(1-r2)λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.16(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-5β8λ+ϵμ4-15λ×(-t8(r2-2)μ25β232λ)])-11×(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)(sn[μ(x-αt+5β232λt])-1±(dn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])×(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])-1)}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2,u3.17(x,t)={-5β8λ+ϵμr4-15λ×(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)(sn[μ(x-αt+5β232λt]±icn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2,u3.18(x,t)={-tμ2(r2-2)85β232λ)])-1)-5β8λ+ϵμr4-15λ×(-tμ2(r2-2)85β232λ)])-1(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)(sn[μ(x-αt+5β232λt])×(1-r2sn[μ(tμ2(r2-2)8x-αt+5β232λt-tμ2(r2-2)85β232λ)]±cn[μ(-tμ2(r2-2)85β232λ)x-αt+5β232λt-tμ2(r2-2)85β232λ)])-1)}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2,u3.19(x,t)={-t8(r2+1)μ25β232λ)]-5β8λ+ϵμ4-15λ×(rsn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)]±idn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])}1/2,μ=±β2(5(1-2r2)+10ϵrr2-13λ)1/2,u3.20(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ4-15λ×(-t8(1-2r2)μ25β232λ)])-1(sn[μ(x-αt+5β232λt-t8(1-2r2)μ2)])×([μ(x-αt+5β232λt1±cn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])-1)}1/2,μ=±β2(5(1-2r2)+10ϵrr2-13λ)1/2,u3.21(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμr4-15λ×(-t8(r2-2)μ25β232λ)])-1(sn[μ(x-αt+5β232λt-t8(r2-2)μ2)])×([μ(x-αt+5β232λt1±dn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])-1)}1/2,μ=±β2r(5(r2-2)+5ϵr4-5r2+43λ)1/2,u3.22(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ415(1-r2)λ×((dn[μ(x-αt+5β232λt-t8(r2+1)μ2)])×([μ(x-αt+5β232λt1±rsn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)15(1-r2)λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.23(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ415(r2-1)λ×((cn[μ(x-αt+5β232λt-t8(r2+1)μ2)])×([μ(x-αt+5β232λt1±sn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)15(r2-1)λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.24(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ(r2-1)4-15λ×((sn[μ(x-αt+5β232λt-t8(r2+1)μ2)])×([μ(x-αt+5β232λtdn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)]±cn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.25(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-5β8λ+ϵμr24-15λ×(-tμ2(r2-2)85β232λ)])-1(cn[μ(x-αt+5β232λt-tμ2(r2-2)8)])×([μ(x-αt+5β232λt1-r2±dn[μ(x-αt+5β232λt-tμ2(r2-2)85β232λ)])-1)}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2.


### 3.4. Soliton and Triangular Function Solutions of (1)

From the expression of2c1c3=c22, we know that r→0 or r→1. According to the properties of Jacobi elliptic functions sn, cn, and dn, the following degenerated soliton and triangular function solutions for (1) can be obtained:
(22)u4.1(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×tanh[βpp+2-2p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.2(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵ2β(2p+1)2λ(p+2)×sech[βpp+22p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.3(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×coth[βpp+2-2p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.4(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+iϵ2β(2p+1)2λ(p+2)×csch[βpp+22p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.5(x,t)={+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1)-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1(cosh(βpp+2-2p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1))±1)×(-2p+1λ(p+1))sinh(βpp+2-2p+1λ(p+1)×(-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1)}1/p,u4.6(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1(sinh(βpp+2-2p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)])±i)×(-2p+1λ(p+1))cosh(βpp+2-2p+1λ(p+1)×[-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1)}1/p,u4.7(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×(-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1(sinh(βpp+2-2p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))×(-2p+1λ(p+1))1±cosh(βpp+2-2p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1)}1/p,u4.8(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵ2β(2p+1)2λ(p+2)×sec[βpp+2-2p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.9(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵ2β(2p+1)2λ(p+2)×csc[βpp+2-2p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.10(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×cot[βpp+22p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.11(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×tan[βpp+22p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.12(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1(1±cos(βpp+22p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))×(-2p+1λ(p+1))sin(βpp+22p+1λ(p+1)×(-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1)}1/p,u4.13(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1(1±sin(βpp+22p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))×(-2p+1λ(p+1))cos(βpp+22p+1λ(p+1)×[-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1)}1/p,u4.14(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1(sin(βpp+22p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))×(-2p+1λ(p+1))1±cos(βpp+22p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1)}1/p,u4.15(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1)-2p+1λ(p+1))))-1(cos(βpp+22p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))×(-2p+1λ(p+1))1±sin(βpp+22p+12λ(p+1)×(-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1)-2p+1λ(p+1))))-1)}1/p.Remark 4.
Whenr→0 and r→1 in the solutions u1.1(x,t) to u3.25(x,t), the soliton and triangular function solutions we obtain can be found in u4.1(x,t) to u4.15(x,t) while p=1,2,1/2, respectively. For example, when r→1, u2.17 and u3.17 can be expressed by
(24)u2.17.1(x,t)={+16β275λt)])-2β5λ+2ϵβ5λ×(tanh[β5-43λ(x-αt+16β275λt)]±isech[β5-43λ(x-αt+16β275λt)])}2,u3.17.1(x,t)={+16β275λt)])-5β8λ+5ϵβ8λ×(tanh[β2-53λ(x-αt+5β248λt)]±isech[β2-53λ(x-αt+5β248λt)])}1/2,
which are the same as u4.6 in the case of p=1/2,2, respectively.Remark 5.
In these expressions of solutions,u4.1(x,t) and u4.2(x,t) are in full agreement with the result which was obtained by Zhang et al. [15].


