# Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients

Authors: Bo Tang; Xuemin Wang; Yingzhe Fan; Junfeng Qu
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of variable-coefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for the KdV-MKdV equation are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving many other nonlinear partial differential equations with variable coefficients in mathematical physics.


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## Body

## 1. Introduction

In the nonlinear science, many important phenomena in various fields can be described by the nonlinear partial differential equations (NPDEs). Searching for exact soliton solutions of NPDEs plays an important and significant role in the study on the dynamics of those phenomena. Various methods have been used to handle nonlinear partial differential equations, such as the Hirota bilinear method [1], inverse scattering method [2, 3] the Bäcklund transformation method [4], subequation method [5–7], F-expansion method [8–10], sine-cosine method [11, 12], sech-tanh method [13, 14], Exp-function method [15, 16], and Jacobi elliptic function method [17–19].It is well known that NPDEs with variable coefficients are more realistic in various physical situations than their constant coefficients counterparts. However, most of the above methods are related to the constant-coefficient NPDEs. The present work is motivated by the desire to establish an auxiliary equation method to construct new and more general exact solutions of variable-coefficient NPDEs, such as soliton and soliton-like solutions, triangular periodic solutions, Jacobi elliptic function solutions, and many exact explicit solutions in form of hyperbolic function solutions and trigonometric function solutions. Being concise and straightforward, this method is applied to the generalized variable-coefficient KdV-MKdV equation as the following form [20]:(1)ut-6f0tuux-6f1tu2ux+f2tuxxx-f3tux+f4tAu+xux=0, where f0(t), f1(t), f2(t), f3(t), and f4(t) are arbitrary functions of t.Equation (1) can be used to describe the propagation of weakly nonlinear long waves in a KdV-typed medium by changing the coefficients of dispersion and nonlinear coefficients. It includes the following three important equations:(a) Settingf0(t), (1) can be degenerated to the following variable-coefficient and nonisospectral KdV equation studied in [21, 22]: (2)ut=K0tuxxx+6uux+4K1tux-ht2u+xux, which models many important nonlinear phenomena, including shallow water waves, dust acoustic solitary structures in magnetized dusty plasmas, and ion acoustic waves in plasmas.(b) Settingf0(t)=0, f1(t)=f2(t)=-h1(t), f3(t)=4h2(t), f4(t)=h0(t), A=1, (1) can be degenerated to the following variable-coefficient MKdV equation studied in [23]: (3)ut=h1tuxxx-6u2ux+4h2tux-h0tu+xux, which models many important nonlinear phenomena, including shallow water waves, dust acoustic solitary structures in magnetized dusty plasmas, and ion acoustic waves in plasmas.(c) Settingf0(t)=-a(t)/6, f1(t)=-b(t)/6, f2(t)=h(t), f3(t)=-d(t), f4(t)=f(t)/A, x=0, (1) becomes the following variable-coefficient Gardner equation: (4)ut+atuux+btu2ux+dtux+ftu+htuxxx=0, which is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory [24, 25]. It also describes a variety of wave phenomena in plasma and solid state.The rest of this paper is organized as follows: in Section2, we will describe the auxiliary equation method for finding out solutions of variable-coefficient NPDEs and give the main steps of the method here. In Section 3, we illustrate the method in detail with the generalized variable-coefficient KdV-MKdV equation. In Section 4, some conclusions are given.


## 2. Description of the Auxiliary Equation Method

Consider a given variable-coefficient nonlinear partial differential equation with independent variableX=(x0=t,x1,x2,…,xn) and dependent variable u:(5)Pu,ut,uxi,uxixj,…=0, where P is in general a polynomial function of its argument and the subscripts denote the partial derivatives.Suppose (5) has the following solution:(6)ut,x=∑i=0laiFiξ, where ai=ai(x1,x2,…,xn,t) and ξ=ξ(x1,x2,…,xn,t) are functions to be determined later and F′(ξ)=dF/dξ satisfies(7)F′2ξ=∑j=0mhiFjξ, where the values of m can be determined by balancing the highest differential term with the nonlinear terms in (5). The main steps by which we get exact solutions to variable-coefficient NPDE are outlined as follows.Step 1.
