# F-Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation

Authors: Yun-Mei Zhao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Based on theF-expansion method, and the extended version of F-expansion
method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation. With
the aid of Maple, more exact solutions expressed by Jacobi elliptic function are obtained.
When the modulus m  of Jacobi elliptic function is driven to the limits 1 and 0, some exact
solutions expressed by hyperbolic function solutions and trigonometric functions can also be
obtained.


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## Body

## 1. Introduction.

In the recent years, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena has become an important toll. Seeking exact solutions of NLEEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as homogeneous balance method [1, 2], auxiliary equation method [3, 4], the Exp-function method [5, 6], Darboux transformation [7, 8], tanh-function method [9], the modified extended tanh-function [10], and Jacobi elliptic function expansion method [11, 12].TheF-expansion method is an effective and direct algebraic method for finding the exact solutions of nonlinear evolution problems [13–15], many nonlinear equations have been successfully solved. Later, the further developed methods named the generalized F-expansion method [16, 17], the modified F-expansion method [18], the extended F-expansion method [19], and the improved F-expansion method [20] have been proposed and applied to many nonlinear problems.Recently, Kudryashov and Sinelshchikov [21] introduced the following equation:
(1)ut+γuux+uxxx-ε(uuxx)x-κuxuxx-νuxx-δ(uux)x=0,
where γ,ε,κ,ν, and δ are real parameters. Equation (1) describes the pressure waves in the liquid with gas bubbles taking into account the heat transfer and viscosity [1]. It was called the Kudryashov-Sinelshchikov equation [22].In practice, analysis of propagation of the pressure waves in a liquid with gas bubbles is an important problem. We know that there are solitary and periodic waves in a mixture of a liquid and gas bubbles, and these waves can be described by the Burgers equation, the Korteweg-de Vries (KdV) equation, and the Burgers-Korteweg-de Vries (BKdV) equation [23–26]. The Kudryashov-Sinelshchikov equation is a generalization of the KdV and the BKdV equations. Indeed, assuming that ε=κ=δ=0, we have the Burgers-Korteweg-de-Vries equation. In the case of ε=κ=λ=δ=0, we get the famous Korteweg-de Vries equation.Recently, the Kudryashov-Sinelshchikov equation has been investigated by different methods and some exact solutions are derived. Ryabov [22] obtained some exact solutions for β=-3 and β=-4 using a modification of the truncated expansion method [27, 28]. Using the bifurcation theory and the method of phase portraits analysis [29, 30], He et al. [31] investigated bifurcations of travelling wave solutions of the Kudryashov-Sinelshchikov equation and proved the existence of the peakon, solitary wave, and smooth and nonsmooth periodic waves. In this paper, we will derive more new exact Jacobian elliptic function solutions of the Kudryashov-Sinelshchikov equation based on the F-expansion method and its extended version.


## 2. TheF-Expansion Method and Its Extended Version

In this section, we will give the detailed description of theF-expansion method and its extended version.Suppose that we have a nonlinear partial differential equation (PDE) foru(x,t) in the form
(2)N(u,ut,ux,utt,uxt,uxx,…)=0,
where N is a polynomial in its arguments.By takingu(x,t)=u(ξ),ξ=x-ct, we look for traveling wave solutions of (2) and transform it to the ordinary differential equation (ODE)
(3)N(u,-cu′,u′,c2u′′,-cu′′,u′′,…)=0.Suppose that the solutionu of (3) can be expressed as a finite series in the form
(4)u=a0+∑i=1n‍(aiFi(ξ)+biFi(ξ)),
where a0, and ai,bi  (i=1,2,…,n) are constants to be determined later, and F(ξ) is a solution of the auxiliary LODE
(5)F′2(ξ)=PF4(ξ)+QF2(ξ)+R,
where P,Q, and R are constants. If the values of P,Q, and R are known, the Jacobian elliptic function solutions F(ξ) can be obtained from (5) which can also be found in Table 1.Table 1
Relations between the coefficients(P,Q,R) and corresponding f(ξ) in f′2=Pf4+Qf2+R [32].
Case
P
Q
R
f
(
ξ
)
1
m
2
-
(
1
+
m
2
)
1
sn
ξ
2
m
2
-
(
1
+
m
2
)
1
cd
ξ
3
-
m
2
2
m
2
-
1
1
-
m
2
cn
ξ
4
-
1
2
-
m
2
m
2
-
1
dn
ξ
5
1
-
(
1
+
m
2
)
m
2
ns
ξ
6
1
-
(
1
+
m
2
)
m
2
dc
ξ
7
1
-
m
2
2
m
2
-
1
-
m
2
nc
ξ
8
m
2
-
1
2
-
m
2
-
1
nd
ξ
9
1
-
m
2
2
-
m
2
1
sc
ξ
10
-
m
2
(
1
-
m
2
)
2
m
2
-
1
1
sd
ξ
11
1
2
-
m
2
1
-
m
2
cs
ξ
12
1
2
m
2
-
1
-
m
2
(
1
-
m
2
)
ds
ξWe may also seek the exact solutions of ODE (3) in the following form:
(6)u=a0+∑i=1n‍(aiFi(ξ)+biFi-1(ξ)F′(ξ)),
where a0,ai,bi(i=1,2,…,n) are constants to be determined later, F(ξ) is a solution of (5).In theF-expansion method, substituting (4) with (5) into ODE (3) and collecting coefficients of Fj(ξ)(j=0,±1,±2,…), we derive a set of overdetermined algebraic equations of a0,ai, and bi for i=1,2,…,n by setting each coefficient to zero. Solving these overdetermined algebraic equations by symbolic computation, we can determine those parameters explicitly.In the extended version ofF-expansion method, by substituting (5) with (6) into ODE (3) and collecting the coefficients of Fj(ξ)F′k(ξ)(j=0,±1,±2,…, k=0,1), we derive a set of overdetermined algebraic equations of a0,ai, and bi for i=1,2,…,n by setting each coefficient to zero. By solving these overdetermined algebraic equations by symbolic computation, we can determine those parameters explicitly.Assuming that the constantsa0,ai, and bi(i=1,2,…,n) can be obtained by solving the algebraic equations, then by substituting these constants and the known general solutions into (4) or (6), we can obtain the explicit solutions of (2) immediately.