## 3.1. The Jacobi Elliptic Function Solutions to (1) in the Case of p=1

In order to make that wave numberμ a real-valued number, we have to choose constants λ and modulus r (c2 depends on r) to satisfy some restrictions. However, in this section, we allow μ to take values in complex number domain. From the expression of g1,g2,μ, and c in (13) and the solutions listed in Table 1, we derive the following Jacobi elliptic function solutions for (1) in the case p=1:
(17)u1.1(x,t)=-β2λ+ϵβr2λ2r2+1×sn[β2-13λ(r2+1)(x-αt+β26λt)],u1.2(x,t)=-β2λ+ϵβr2λ2r2+1×cd[β2-13λ(r2+1)(x-αt+β26λt)],u1.3(x,t)=-β2λ+ϵβr2λ22r2-1×cn[β213λ(2r2-1)(x-αt+β26λt)],u1.4(x,t)=-β2λ+ϵβ2λ22-r2×dn[β213λ(2-r2)(x-αt+β26λt)],u1.5(x,t)=-β2λ+ϵβ2λ2(r2-1)2r2-1×nc[β213λ(2r2-1)(x-αt+β26λt)],u1.6(x,t)=-β2λ+ϵβ2λ2r2+1×ns[β2-13λ(r2+1)(x-αt+β26λt)],u1.7(x,t)=-β2λ+ϵβ2λ2r2+1×dc[β2-13λ(r2+1)(x-αt+β26λt)],u1.8(x,t)=-β2λ+ϵβ2λ2(r2-1)r2-2×nd[β213λ(2-r2)(x-αt+β26λt)],u1.9(x,t)=-β2λ+ϵβ2λ2r2-2×cs[β213λ(2-r2)(x-αt+β26λt)],u1.10(x,t)=-β2λ+ϵβ2λ2(1-r2)r2-2×sc[β213λ(2-r2)(x-αt+β26λt)],u1.11(x,t)=-β2λ+ϵβr2λ2(r2-1)1-2r2×sd[β2-13λ(1-2r2)(x-αt+β26λt)],u1.12(x,t)=-β2λ+ϵβ2λ21-2r2×ds[β2-13λ(1-2r2)(x-αt+β26λt)],u1.13(x,t)=-β2λ+ϵβ2λ11+r2×{rcn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(r2+1)]±nd[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(r2+1)]},u1.14(x,t)=-β2λ+ϵβ2λ12r2-1×{×(x-αt+β2t6λ)[β223λ(1-2r2)])-1)1×(sn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1±(cn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])×(sn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1},u1.15(x,t)=-β2λ+ϵβ2λr2-11+r2×{×(x-αt+β2t6λ)[β223λ(1-2r2)])-11×(cn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1±(sn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])×(cn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1},u1.16(x,t)=-β2λ+ϵβ2λ12-r2×{×(x-αt+β2t6λ)[β223λ(1-2r2)])-11×(sn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1±(dn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])×(sn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1},u1.17(x,t)=-β2λ+ϵβr2λ12-r2×{sn[β223λ(r2-2)(x-αt+β2t6λ)]±icn[β223λ(r2-2)(x-αt+β2t6λ)]},u1.18(x,t)=-β2λ+ϵβr2λ12-r2×((x-αt+β2t6λ)[β223λ(1-2r2)])-1(dn[β223λ(r2-2)(x-αt+β2t6λ)])×(1-r2sn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)]±cn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.19(x,t)=-β2λ+ϵβ2λ12r2-1×{rsn[β223λ(1-2r2)(x-αt+β2t6λ)]±idn[β223λ(1-2r2)(x-αt+β2t6λ)]},u1.20(x,t)=-β2λ+ϵβ2λ12r2-1×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(sn[β213λ(1-2r2)(x-αt+β2t6λ)])×(1±cn[β223λ(1-2r2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.21(x,t)=-β2λ+ϵβ2λr22-r2×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(sn[β213λ(r2-2)(x-αt+β2t6λ)])×(1±dn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.22(x,t)=-β2λ+ϵβ2λ1-r2r2+1×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(dn[β223λ(r2+1)(x-αt+β2t6λ)])×(1±rsn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.23(x,t)=-β2λ+ϵβ2λr2-11+r2×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(cn[β223λ(r2+1)(x-αt+β2t6λ)])×(1±sn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.24(x,t)=-β2λ+ϵβ(1-r2)2λ-1r2+1×((x-αt+β2t6λ)[β223λ(1-2r2)])-1(sn[β223λ(r2+1)(x-αt+β2t6λ)])×(dn[β223λ(r2+1)×(x-αt+β2t6λ)]±cn[β223λ(r2+1)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1),u1.25(x,t)=-β2λ+ϵβr22λ2-r2×(×(x-αt+β2t6λ)[β223λ(1-2r2)])-1(cn[β223λ(r2-2)(x-αt+β2t6λ)])×(1-r2±dn[β223λ(r2-2)×(x-αt+β2t6λ)[β223λ(1-2r2)])-1).Remark 1.
From the properties of Jacobi elliptic functions, we know that when the moduler→0 or r→1, sn, cn, and dn degenerate into triangular or hyperbolic functions, from which we can obtain soliton and triangular function solutions of (1) in the case p=1. In other word, when p=1, we can deduce Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).Remark 2.
Whenα=0, k=(β/2)2/3λ(1-2r2), a0=-β/2λ, and a1=±k-3/2λ, u1.14(x,t) turns into
(18)u1.14(x,t)=a0+a1{nsk[x-(1-2r22k2+λa02+βa0)t]+csk[x-(1-2r22k2+λa02+βa0)t]},
which was obtained by Yomba [17].Remark 3.
Supposingk=(β/6)-1/qνλ, p=(2+3/3)-6qνλ (q and ν are arbitrary constants). When r→1 in u1.17(x,t), we obtain
(19)u1.17.1(x,t)=a0-ϵβ12λq-6qνλ×{±-2qνλ+-6qνλ(tanh-6qνλξ)±isech-6qνλξ},
where a0=-(βp/2λ)-6qνλ+β/2λ,ξ=k(x-(λa02+βa0)t). Solution (19) is in full agreement with that presented in [17] by Yomba.