Substituting (6) along with (7) into (5) and then setting all coefficients of xpFj(ξ)F′q(ξ) (p,q=0,1; j=0,1,2,…) of the resulting equation to zero, we get an overdetermined PDEs system for ai (i=0,1,2,…,k) and ξ.Step 2.
Solving the set of overdetermined PDEs by use of Mathematica can permit obtention of explicit expressions ofai (i=0,1,2,…,l) and ξ.Step 3.
Substitutingai (i=0,1,2,…,l) and ξ obtained in Step 2 into (6) along with the solutions of (7), we can get the exact solutions of (5).Remark 1.
In this paper, we consider the case ofm=4. As we know the more solutions of (7) we find, the more exact solutions of (5) may be obtained. However, the general solutions are difficult to be listed because of the complexity of (7). Some special solutions [19, 26–29] are listed as follows.Case 1.
Suppose thath1=h3=0; the solutions of (7) are given in Tables 1 and 2.Table 1
Solutions of Case1. m (0<m<1) denotes the modulus of the Jacobi elliptic function, k1=1-m2, and A,B,C (ABC≠0), and D are arbitrary constants.
h
0
h
2
h
4
F
ξ
0
>0
<0
-
h
2
h
4
sech
⁡
h
2
ξ
0
>0
>0
h
2
h
4
csch
⁡
h
2
ξ
h
2
2
4
h
4
<0
>0
-
h
2
2
h
4
tanh
⁡
-
h
2
ξ
2
0
<0
>0
-
h
2
h
4
sec
⁡
-
h
2
ξ,-h2h4csc⁡-h2ξ
h
2
2
4
h
4
>0
>0
h
2
2
h
4
tan
⁡
h
2
ξ
2
0
0
>0
-
1
h
4
ξ
1
-
(
1
+
m
2
)
m
2
sn
ξ, cdξ=cnξdnξ
(
1
-
m
2
)
2
m
2
-
1
-
m
2
cn
ξ
m
2
-
1
2
-
m
2
-
1
dn
ξ
m
2
-
(
1
+
m
2
)
1
ns
ξ
=
1
sn
ξ, dcξ=dnξcnξ
-
m
2
2
m
2
-
1
1
-
m
2
nc
ξ
=
1
cn
ξ
-
1
2
-
m
2
m
2
-
1
nd
ξ
=
1
dn
ξ
1
2
-
m
2
1
-
m
2
sc
ξ
=
sn
ξ
cn
ξTable 2
Solutions of Case1 (continued). m (0<m<1) denotes the modulus of the Jacobi elliptic function, k1=1-m2, and A,B,C (ABC≠0), and D are arbitrary constants.
h
0
h
2
h
4
F
(
ξ
)
1
2
m
2
-
1
-
m
2
1
-
m
2
sd
⁡
ξ
=
sn
ξ
dn
ξ
1
-
m
2
2
-
m
2
1
cs
ξ
=
cn
ξ
sn
ξ
-
m
2
1
-
m
2
2
m
2
-
1
1
ds
ξ
=
dn
ξ
sn
ξ
1
4
1
-
2
m
2
2
1
4
ns
ξ
±
cs
ξ
1
-
m
2
4
1
