# Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

Authors: Xiaoxiao Zheng; Yadong Shang; Yong Huang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions
of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle
function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic
function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid
of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.


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## Body

## 1. Introduction

It is well known that the KdV equation plays an important role in the soliton theory. Many properties of the KdV equation, such as symmetry, Bäcklund transformation, infinite conservation laws, Lax pairs, and Painleve analysis, have been studied. Miura transformation links the KdV equation with the mKdV equation. Therefore, as the KdV equation, mKdV equation is also important in mathematical physics field. In recent years, some authors considered the constant coefficients mKdV equation [1–4]. However, in practical applications, the coefficients of nonlinear evolution equations vary with time and space. Therefore, the exact solution of the variable coefficient nonlinear evolution equations has a greater application value.This paper will discuss the variable coefficients mKdV equation (VC-mKdV):(1)ut+a(t)ux+b(t)u2ux+r(t)uxxx=0,
where a(t),b(t), and r(t) are arbitrary function of variable t. More recently, some properties of the variable coefficients mKdV equation have been studied [5–18]. The aim of this paper is to apply the first integral method and the extended hyperbolic function method for constructing a series of explicit exact solutions to the VC-mKdV equation (1), such as rational function solutions, periodic wave solutions of triangle functions, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, Weierstrass elliptic function solution, and many exact explicit solutions in form of the rational function of hyperbolic function and the rational function of triangle function.The rest of this paper is organized as follows. In Section2, the outline of the first integral method will be given. In Section 3 we introduce a transformation to transform the VC-mKdV equation (1) into a constant coefficients mKdV equation. Section 4 is the main part of this paper; the methods are employed to seek the explicit and exact solutions of the VC-mKdV equation (1). In the last section, some conclusion is given.


## 2.  The First Integral Method

The first-integral method, which is based on the ring theory of commutative algebra, was first proposed by Professor Feng Zhaosheng [19] in 2002. The method has been applied by Feng to solve Burgers-KdV equation, the compound Burgers-KdV equation, an approximate Sine-Gordon equation in (n+1)-dimensional space, and two-dimensional Burgers-KdV equation [20–24].In the recent years, many authors employed this method to solve different types of nonlinear partial differential equations in physical mathematics. More information about these applications can be found in [25] and references therein. The most advantage is that the first integral method has not many sophisticated computation in solving nonlinear algebra equations compared to other direct algebra method. For completeness, we briefly outline the main steps of this method.The main steps of this method are summarized as follows.Given a system of nonlinear partial differential equations, for example, in two independent variables,(2)P(ut,ux,uxx,uxt,…)=0.
Using traveling wave transformation u(x,t)=f(ξ),ξ=kx+ωt+ξ0 and some other mathematical operations, the systems (2) can be reduced to a second-order nonlinear ordinary differential equation:
(3)D(f,f′,f′′)=0.
By introducing new variables X=f(ξ),Y=f′(ξ) or making some other transformations, we reduce ordinary differential equation (3) to a system of the first order ordinary differential equation:
(4)X′=YY′=H(X,Y).
Suppose that the first integral of (4) has a form as follows:
(5)P(X,Y)=∑i=0m‍ai(X)Yi=0.
(In general m=1 or m=2), where ai(X)(i=0,1,…,m) are real polynomials of X.According to the Division theorem from ring theory of commutative algebra, there exists polynomialsα(X),β(X) of variable X in ℜ[X] such that
(6)dPdξ=[α(X)+β(X)Y]P(X,Y).We determine polynomialsα(X),β(X),ai(X)(i=0,1,2,…) from (6), furthermore, obtain P(X,Y).Then substitutingX=f(ξ),Y=f′(ξ) or other transformations into (5), exact solutions to (2) are established, through solving the resulting first-order integrable differential equation.