## 3.2. The Jacobi Elliptic Function Solutions to (1) in the Case of p=1/2

Similarly as in Section3.1, we allow μ to take values in complex number field. Making use of (14) and the solutions listed in Table 1, we obtain the following Jacobi elliptic function solutions for (1) in the case p=1/2:
(20)u2.1(x,t)={-2β5λ+2ϵβr5λ2r2+1×sn[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.2(x,t)={-2β5λ+2ϵβr5λ2r2+1×cd[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.3(x,t)={[2β275λ(2r2-1)(x-αt+16β275λt)]-2β5λ+2ϵβr5λ22r2-1×cn[2β275λ(2r2-1)(x-αt+16β275λt)]}2,u2.4(x,t)={-2β5λ+2ϵβ5λ22-r2×dn[β523λ(2-r2)(x-αt+16β275λt)]}2,u2.5(x,t)={-2β5λ+2ϵβ5λ2r2-22r2-1×nc[2β275λ(2r2-1)(x-αt+16β275λt)]}2,u2.6(x,t)={-2β5λ+2ϵβ5λ2r2+1×ns[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.7(x,t)={-2β5λ+2ϵβ5λ2r2+1×dc[β5-23λ(r2+1)(x-αt+16β275λt)]}2,u2.8(x,t)={[β523λ(2-r2)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ2-2r22-r2×nd[β523λ(2-r2)(x-αt+16β275λt)]2-2r22-r2}2,u2.9(x,t)={-2β5λ+2ϵβ5λ-22-r2×cs[β523λ(2-r2)(x-αt+16β275λt)]}2,u2.10(x,t)={[β523λ(2-r2)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ2r2-22-r2×sc[β523λ(2-r2)(x-αt+16β275λt)]2r2-22-r2}2,u2.11(x,t)={[β523λ(2r2-1)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ2r2(1-r2)2r2-1×sd[β523λ(2r2-1)(x-αt+16β275λt)]2r2(1-r2)2r2-1}2,u2.12(x,t)={[2β275λ(2r2-1)(x-αt+16β275λt)]-2β5λ+2ϵβ5λ21-2r2×ds[2β275λ(2r2-1)(x-αt+16β275λt)]}2,u2.13(x,t)={-2β5λ+2ϵβ5λ1r2+1×(rcn[β543λ(r2+1)(x-αt+16β275λt)]±dn[β543λ(r2+1)(x-αt+16β275λt)])}2,u2.14(x,t)={×(x-αt+16β275λt)43λ(-2r2+1)])-1)×(x-αt+16β275λt)])-1)-2β5λ+2ϵβ5λ-11-2r2×(×(x-αt+16β275λt)43λ(-2r2+1)])-11×(sn[β543λ(-2r2+1)×(x-αt+16β275λt)43λ(-2r2+1)])-1±(cn[β543λ(-2r2+1)×(x-αt+16β275λt)43λ(-2r2+1)])×(sn[β543λ(-2r2+1)×(x-αt+16β275λt)43λ(-2r2+1)])-1)}2,u2.15(x,t)={×(x-αt+16β275λt)])-1)-2β5λ+2ϵβ5λ-1-r2r2+1×(×(x-αt+16β275λt)43λ(r2+1)])-11×(cn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1±(sn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])×(cn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)}2,u2.16(x,t)={×(x-αt+16β275λt)43λ(r2-2)])-1)-2β5λ+2ϵβ5λ-1r