+
m
2
2
1
-
m
2
4
nc
ξ
±
sc
ξ
m
2
4
m
2
-
2
2
1
4
ns
ξ
±
ds
ξ
m
2
4
m
2
-
2
2
m
2
4
sn
ξ
±
i
cn
ξ,dnξ1-m2snξ±cnξ
1
4
1
-
2
m
2
2
1
4
m
sn
ξ
±
i
dn
ξ,snξ1±cnξ
1
4
m
2
-
2
2
m
2
4
sn
ξ
1
±
dn
ξ
m
2
-
1
4
m
2
+
1
2
m
2
-
1
4
dn
ξ
1
±
m
sn
ξ,msd⁡ξ±ndξ
1
-
m
2
4
m
2
+
1
2
1
-
m
2
4
cn
ξ
1
±
sn
ξ,ncξ±scξ
1
4
m
2
+
1
2
(
1
-
m
2
)
2
4
sn
ξ
dn
ξ
±
cn
ξ
1
4
m
2
-
2
2
m
4
4
cn
ξ
1
-
m
2
±
dn
ξ
-
1
-
m
2
2
4
m
2
+
1
2
-
1
4
m
cn
ξ
±
dn
ξ
1
4
1
4
dn
ξ
m
cn
ξ
±
i
1
-
m
2
1
-
2
m
2
2
sn
ξ
1
±
cn
ξ,cnξ1-m2snξ±dnξ
1
2
-
4
m
2
1
sn
ξ
dn
ξ
cn
ξ
m
-
1
2
4
A
2
1
+
m
2
+
6
m
2
A
2
m
-
1
2
4
dn
ξ
cn
ξ
A
1
+
sn
ξ
1
+
m
sn
ξ
m
+
1
2
4
A
2
1
+
m
2
-
6
m
2
A
2
m
+
1
2
4
dn
ξ
cn
ξ
A
1
+
sn
ξ
1
-
m
sn
ξ
-
2
m
3
+
m
4
+
m
2
6
m
-
m
2
-
1
-
4
m
m
dn
ξ
cn
ξ
1
+
m
s
n
2
ξ
2
m
3
+
m
4
+
m
2
-
6
m
-
m
2
-
1
4
m
m
dn
ξ
cn
ξ
-
1
+
m
s
n
2
ξ
2
+
2
k
1
-
m
2
6
k
1
-
m
2
+
2
4
k
1
m
2
sn
ξ
cn
ξ
k
1
-
d
n
2
ξ
2
-
2
k
1
-
m
2
-
6
k
1
-
m
2
+
2
-
4
k
1
-
m
2
sn
ξ
cn
ξ
k
1
+
d
n
2
ξ
m
2
-
1
4
C
2
m
2
-
B
2
m
2
+
1
2
C
2
m
2
-
B
2
m
2
-
1
4
B
2
-
C
2
/
B
2
-
C
2
m
2
+
sn
ξ
B
cn
ξ
+
C
dn
ξ
m
4
4
C
2
+
B
2
m
2
2
-
1
C
2
+
B
2
4
B
2
+
C
2
-
C
2
m
2
/
B
2
+
C
2
+
dn
ξ
B
sn
ξ
+
C
cn
ξ
2
m
-
m
2
-
1
B
2
2
m
2
+
2
-
B
2
m
2
-
B
2
-
2
B
2
m
m
s
n
2
ξ
-
1
B
m
s
n
2
ξ
+
1
-
2
m
+
m
2
+
1
B
2
2
m
2
+
2
-
B
2
m
2
-
B
2
-
2
B
2
m
m
s
n
2
ξ
+
1
B
m
s
n
2
ξ
-
1
0
1
1
2
±
2
-
2
t
a
n
h
2
D
-
ξ
t
a
n
h
D
-
ξCase 2.
Suppose thath0=h1=0; the solutions of (7) are given in Table 3.Table 3
Solutions of Case2. Δ2=h32-4h2h4, ϵ=±1, and a,b,c, and d are the arbitrary constants.
h
2
h
3
h
4
Δ
2
F
ξ
1
-
2
c
a
c
2
-
b
2
a
2
a
sech
⁡
ξ
b
+
c
sech
⁡
ξ
1
-
2
c
a
c
2
+
b
2
a
2
a
csch
⁡
ξ
b
+
c
csch
⁡
ξ
4
-
4
2
b
+
d
a
c
2
+
4
b
2
+
4
b
d
a
2
a
sech
2
⁡
ξ
b
sech
2
⁡
ξ
+
c
tanh
⁡
ξ
+
d
4
4
d
-
2
b
a
c
2
+
4
b
2
-
4
b
d
a
2
a
csch
2
⁡
ξ
b
csch
2
⁡
ξ
+
c
c
o
t
h
ξ
+
d
a
2
2
a
b
b
2
-
a
c
b
c