## 3.  A Transformation to the VC-mKdV Equation

In order to transfer (1) into the form of (3), we firstly do some transformations for (1). Since (1) is a variable coefficients equation and we need to transform it to the constant coefficients mKdV equation, we introduce a transformation
(7)u(x,t)=U(X,T)p(x,t),
where X=X(x,t),T=T(x,t). Through this transformation, we hope that (1) be changed into the form of constant coefficients mKdV equation:
(8)UT+24U2UX+UXXX=0.
In order to obtain the above transformation equation, substituting (7) into (1) and assuming Tx=0, and simultaneously on both side of the formulas by dividing r(t)Xx3, we have
(9)pbU3pxrXx3+p2bU2UXrXx2+(ptrXx3p+pxxxXx3p+apxrXx3p)U+(arXx2+XtrXx3+XxxxXx3+2pxxXx2p+3XxxpxXx3p)UX+TtrXx3UT+(3XxxXx2+3pxXxp)UXX+UXXX=0.
Comparing the coefficients of (9) with (8), such as U2UX,U,Ux, and so on, we have
(10)pbpxrXx3=0,(11)p2brXx2=24,(12)ptrXx3p+pxxxXx3p+apxrXx3p=0,(13)arXx2+XtrXx3+XxxxXx3+2pxxXx2p+3XxxpxXx3p=0,(14)TtrXx3=1,(15)3XxxXx2+3pxXxp=0.
From (11) and (14), we have Xx≠0 and
(16)b=24rXx2p2,r=TtXx3.
Substituting (16) into (10), (12), (13), and (15), we obtain
(17)24pxXxp=0,(18)Xx3pt+pxxxTt+Xx3apxTtXx3p=0,(19)aXx4P+Xx3Xtp+XxxxTtp+2pxxTtXx+3XxxpxTtTtXx3p=0,(20)3(Xxxp+pxXx)Xx2p=0.
Form (18), we have Tt≠0. Also from (20), we have Xxx=0 and Xxxx=0. From (17) and (18), we have that p can only be a constant. For simplicity, we take p=C. Substituting this into (19), we obtain
(21)X=F(x-∫‍a(t)dt).
For simplicity, we take
(22)X=x-∫‍a(t)dt.
Substituting (23) into (16), we have
(23)T=∫‍r(t)dt+C1,
where C1 is an arbitrary constant. Thus we get s transform between VC-mKdV equation (1) with constant coefficients mKdV equation (8).