2-2×(×(x-αt+16β275λt)])-11×(sn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])-1±(dn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])×(sn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])-1)}2,u2.17(x,t)={β543λ(r2-2)-2β5λ+2ϵβr5λ-1r2-2×(sn[β543λ(r2-2)×(x-αt+16β275λt)[β223λ(1-2r2)]±icn[β543λ(r2-2)×(x-αt+16β275λt)β543λ(r2-2)])}2,u2.18(x,t)={4β275λ(r2-2)×(x-αt+16β275λt)4β275λ(r2-2)])-1-2β5λ+2ϵβr5λ-1r2-2×(×(x-αt+16β275λt)4β275λ(r2-2)])-1(dn[β543λ(r2-2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1-r2sn[4β275λ(r2-2)×(x-αt+16β275λt)4β275λ(r2-2)]±cn[4β275λ(r2-2)×(x-αt+16β275λt)4β275λ(r2-2)])-1)}2,u2.19(x,t)={×(x-αt+16β275λt)])-1×(x-αt+16β275λt)43λ(1-2r2)])-1)-2β5λ+2ϵβ5λ2r2-1×(×(x-αt+16β275λt)43λ(1-2r2)])-1(sn[β543λ(1-2r2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±cn[β543λ(1-2r2)×(x-αt+16β275λt)43λ(1-2r2)])-1)}2,u2.20(x,t)={β543λ(r2-2)-2β5λ+2ϵβ5λ12r2-1×(rsn[β543λ(1-2r2)×(x-αt+16β275λt)[β223λ(1-2r2)]±idn[β543λ(1-2r2)×(x-αt+16β275λt)β543λ(r2-2)])}2,u2.21(x,t)={×(x-αt+16β275λt)])-1×(x-αt+16β275λt)43λ(2-r2)])-1)-2β5λ+2ϵβr5λ12-r2×(×(x-αt+16β275λt)43λ(2-r2)])-1(sn[β543λ(2-r2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±dn[β543λ(2-r2)×(x-αt+16β275λt)43λ(2-r2)])-1)}2,u2.22(x,t)={-2β5λ+2ϵβ5λ1-r2r2+1×(×(x-αt+16β275λt)43λ(r2+1)])-1(dn[β543λ(r2+1)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±rsn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)}2,u2.23(x,t)={-2β5λ+2ϵβ5λr2-1r2+1×((cn[β543λ(r2+1)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1±sn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)r2-1r2+1}2,u2.24(x,t)={×(x-αt+16β275λt)43λ(r2+1)])-1)-2β5λ+2ϵβ(1-r2)5λ-1r2+1×(×(x-αt+16β275λt)43λ(r2+1)])-1(sn[β543λ(r2+1)×(x-αt+16β275λt)[β223λ(1-2r2)])×(dn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)]±cn[β543λ(r2+1)×(x-αt+16β275λt)43λ(r2+1)])-1)}2,u2.25(x,t)={×(x-αt+16β275λt)43λ(r2-2)])-1)-2β5λ+2ϵβr25λ12-r2×((cn[β543λ(r2-2)×(x-αt+16β275λt)[β223λ(1-2r2)])×(1-r2±dn[β543λ(r2-2)×(x-αt+16β275λt)43λ(r2-2)])-1)}2.Whenp=1/2, we can also obtain Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).