+
c
o
s
h
a
ξ
-
s
i
n
h
a
ξ
a
2
2
a
b
b
2
-
a
c
o
s
h
a
ξ
+
s
i
n
h
a
ξ
b
c
+
c
o
s
h
a
ξ
+
s
i
n
h
a
ξ
-
1
2
c
a
-
c
2
-
b
2
a
2
a
s
e
c
ξ
b
+
c
s
e
c
ξ
-
1
2
c
a
-
c
2
-
b
2
a
2
a
c
s
c
ξ
b
+
c
c
s
c
ξ
-
4
4
2
b
+
d
a
-
c
2
+
4
b
2
+
4
b
d
a
2
a
s
e
c
2
ξ
b
s
e
c
2
ξ
+
c
t
a
n
ξ
+
d
-
4
4
2
b
+
d
a
-
c
2
+
4
b
2
+
4
b
d
a
2
a
c
s
c
2
ξ
b
c
s
c
2
ξ
+
c
c
o
t
ξ
+
d
>0
-
h
2
h
3
sech
2
⁡
h
2
/
2
ξ
h
3
2
-
h
2
h
4
1
+
ϵ
t
a
n
h
h
2
/
2
ξ
2
>0
h
2
h
3
csch
2
⁡
h
2
/
2
ξ
h
3
2
-
h
2
h
4
1
+
ϵ
c
o
t
h
h
2
/
2
ξ
2
>0
4
h
2
e
ϵ
h
2
ξ
e
ϵ
h
2
ξ
2
-
4
h
2
h
4
>0
>0
2
h
2
sech
⁡
h
2
ξ
ϵ
Δ
2
-
h
3
sech
⁡
h
2
ξ
<0
>0
2
h
2
s
e
c
-
h
2
ξ
ϵ
Δ
2
-
h
3
s
e
c
-
h
2
ξ
>0
<0
2
h
2
csch
⁡
h
2
ξ
ϵ
-
Δ
2
-
h
3
csch
⁡
h
2
ξ
<0
>0
2
h
2
c
s
c
-
h
2
ξ
ϵ
Δ
2
-
h
3
c
s
c
-
h
2
ξ
>0
=0
-
h
2
h
3
1
±
t
a
n
h
h
2
ξ
2
>0
=0
-
h
2
h
3
1
±
c
o
t
h
h
2
ξ
2
>0
>0
-
h
2
sech
2
⁡
h
2
/
2
ξ
h
3
+
2
ϵ
h
2
h
4
t
a
n
h
h
2
/
2
ξ
>0
>0
h
2
csch
2
⁡
h
2
/
2
ξ
h
3
+
2
ϵ
h
2
h
4
c
o
t
h
h
2
/
2
ξ
<0
>0
-
h
2
s
e
c
2
-
h
2
/
2
ξ
h
3
+
2
ϵ
-
h
2
h
4
t
a
n
h
-
h
2
/
2
ξ
<0
>0
-
h
2
c
s
c
2
-
h
2
/
2
ξ
h
3
+
2
ϵ
-
h
2
h
4
c
o
t
-
h
2
/
2
ξ
>0
=0
±
4
h
2
e
ϵ
h
2
ξ
1
-
4
h
2
h
4
e
2
ϵ
h
2
ξCase 3.
Ifh0=k, h1=-4k, h2=-1-k2+6k, h3=2(k-1)2, h4=-(k-1)2, (7) has the following solutions:(8)Fξ=ksnξ+12ksnξ+ksnξ2+1,Fξ=k3/2snξ+1k3/2snξ-dn2ξ+1.Case 4.
Ifh0=-h1/4, h1<0, h2=-h1(2-k2), h3=2h1(1-k2), h4=-h1(1-k2), (7) has the following solutions:(9)Fξ=sn-h1ξsn-h1ξ+cn-h1ξ+1,Fξ=1+cn-h1ξ-sn-h1ξ+cn-h1ξ+1.Case 5.
Ifh0>0,h1=-4h0,h2=4h0(2-k2),h3=-8h0(1-k2),h4=4h0(1-k2), (7) has the following solutions:(10)Fξ=cn2h0ξ1-k2sn2h0ξ+cn2h0ξ+dn2h0ξ,Fξ=1-k2sn2h0ξ+dn2h0ξ1-k2sn2h0ξ+cn2h0ξ+dn2h0ξ.Case 6.
Ifh0=-h1/4, h1=-(kα)2, h2=-(2h1+α2), h3=2(h1+α2), h4=-(h1+α2), (7) has the following solutions:(11)Fξ=ksnαξksnαξ+dnαξ-1,Fξ=dnαξ+1ksnαξ-dnαξ-1,Fξ=kcnαξkcnαξ+dnαξ+1-k2,Fξ=dnαξ+1-k2kcnαξ+dnαξ+1-k2.