## 4.  Explicit and Exact Solutions of the VC-mKdV Equation

We firstly obtain explicit and exact solutions of the constant coefficients mKdV equation (8) and then obtain explicit and exact solutions of the constant coefficients mKdV equation (1). In the view of (8), we suppose ξ=kX+ωT+ξ0, and then we have
(24)ωU′+24kU2U′+k3U′′′=0.
Integrating (24) once with respect to ξ, we obtain
(25)U′′+mU3+lU=E,
where l=ω/k3,m=8/k2, and E is arbitrary integration constant. Let R=U(ξ),S=R′; (25) can be converted to a system of nonlinear ODEs as follows:
(26)R′=S,S′=-mR3-lR+E.
Now we employ the Division theorem to seek the first integral to (26). Suppose that R=R(ξ),S=S(ξ) are the nontrivial solution to the system (26), and its first integral is an irreducible polynomial in ℜ[R,S]:
(27)P(R(ξ),S(ξ))=∑i=02‍ai(R)Si=0,
where ai, i=0,1,2 are polynomial of R. According to the Division theorem, there exists polynomials α(R),β(R) of variable R in ℜ[R] such that
(28)dPdξ=a2′(R)S3+2a2(R)SS′+a1′(R)S2+a1(R)S′+a0′(R)S=(α(R)+β(R)S)(a2(R)S2+a1(R)S+a0(R)).
Collecting all the terms with the same power of S together and equating each coefficient to zero yield a set of nonlinear algebraic equations as follows:
(29)a2′(R)=β(R)a2(R),(30)a1′(R)=β(R)a1(R)+α(R)a2(R),(31)a0′(R)=β(R)a0(R)+α(R)a1(R)+2a2(R)(mR3+lR-E),(32)a1(R)(-mR3-lR+E)=α(R)a0(R).
Because ai(R), (i=1,2) are polynomials, from (29) we can deduce deg[a2(R)]=0, β(R)=0; that is a2(R) is a constant. For simplicity, we take β(R)=0, a2(R)=1. Then we determine a0(R), a1(R), and α(R). From (30), we have deg[a1(R)]-1=deg[α(R)] or a1(R)=α(R)=0. In what follows we will discuss these two situations.(1)
In the case ofα(R)=0,a1(R)=0.In this case, (30) and (32) are satisfied. From (31), we can derive a0(R)=(m/2)R4+lR2-2ER+d, where d is an integral constant. Substituting a2(R),a1(R),a0(R) into (27), one obtains that
(33)R′=±-m2R4-lR2+2ER-d.
Based on the discussion for different parameters, we can obtain the solutions of the nonlinear ordinary differential equation (33).(a)
Ford=E=0, (33) admits the following three general solutions:
(34)R1=±2lmcsch-lξ,lm>0R2=±-2lmsech-lξ,l<0,m>0R3=±1-m/2ξ,l=0.Combining (7), (22), (23), (34), and p(x,t)=C,R=U(ξ), one can get the following three sets of explicit exact solutions to (1):
(35)u1(x,t)=±iCω4kcschωk3ξ,kω>0,u2(x,t)=±C-ω4ksech-ωk3ξ,kω<0,u3(x,t)=±Ck2i[k(x-∫‍a(t)dt)+ξ0],ω=0,
where ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0, C,C1 are arbitrary parameters, and ξ0 is an arbitrary constant.(b)
ForE=0, d=l2/2m, we can obtain the following four sets of explicit exact solutions to (33)
(36)R4=±lmtan-l2ξ,lm>0,R5=±lmcot-l2ξ,lm>0,R6=±-lmtanhl2ξ,lm<0,R7=±-lmcothl2ξ,lm<0. Combining (7), (22), (23), (36), and p(x,t)=C,R=U(ξ), we can get the following four explicit exact solutions of (1):
(37)u4(x,t)=±Cω8ktan-ω2k3ξ,kω>0,u5(x,t)=±Cω8kcot-ω2k3ξ,kω>0,u6(x,t)=±C-ω8ktanhω2k3ξ,kω<0,u7(x,t)=±C-ω8kcothω2k3ξ,kω<0,
where ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0, C,C1 are arbitrary parameters, and ξ0 is an arbitrary constant.(c)
ForE=l=0, from (33) we have
(38)R′=±-m2R4-d. Let Z=R2; (38) becomes
(39)Z′=±-m2Z3-dZ.
While m<0, the above equation possesses a Weierstrass elliptic function doubly periodic wave type solution:
(40)Z=℘(-m8ξ,-8dm,0).
Combining (7), (22), (23), (40), Z=R2,p(x,t)=C, and R=U(ξ), we derive that (1) admits a Weierstrass elliptic function doubly periodic wave type solution:
(41)u8(x,t)=±C℘(±1kiξ,-dk2,0),
where ξ=k(x-∫‍a(t)dt), C,C1 are arbitrary parameters, and ξ0 is an arbitrary constant.(d)
ForE=0,d≠0, we obtain elliptic function solutions for (33) as follows:
(42)R9=±1-lmcn1-l2ξd=l2-12mR10=±2mdn2+lξd=-2(l+1)mR11=±-1+lmncl-l2ξd=l2-12mR12=±-2(1+l)mnd2+lξd=-2(l+1)mR13=±1-l22msd1-l2ξd=l2-12mR14=±2(1+l)mscl+2ξd=-2(l+1)m.Combining (7), (22), (23), the above result (42), and p(x,t)=C,R=U(ξ), we can get the following six Jacobi elliptic doubly periodic wave solutions of (1):
(43)u9(x,t)=±k3-ω8kcnk3-ω2k3ξd=ω3-k616k4u10(x,t)=±k2dn2k3+ωk3ξd=-ω+k34ku11(x,t)=±-k3+ω8knck3-ω2k3ξd=ω2-k616k4u12(x,t)=±-k3+ω4knd2k3+ωk3ξd=-k3+ω4ku13(x,t)=±k6-ω216k4sdω-k32k3ξd=ω2-k616k4u14(x,t)=±ω+k34k3scω+2k3k3ξd=-ω+k34k3,
where ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0, C,C1 are arbitrary parameters, and ξ0 is an arbitrary constant.(2)