## 3.3. The Jacobi Elliptic Function Solutions to (1) in the Case ofp=2

From the expressions of (15), we get the following jacobi elliptic function solutions to (1):
(21)u3.1(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμr2-15λ×sn[μ(x-αt+5β232λt+t4(r2+1)μ2)]}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.2(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμr2-15λ×cd[μ(x-αt+5β232λt+t4(r2+1)μ2)]}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.3(x,t)={-t4(2r2-1)μ25β232λ)-5β8λ+ϵμr215λ×cn[μ(x-αt+5β232λt-t4(2r2-1)μ2)]}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.4(x,t)={-t4(2-r2)μ25β232λ)]-5β8λ+ϵμ215λ×dn[μ(x-αt+5β232λt-t4(2-r2)μ2)]15λ}1/2,μ=±β4(5(2-r2)+5ϵr26(1-r2)λ)1/2,u3.5(x,t)={-t4(2r2-1)μ25β232λ)]-5β8λ+ϵμ2-15(1-r2)λ×nc[μ(x-αt+5β232λt-t4(2r2-1)μ2)]}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.6(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμ2-15λ×ns[μ(x-αt+5β232λt+t4(r2+1)μ2)]-15λ}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.7(x,t)={+t4(r2+1)μ25β232λ)]-5β8λ+ϵμ2-15λ×dc[μ(x-αt+5β232λt+t4(r2+1)μ2)]-15λ}1/2,μ=±β4r(-5(r2+1)+5ϵ(r2-1)6λ)1/2,u3.8(x,t)={-5β8λ+ϵμ2-15(r2-1)λ×nd[μ(x-αt+5β232λt-t4(2-r2)μ2)]-15(r2-1)λ}1/2,μ=±β4(5(2-r2)+5ϵr26(1-r2)λ)1/2,u3.9(x,t)={-5β8λ+ϵμ2-15λ×cs[μ(x-αt+5β232λt-t4(2-r2)μ2)]}1/2,μ=±β4(5(2-r2)+5ϵr26λ(1-r2))1/2,u3.10(x,t)={-5β8λ+ϵμ2-15(1-r2)λ×sc[μ(x-αt+5β232λt-t4(2-r2)μ2)]-15(1-r2)λ}1/2,μ=±β4(5(2-r2)+5ϵr26λ(1-r2))1/2,u3.11(x,t)={-t4(2r2-1)μ25β232λ)]-5β8λ+ϵμr215(1-r2)λ×sd[μ(x-αt+5β232λt-t4(2r2-1)μ2)]15(1-r2)λ}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.12(x,t)={-5β8λ+ϵμ2-15λ×ds[μ(x-αt+5β232λt-t4(2r2-1)μ2)]}1/2,μ=±β4r(5(2r2-1)+5ϵ6(r2-1)λ)1/2,u3.13(x,t)={-t8(r2+1)μ25β232λ)]-5β8λ+ϵμ415λ×(rcn[μ(x-αt+5β232λt-t8(r2+1)μ2)]±dn[μ(x-αt+5β232λt-t8(r2+1)μ2)])15λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.14(x,t)={-t8(1-2r2)μ25β232λ)])-1)-5β8λ+ϵμ4-15λ×(-t8(1-2r2)μ25β232λ)])-11×(sn[μ(x-αt+5β232λt-t8(1-2r2)5β232λ)])-1±(cn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])×(sn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])-1)-15λ}1/2,μ=±β2(5(1-2r2)+10ϵrr2-13λ)1/2,u3.15(x,t)={-5β8λ+ϵμ4-15(1-r2)λ×(-t8(r2+1)μ25β232λ)])-11×(cn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1±(dn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])×(cn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)-15(1-r2)λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.16(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