## 3. Solutions for the Generalized Variable-Coefficient KdV-MKdV Equation

To solve (1), we first make the transformation(12)u=uξ,ξ=ptx+qt, where p(t) and q(t) are functions to be determined later. We consider that the solutions of (1) can be expressed as follows:(13)uξ=∑i=0lait,xFiξ, where ai(t,x) (i=0,1,…,l) are functions to be determined later.Substituting (13) into (1), we can easily find that l=1 by balancing uxxx and u2ux in (1). Therefore,(14)uξ=a0t,x+a1t,xFξ.With the aid of Mathematica, substituting (14) along with (7) into (1) and then setting each coefficient of F(ξ) to zero, we obtain a set of nonlinear and parameterized partial differential equations with respect to unknowns a0, a1, p, and q as follows: (15)a0t+Af4ta0=0,a1t+Af4ta1=0,-6f0ta12pt-12pta0a12+3a1h3f2tpt3=0,a1q′t-6a0a1f0tpt-6a02a1f1tpt+a1h2f2tpt3-a1f3tpt=0,a1p′t+a1f4tpt=0,-6f1ta13pt+6h4f2ta1pt3=0,a0xf4t=0,a1xf4t=0, where a0t=∂a0/∂t, a1t=∂a1/∂t, a0x=∂a0/∂x, a1x=∂a1/∂x.For the sake of simplicity, in the rest of this paper, we introduce four notations:(16)φt=exp⁡-∫tf4sds,φs=exp⁡-∫sf4τdτ,ϕt=exp⁡-∫tAf4sds,ϕs=exp⁡-∫sAf4τdτ.Solving the system by Mathematica, we get the following solution set:(17)a0=-a12a1f0t-p2th3f2t4p2th4f2t,a1=c2ϕt,pt=c1φt,f1t=h4p2tf2ta12,qt=c3+∫18h4f2sps-12a12f02s+3h32-8h2h4p4sf22s+8h4p2sf2sf3sds, where c1, c2, and c3 are arbitrary constants.Therefore, from Cases1–6, we obtain many kinds of explicit nontraveling solutions of (1). Taking Case 1 as an example, we have the following.Case 7.
Soliton and soliton-like solutions are as follows (Figures1 and 2):(18)u1=-a12f0t2p2th4f2t+c2ϕt-h2h4sech⁡h2ξ,where ξ=c1φ(t)x+∫‍[-3a12f02(s)-2h2h4p4(s)f22(s)+2h4p2(s)f2(s)f3(s)]/2h4f2(s)p(s)ds+c3. Consider(19)u2=-a12f0t2p2th4f2t+c2ϕth2h4csch⁡h2ξ, where ξ=c1φ(t)x+∫‍[-3a12f02(s)-2h2h4p4(s)f22(s)+2h4p2(s)f2(s)f3(s)]/2h4f2(s)p(s)ds+c3. Consider(20)u3=-a12f0t2p2th4f2t+c2ϕt-h22h4tanh⁡-h22ξ, where ξ=c1φ(t)x+∫‍[-3a12f02(s)-2h2h4p4(s)f22(s)+2h4p2(s)f2(s)f3(s)]/2h4f2(s)p(s)ds+c3. Consider(21)u4=-a12f0tp2tf2t±c2ϕt2-2tanh2⁡D-ξtanh⁡D-ξ, where ξ=c1φ(t)x+∫‍1/2h4f2(s)p(s)[-3a12f02(s)-p4(s)f22(s)+p2(s)f2(s)f3(s)]ds+c3.Figure 1
The soliton-like solution (18) with h2=c1=c2=A=1, h4=-1, c3=0, and f0(t)=f2(t)=f3(t)=t;  f1(t)=-t.Figure 2
The soliton-like solution (20) with h2=-2, h4=h0=c1=c2=A=1, c3=0, and f0(t)=f1(t)=f2(t)=f3(t)=t.Case 8.