## 3. Exact Solutions of the Kudryashov-Sinelshchikov Equation in the Case ofν=δ=0

In this section, we solve the Kudryashov-Sinelshchikov equation in case ofν=δ=0 by F-expansion method the in order to find the exact solutions of the Kudryashov-Sinelshchikov equation. Using scale transformation:
(7)x=x′,t=t′,u=1εU,
the Kudryashov-Sinelshchikov equation is written in the form
(8)Ut+αUUx+Uxxx-(UUxx)x-βUxUxx=0,
where α=γ/ε and β=κ/ε.We let(9)U(x,t)=1-ϕ(ξ),ξ=x-ct.
Under this transformation, (8) can be reduced to the following ordinary differential equation (ODE):
(10)cϕ′-α(1-ϕ)ϕ′-ϕ′′′+((1-ϕ)ϕ′′)′-βϕ′ϕ′′=0.By integrating (10) once with respect to ξ, we have
(11)12αϕ2+(c-α)ϕ-ϕϕ′′-12β(ϕ′)2+C=0,
where C is an integration constant.Based on theF-expansion method, we take the solution of ODE (11) as follows:
(12)ϕ=a0+a1F(ξ)+b1F(ξ)+a2F2(ξ)+b2F2(ξ),
where a0,a1,b1,a2, and b2 are constants to be determined and F(ξ) satisfies the elliptic Equation (5).By substituting (12) and (5) into (11), the left-hand side of (11) becomes a polynomial in F(ξ). Setting their coefficients to zero yields a system of algebraic equations in a0,a1,b1,a2, and b2. By solving the overdetermined algebraic equations by Maple, we can obtain the following six sets of solutions:(1)
(13)a1=a2=b1=0,c=-4(b22P+b2Q(1-2a0)+3Ra0(a0-1))b2,α=4(3Ra0-b2Q)b2,C=6a0(b22P+Ra02-a0b2Q)b2,β=-3,(2)
(14)a1=b1=b2=0,c=-4(a22R+a2Q(1-2a0)+3Pa0(a0-1))a2,α=4(3Pa0-a2Q)a2,C=6a0(a22R+a02P-a0a2Q)a2,β=-3,(3)
(15)a1=b1=0,b2=a2RP,c=-4(-4a22R+a2Q(1-2a0)+3Pa0(a0-1))a2,α=4(3Pa0-a2Q)a2,C=6(-4a0a22RP+4a23RQ+a03P2-a02Pa2Q)a2P,β=-3,(4)
(16)a0=a1=a2=b2=0,c=-2Q,α=-2Q,C=-2b12P,β=-4,(5)
(17)a0=a2=b1=b2=0,c=-2Q,α=-2Q,C=-2a12R,β=-4,(6)
(18)a0=a2=b2=0,c=-2Q+12RPR,α=-2Q+12RPR,C=8PRb12Q-16Pb12,β=-4.Substituting (13)–(18) into (12) with (9), we have the following formal solution of (8):
(19)U=1-a0-b2F2(ξ),
where ξ=x+(4(b22P+b2Q(1-2a0)+3Ra0(a0-1))/b2)t, α=4(3Ra0-b2Q)/b2, β=-3,
(20)U=1-a0-a2F2(ξ),
where ξ=x+(4(a22R+a2Q(1-2a0)+3Pa0(a0-1))/a2)t, α=4(3Pa0-a2Q)/a2,  β=-3,
(21)U=1-a0-a2F2(ξ)-a2RPF2(ξ),
where ξ=x+(4(-4a22R+a2Q(1-2a0)+3Pa0(a0-1))/a2)t, α=4(3Pa0-a2Q)/a2, β=-3. 						(22)U=1-b1F(ξ),
where ξ=x+2Qt, α=-2Q, β=-4,
(23)U=1-a1F(ξ),
where ξ=x+2Qt, α=-2Q, β=-4, and