-5β8λ+ϵμ4-15λ×(-t8(r2-2)μ25β232λ)])-11×(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)(sn[μ(x-αt+5β232λt])-1±(dn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])×(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])-1)}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2,u3.17(x,t)={-5β8λ+ϵμr4-15λ×(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)(sn[μ(x-αt+5β232λt]±icn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2,u3.18(x,t)={-tμ2(r2-2)85β232λ)])-1)-5β8λ+ϵμr4-15λ×(-tμ2(r2-2)85β232λ)])-1(sn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)(sn[μ(x-αt+5β232λt])×(1-r2sn[μ(tμ2(r2-2)8x-αt+5β232λt-tμ2(r2-2)85β232λ)]±cn[μ(-tμ2(r2-2)85β232λ)x-αt+5β232λt-tμ2(r2-2)85β232λ)])-1)}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2,u3.19(x,t)={-t8(r2+1)μ25β232λ)]-5β8λ+ϵμ4-15λ×(rsn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)]±idn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])}1/2,μ=±β2(5(1-2r2)+10ϵrr2-13λ)1/2,u3.20(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ4-15λ×(-t8(1-2r2)μ25β232λ)])-1(sn[μ(x-αt+5β232λt-t8(1-2r2)μ2)])×([μ(x-αt+5β232λt1±cn[μ(x-αt+5β232λt-t8(1-2r2)μ25β232λ)])-1)}1/2,μ=±β2(5(1-2r2)+10ϵrr2-13λ)1/2,u3.21(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμr4-15λ×(-t8(r2-2)μ25β232λ)])-1(sn[μ(x-αt+5β232λt-t8(r2-2)μ2)])×([μ(x-αt+5β232λt1±dn[μ(x-αt+5β232λt-t8(r2-2)μ25β232λ)])-1)}1/2,μ=±β2r(5(r2-2)+5ϵr4-5r2+43λ)1/2,u3.22(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ415(1-r2)λ×((dn[μ(x-αt+5β232λt-t8(r2+1)μ2)])×([μ(x-αt+5β232λt1±rsn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)15(1-r2)λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.23(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ415(r2-1)λ×((cn[μ(x-αt+5β232λt-t8(r2+1)μ2)])×([μ(x-αt+5β232λt1±sn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)15(r2-1)λ}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.24(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-t8(r2+1)μ25β232λ)])-1-5β8λ+ϵμ(r2-1)4-15λ×((sn[μ(x-αt+5β232λt-t8(r2+1)μ2)])×([μ(x-αt+5β232λtdn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)]±cn[μ(x-αt+5β232λt-t8(r2+1)μ25β232λ)])-1)}1/2,μ=±β2(1-r2)(5(r2+1)+10ϵr3λ)1/2,u3.25(x,t)={-t8(r2-2)μ25β232λ)])-1)-t8(r2-2)μ2)])-1)-5β8λ+ϵμr24-15λ×(-tμ2(r2-2)85β232λ)])-1(cn[μ(x-αt+5β232λt-tμ2(r2-2)8)])×([μ(x-αt+5β232λt1-r2±dn[μ(x-αt+5β232λt-tμ2(r2-2)85β232λ)])-1)}1/2,μ=±β2r2(5(r2-2)+10ϵ1-r23λ)1/2.