Triangular periodic solutions are as follows:(22)u5=-a12f0t2p2th4f2t+c2ϕt-h2h4sec⁡-h2ξ,where ξ=c1φ(t)x+∫‍1/2h4f2(s)p(s)[-3a12f02(s)-2h2h4p4(s)f22(s)+2h4p2(s)f2(s)f3(s)]ds+c3. Consider(23)u6=-a12f0t2p2th4f2t+c2ϕth2h4csc⁡-h2ξ,where ξ=c1φ(t)x+∫1/2h4f2(s)p(s)‍[-3a12f02(s)-2h2h4p4(s)f22(s)+2h4p2(s)f2(s)f3(s)]ds+c3. Consider(24)u7=-a12f0t2p2th4f2t+c2ϕt-h22h4tan⁡-h22ξ,where ξ=c1φ(t)x+∫‍1/2h4f2(s)p(s)[-3a12f02(s)-2h2h4p4(s)f22(s)+2h4p2(s)f2(s)f3(s)]ds+c3.Case 9.
Jacobi elliptic function solutions and combined Jacobi elliptic function solutions are as follows:(25)u8=-a12f0t2p2tm2f2t+c2ϕtcdξ,where ξ=c1φ(t)x+∫‍[-3a12f02(s)+2(m4+m2)p4(s)f22(s)+2m2p2(s)f2(s)f3(s)]/2m2f2(s)p(s)ds+c3. Consider(26)u9=-a12f0t2p2tf2t+c2ϕtnsξ,where ξ=c1φ(t)x+∫‍[-3a12f02(s)+2(m2+1)p4(s)f22(s)+2p2(s)f2(s)f3(s)]/2f2(s)p(s)ds+c3. Consider(27)u10=a12f0t2p2tf2t+c2ϕtdnξ,where ξ=c1φ(t)x+∫‍[3a12f02(s)+2(m2-2)p4(s)f22(s)+2p2(s)f2(s)f3(s)]/2f2(s)p(s)ds+c3. Consider(28)u11=a12f0t2m2p2tf2t+c2ϕtcnξ,where ξ=c1φ(t)x+∫‍3a12f02s+2m2-2m4p4sf22s+2m2p2sf2sf3s/2m2f2(s)p(s)ds+c3. Consider(29)u12=-a12f0t2p2t1-m2f2t+c2ϕtncξ,where ξ=c1φ(t)x+∫([-3a12f02s+4m4-6m2+2p4sf22s+(2-2m2)p2sf2sf3s]/2(1  −  m2)f2(s)p(s))ds+c3. Consider(30)u13=-a12f0t2p2tm2-1f2t+c2ϕtndξ,where ξ=c1φ(t)x+∫([-3a12f02s+2m4-3m2+2p4sf22s+(2m2-2)p2sf2sf3s]/(2m2  −  2)f2(s)p(s))ds+c3. Consider(31)u14=-a12f0t2p2tf2t+c2ϕtcsξ,where ξ=c1φ(t)x+∫‍-3a12f02s+2m2-4p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(32)u15=a12f0t2p2t-1+m2f2t+c2ϕtscξ,where ξ=c1φ(t)x+∫‍([-3a12f02s-2m4-3m2+2p4sf22s+(2-2m2)p2sf2sf3s]/2(1  −  m2)f2(s)p(s))ds+c3. Consider(33)u16=-a12f0t2p2tm2m2-1f2t+c2ϕtsdξ,where ξ=c1φ(t)x+∫([-3a12f02s-2m22m4-3m2+1p4sf22s+2m2m2-1p2sf2sf3s]/2m2(m2  −  1)f2(s)p(s))ds+c3. Consider(34)u17=-a12f0t2p2tf2t+c2ϕtdsξ,where ξ=c1φ(t)x+∫‍-3a12f02s-22m2-1p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(35)u18=-2a12f0tp2tf2t+c2ϕtnsξ±csξ,where ξ=c1φ(t)x+∫‍-12a12f02s+2m2-1p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(36)u19=2a12f0tp2tm2-1f2t+c2ϕtncξ±scξ,where ξ=c1φ(t)x+∫(-12a12f02s+m4-1p4sf22s+21-m2p2sf2sf3s/2(1  −  m2)f2(s)p(s))ds+c3. Consider(37)u20=-2a12f0tp2tf2t+c2ϕtnsξ+dsξ,where ξ=c1φ(t)x+∫‍-12a12f02s-m2-1p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(38)u21=-2a12f0tp2tm2f2t+c2ϕtsnξ±icnξ,where