In the case ofdeg[a1(R)]-1=deg[α(R)].In this case, we assume thatdeg[α(R)]=k1, deg[a0(R)]=k2; then we have deg[a1(R)]=k1+1. Now, by balancing the degrees of both sides of (32), we can deduce that k2=4. By balancing the degrees of both sides of (31), we can also conclude that k1=1 or k1=0. If k1=0, assuming that α(R)=A0, a1(R)=A1R+A2, a0(R)=C4R4+C3R3+C2R2+C1R+C0 and substituting them into (30)–(32), by equating the coefficients of the different powers of X on both sides of (30) to (32), we can get that α(R)=a1(R)=0. This is contradicting with our assumption. It indicates that k1≠0. While k1=1, assuming that a0(R)=C4R4+C3R3+C2R2+C1R+C0, a1=A2R2+A1R+A0, α(X)=B1R+B0, then substituting these representations into (30)–(32), and by equating the coefficients of the different powers of R on both sides of (30) to (32), we can obtain an overdetermined system of nonlinear algebraic equations:
(44)2A2=B1A1=B04C4-2m=A2B13C3=A1B1+A2B02C2-2l=A1B0+A0B1C1=A2B0-2E-mA2=B1C4-mA1=B1C3+C4B0-mA0-lA2=B1C2+B0C3EA2-lA1=B1C1+B0C2EA1-lA2=B1C0+B0C1EA0=B0C0.By analyzing all kinds of possibilities, we have the following.(a)
WhileB0=C0=0, it leads to a contradiction.(b)
WhileB0≠0,C0=0, it also leads to a contradiction.(c)
WhileB0=0,C0≠0, we can derive that
(45)E=0,A1=C1=C3=0,C4=-m2,C2=-l,C0=-l22m,A22=-2m,A0=lmA2.Setting (45) in (27) yields
(46)dRR2+(l/m)=±-2m2dξ.
Solving (46), we can obtain solutions R4,R5,R6, and R7 again. Consequently, we obtain explicit exact solutions u4,u5,u6, and u7 to (1). Here we will not list them one by one.We obtain various of explicit and exact solutions of (1) by using the extended hyperbolic functions method presented in [26] by author. we can get the following explicit exact solutions to (1):
(47)R15=±-2masinh(ξ)+bcosh(ξ)acosh(ξ)+bsinh(ξ),
when l=2, where a,b, and r are arbitrary constants
(48)R16=±2(a2-b2)m1acosh(ξ)+bsinh(ξ),
when l=-1, where a,b, and r are arbitrary constants such that (a2-b2)m>0(49)R17=±a2-b2-r22m1acosh(ξ)+bsinh(ξ)±-12masinh(ξ)+bcosh(ξ)acosh(ξ)+bsinh(ξ)+r,
when l=1/2, where a,b, and r are arbitrary constants
(50)R18=±a2-b2-r22m1acosh(ξ)+bsinh(ξ)±-12mbcosh(ξ)-asinh(ξ)acosh(ξ)+bsinh(ξ)+r,
when l=-1/2, where a,b, and r are arbitrary constants
(51)R19=±-2(a2+b2)m1acosh(ξ)+bsinh(ξ),
when l=1, where a,b, and r are arbitrary constants
(52)R20=±-2mbcosh(ξ)-asinh(ξ)acosh(ξ)+bsinh(ξ),
when l=-2, where a,b, and r are arbitrary constantsCombining (7), (22), (23), the above result (47)–(52), and p(x,t)=C,R=U(ξ), the VC-mKdV equation (1) has explicit and exact solitary wave solutions:
(53)u15(x,t)=±ki(asinh(ξ)+bcosh(ξ))2(acosh(ξ)+bsinh(ξ)),
where k,ω,a, and b are arbitrary constants such that ω=2k3, ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0(54)u16(x,t)=±ka2-b221acosh(ξ)+bsinh(ξ),
where k,ω,a, and b are arbitrary constants such that ω=-k3, ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0(55)u17(x,t)=±ka2-b2-r241acosh(ξ)+bsinh(ξ)±-12masinh(ξ)+bcosh(ξ)acosh(ξ)+bsinh(ξ)+r,
where k,ω,a, and b are arbitrary constants such that 2ω=k3, ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0(56)u18(x,t)=±ka2-b2-r22(acosh(ξ)+bsinh(ξ))±-12mbcosh(ξ)-asinh(ξ)acosh(ξ)+bsinh(ξ)+r,
where k,ω,a, and b are arbitrary constants such that -2ω=k3, ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0(57)u19(x,t)=±-2(a2+b2)m1acosh(ξ)+bsinh(ξ),
where k,ω,a, and b are arbitrary constants such that ω=k3, ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0(58)u20(x,t)=±ki(bcosh(ξ)-asinh(ξ))2(acosh(ξ)+bsinh(ξ)),
where k,ω,a, and b are arbitrary constants such that ω=-2k3, ξ=k(x-∫‍a(t)dt)+ω(∫‍r(t)dt+C1)+ξ0.


## 5.  Summary and Conclusions

In summary, motivated by [27], we establish a transform from VC-mKdV equation to the constant coefficient mKdV equation firstly. Then we employ the first integral method and the extended hyperbolic function method to uniformly construct a series of explicit exact solutions for VC-mKdV equations. Abundant explicit exact solutions to VC-mKdV equations are obtained through an exhaustive analysis and discussion for different parameters. The exact solutions obtained in this paper include that of the solitary wave solutions of kink-type, singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, Weierstrass elliptic function doubly periodic wave solutions, and so forth. In particular, the six explicit exact Jacobi elliptic function doubly periodic solutions R9–R14 and Weierstrass elliptic function doubly periodic wave solution R8 are uniformly obtained. Some known results of previous references are enriched greatly. The results indicate that the first integral method and extended hyperbolic function method are very effective methods to solve nonlinear differential equation. The methods also are readily applicable to a large variety of other nonlinear evolution equations in physical mathematics.



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