## 3.4. Soliton and Triangular Function Solutions of (1)

From the expression of2c1c3=c22, we know that r→0 or r→1. According to the properties of Jacobi elliptic functions sn, cn, and dn, the following degenerated soliton and triangular function solutions for (1) can be obtained:
(22)u4.1(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×tanh[βpp+2-2p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.2(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵ2β(2p+1)2λ(p+2)×sech[βpp+22p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.3(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×coth[βpp+2-2p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.4(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+iϵ2β(2p+1)2λ(p+2)×csch[βpp+22p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.5(x,t)={+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1)-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1(cosh(βpp+2-2p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1))±1)×(-2p+1λ(p+1))sinh(βpp+2-2p+1λ(p+1)×(-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1)}1/p,u4.6(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1(sinh(βpp+2-2p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)])±i)×(-2p+1λ(p+1))cosh(βpp+2-2p+1λ(p+1)×[-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1)}1/p,u4.7(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+ϵβ(2p+1)2λ(p+2)×(-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1(sinh(βpp+2-2p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))×(-2p+1λ(p+1))1±cosh(βpp+2-2p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1)}1/p,u4.8(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵ2β(2p+1)2λ(p+2)×sec[βpp+2-2p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.9(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+ϵ2β(2p+1)2λ(p+2)×csc[βpp+2-2p+12λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.10(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×cot[βpp+22p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.11(x,t)={-2p+14λ(p+1)]-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×tan[βpp+22p+14λ(p+1)×(x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+14λ(p+1)]}1/p,u4.12(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1(1±cos(βpp+22p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))×(-2p+1λ(p+1))sin(βpp+22p+1λ(p+1)×(-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))-1)}1/p,u4.13(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1(1±sin(βpp+22p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))×(-2p+1λ(p+1))cos(βpp+22p+1λ(p+1)×[-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1)]-2p+1λ(p+1)))-1)}1/p,u4.14(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1(sin(βpp+22p+1λ(p+1)×(β2(2p+1)tλ(p+2)2(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1))-2p+1λ(p+1)))×(-2p+1λ(p+1))1±cos(βpp+22p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))-1)}1/p,u4.15(x,t)={-2p+1λ(p+1)))-1-β(2p+1)2λ(p+2)+iϵβ(2p+1)2λ(p+2)×(+β2(2p+1)tλ(p+2)2(p+1)-2p+1λ(p+1))))-1(cos(βpp+22p+1λ(p+1)×[β2(2p+1)tλ(p+2)2(p+1)x-αt-2p+1λ(p+1)+β2(2p+1)tλ(p+2)2(p+1)]))×(-2p+1λ(p+1))1±sin(βpp+22p+12λ(p+1)×(-2p+1λ(p+1)x-αt+β2(2p+1)tλ(p+2)2(p+1)-2p+1λ(p+1))))-1)}1/p.Remark 4.
Whenr→0 and r→1 in the solutions u1.1(x,t) to u3.25(x,t), the soliton and triangular function solutions we obtain can be found in u4.1(x,t) to u4.15(x,t) while p=1,2,1/2, respectively. For example, when r→1, u2.17 and u3.17 can be expressed by
(24)u2.17.1(x,t)={+16β275λt)])-2β5λ+2ϵβ5λ×(tanh[β5-43λ(x-αt+16β275λt)]±isech[β5-43λ(x-αt+16β275λt)])}2,u3.17.1(x,t)={+16β275λt)])-5β8λ+5ϵβ8λ×(tanh[β2-53λ(x-αt+5β248λt)]±isech[β2-53λ(x-αt+5β248λt)])}1/2,
which are the same as u4.6 in the case of p=1/2,2, respectively.Remark 5.
In these expressions of solutions,u4.1(x,t) and u4.2(x,t) are in full agreement with the result which was obtained by Zhang et al. [15].


## 4. Conclusion

In this paper, by using the auxiliary differential equation method, we obtain the Jacobi function solutions, the degenerated solitons, and the triangular function solutions to the generalized dispersive KdV-mKdV equation (1) in the case of p=1, p=2, and p=1/2 with the help of symbolic computation system Matlab. However, for arbitrary exponent p, we only obtain the degenerated solitons and triangle function solutions to (1). It may need other technique to deal with the Jacobi elliptic function solutions for the generalized dispersive KdV-mKdV equation with arbitrary exponent p.



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*Source: 249043-2013-07-10.xml*