ξ=c1φ(t)x+∫‍-12a12f02s-m4-2m2p4sf22s+2m2p2sf2sf3s/2m2f2(s)p(s)ds+c3. Consider(39)u22=-2a12f0tp2tm2f2t+c2ϕtdnξi1-m2snξ±cnξ,where ξ=c1φ(t)x+∫‍-12a12f02s-m4-2m2p4sf22s+2m2p2sf2sf3s/2m2f2(s)p(s)ds+c3. Consider(40)u23=-a12f0t2p2tf2t+c2ϕtsnξdnξcnξ,where ξ=c1φ(t)x+∫‍-3a12f02s-22-4m2p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(41)u24=-2a12f0tp2tA2-1+m2f2t+c2ϕtdnξcnξA1+snξ1+msnξ,where ξ=c1φ(t)x+∫([-12a12f02s--1+m2m2+6m+1p4sf22s+2A2(-1  +  m2)p2sf2sf3s]/2A2(-1  +  m2)f2(s)p(s))ds+c3. Consider(42)u25=-2a12f0tp2tA21+m2f2t+c2ϕtdnξcnξA1+snξ1-msnξ,where ξ=c1φ(t)x+∫([-12a12f02s-1+m2m2-6m+1p4sf22s+2A21+m2p2sf2sf3s]/2A2(1  +  m2)f2(s)p(s))ds+c3. Consider(43)u26=ma12f0t8p2tf2t+c2ϕtmdnξcnξ1+msn2ξ,where ξ=c1φ(t)x+∫‍m23ma12f02s-8m2-6m+1p4sf22s+8p2sf2sf3s/8f2(s)p(s)ds+c3. Consider(44)u27=-ma12f0t8p2tf2t+c2ϕtmdnξcnξmsn2ξ-1,where ξ=c1φ(t)x+∫‍m2-3ma12f02s+8m2+6m+1p4sf22s-8p2sf2sf3s/8f2(s)p(s)ds+c3. Consider(45)u28=-a12f0t8p2tk1f2t+c2ϕtm2snξcnξk1-dn2ξ,where ξ=c1φ(t)x+∫‍-3a12f02s-8k16k1-m2+2p4sf22s+8k1p2sf2sf3s/8k1f2(s)p(s)ds+c3. Consider(46)u29=a12f0t8k1p2tf2t-c2ϕtm2snξcnξk1+dn2ξ,where ξ=c1φ(t)x+∫‍3a12f02s+8k16k1+m2-2p4sf22s+8k1p2sf2sf3s/8k1f2(s)p(s)ds+c3. Consider(47)u30=-2a12f0tp2tm2-1C2m2-B2f2t+c2ϕtB2-C2/B2-C2m2+snξBcnξ+Cdnξ,where ξ=c1φ(t)x+∫([-12a12f02s-m4-1C2m2-B2p4sf22s+2m2-1(C2m2  −  B2)p2sf2sf3s]/(m2  −  1)(C2m2-B2)f2(s)p(s))ds+c3. Consider(48)u31=-2a12f0tp2tC2+B2f2t+c2ϕtB2+C2-C2m2/B2+C2+dnξBsnξ+Ccnξ,where ξ=c1φ(t)x+∫([-12a12f02s-C2+B2m2-2p4sf22s+2(C2  +  B2)p2sf2sf3s]/(C2  +  B2)f2(s)p(s))ds  +  c3. Consider(49)u32=a12f0t2p2tB2m+12f2t+c2ϕtmsn2-1Bmsn2+1,where ξ=c1φ(t)x+∫([3a12f02(s)-4B2(m+1)2(m2+1)p4(s)f22(s)+2B2(m+1)2p2(s)f2(s)f3(s)]/2B2(m  +  1)2f2(s)p(s))ds+c3. Consider(50)u33=a12f0t2p2tB2m-12f2t+c2ϕtmsn2+1Bmsn2-1,where ξ=c1φ(t)x+∫([3a12f02(s)-4B2(m-1)2(m2+1)p4(s)f22(s)+2B2(m-1)2p2(s)f2(s)f3(s)]/2B2(m  −  1)2f2(s)p(s))ds+c3. Consider(51)u34=-2a12f0tp2tf2t+c2ϕtdnξmcnξ±i1-m2,where ξ=c1φ(t)x+∫(-12a12f02s+2m2-1p4sf22s+2p2sf2sf3s/2f2(s)p(s))ds+c3. Consider(52)u35=-2a12f0tp2tf2t+c2ϕtmsnξ±idnξ,where ξ=c1φ(t)x+∫‍[-12a12f02(s)+(2m2-1)p4(s)f22(s)+2p2(s)f2(s)f3(s)]/2f2(s)p(s)ds+c3. Consider(53)u36=-2a12f0tp2tf2t+c2ϕtsnξ1±cnξ,where ξ=c1φ(t)x+∫‍-12a12f02s+2m2-1p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(54)u37=-2a12f0tp2tf2t+c2ϕtcnξ1-m2snξ±dnξ,where ξ=c1φ(t)x+∫‍-12a12f02s+2m2-1p4sf22s+2p2sf2sf3s/2f2(s)p(s)ds+c3. Consider(55)u38=-2a12f0tp2tm2-1f2t+c2ϕtdnξ1±msnξ,where ξ=c1φ(t)x+∫‍-12a12f02s-m4-1p4sf22s+2m2-1p2sf2sf3s/2(m2-1)f2(s)p(s)ds+c3. Consider(56)u39=-2a12f0tp2tm2-1f2t+c2ϕtmsdξ±ndξ,where ξ=c1φ(t)x+∫‍-12a12f02s-m4-1p4sf22s+2m2-1p2sf2sf3s/2(m2-1)f2(s)p(s)ds+c3. Consider(57)u40=2a12f0tp2tm2-1f2t+c2ϕtcnξ1±msnξ,where ξ=c1φ(t)x+∫-12a12f02s+m4-1p4sf22s+21-m2p2sf2sf3s/2(1-m2)f2(s)p(s)ds+c3. Consider(58)u41=2a12f0tp2tm2-1f2t+c2ϕtncξ±scξ,where ξ=c1φ(t)x+∫-12a12f02s+m4-1p4sf22s+21-m2p2sf2sf3s/2(1-m2)f2(s)p(s)ds+c3. Consider(59)u42=-2a12f0tp2tm2-12f2t+c2ϕtsnξdnξ±cnξ,where ξ=c1φ(t)x+∫([-12a12f02(s)-(m2+1)(m2-1)2p4(s)f22(s)+2(m2-1)2p2(s)f2(s)f3(s)]/2(m2  −  1)2f2(s)p(s))ds+c3. Consider(60)u43=-2a12f0tp2tm4f2t+c2ϕtcnξ1-m2±dnξ,where ξ=c1φ(t)x+∫‍[-12a12f02(s)-m4(m2-2)p4(s)f22(s)+2m4p2(s)f2(s)f3(s)]/2m4f2(s)p(s)ds+c3.Remark 2.
There are some other hyperbolic function solutions and trigonometric function solutions which can be obtained from solutions (25)–(60) at the limit case when m→1 and m→0, but we omit them for simplicity.Remark 3.
By using our method, we can obtain many more general exact solutions including all the solutions given in [20] as special cases. More precisely, setting h0=h1=h3=0, h2=1, and h4=-1, solution (18) is equivalent to (22) in [20]. Similarly, solution (20) is equivalent to (38) obtained in [20] setting h0=h4=1, h1=h3=0, and h2=-2. Setting h0=h4=1/4, h1=h3=0, h2=-1/2, and m→1, solution (52) is equivalent to (62) in [20]. The comparison between our method and the well known results in [20] shows that our method is more powerful than the method in [20] in constructing exact solutions.


## 4. Conclusion

With the help of symbolic computation, we employ the auxiliary equation method to find many new and more general exact solutions of the generalized KdV-MKdV equation with variable coefficients. This paper showed that the auxiliary equation method not only gives a unified formulation to construct a series of exact solutions but also provides a guideline to classify the types of solutions according to the given parameters. Applying the more generalized ansätz and auxiliary equation method to other variable-coefficients NPDEs is our future work.



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