(24)U=1-PRb1F(ξ)-b1F(ξ),
where ξ=x+(2Q-12RP/R)t, α=-2Q+12RP/R, β=-4.Combining (19)–(24) with Table 1, some exact solutions of (8) are obtained.WhenP=m2,Q=-(1+m2), and R=1, the solutions of elliptic equation (5) are F(ξ)=sn(ξ,m) and F(ξ)=cd(ξ,m) from Table 1, so the exact solutions for the Kudryashov-Sinelshchikov equation are obtained.From  (19), we have
(25)U1=1-a0-b2ns2(ξ,m),(26)U2=1-a0-b2dc2(ξ,m),
where ξ=x+(4(b22m2-b2(1+m2)(1-2a0)+3a0(a0-1))/b2)t, α=4(3a0+b2(1+m2))/b2,  β=-3.Whenm→1, from (25), the exact solution of (8) is
(27)U3=1-a0-b2coth2(ξ),
where ξ=x+(4(b22-2b2(1-2a0)+3a0(a0-1))/b2)t,  α=4(3a0+2b2)/b2,    β=-3.Whenm→0, from (25) and (26), the exact solutions of (8) are
(28)U4=1-a0-b2csc2(ξ),U5=1-a0-b2sec2(ξ),
where ξ=x+(4(-b2(1-2a0)+3a0(a0-1))/b2)t, α=4(3a0+b2)/b2, β=-3.From (20), we have
(29)U6=1-a0-a2sn2(ξ,m),(30)U7=1-a0-a2cd2(ξ,m),
where ξ=x+(4(a22-a2(1+m2)(1-2a0)+3a0m2(a0-1))/a2)t, α=4(3a0m2+a2(1+m2))/a2, β=-3.Whenm→1, from (29), the exact solution of (8) is
(31)U8=1-a0-a2tanh2(ξ),
where ξ=x+(4(a22-2a2(1-2a0)+3a0(a0-1))/a2)t, α=4(3a0+2a2)/a2, β=-3.Whenm→0, from (29) and (30), the exact solutions of (8) are
(32)U9=1-a0-a2sin2(ξ),U10=1-a0-a2cos2(ξ),
where ξ=x+(4(-a2+2a0a2+a22)/a2)t, α=4, β=-3.From (21), we have
(33)U11=1-a0-a2sn2(ξ,m)-a2m2ns2(ξ,m),(34)U12=1-a0-a2cd2(ξ,m)-a2m2dc2(ξ,m),
where ξ=x+(4(-4a22-a2(1+m2)(1-2a0)+3a0m2(a0-1))/a2)t,    α=4(3a0m2+a2(1+m2))/a2,  β=-3.Whenm→1, from (33) the exact solution of (8) is
(35)U13=1-a0-a2tanh2(ξ)-a2coth2(ξ),
where ξ=x+(4(-4a22-2a2(1-2a0)+3a0(a0-1))/a2)t,    α=4(3a0+2a2)/a2,    β=-3.From (22), we have
(36)U14=1-b1ns(ξ,m),(37)U15=1-b1dc(ξ,m),
where ξ=x-2(1+m2)t, α=2(1+m2), β=-4.Whenm→1, from (36), the exact solution of (8) is
(38)U16=1-b1coth(ξ),
where ξ=x-4t, α=4, β=-4.Whenm→0, from (36) and (37), the exact solutions of (8) are
(39)U17=1-b1csc(ξ),U18=1-b1sec(ξ),
where ξ=x-2t, α=2, β=-4.From (23), we have
(40)U19=1-a1sn(ξ,m),(41)U20=1-a1cd(ξ,m),
where ξ=x-2(1+m2)t, α=2(1+m2), β=-4.Whenm→1, from (40) the exact solution of (8) is
(42)U21=1-a1tanh(ξ),
where ξ=x-4t, α=4, β=-4.Whenm→0, from (40) and (41), the exact solutions of (8) are
(43)U22=1-a1sin(ξ),U23=1-a1cos(ξ),
where ξ=x-2t, α=2, β=-4.From (24), we have
(44)U24=1-mb1sn(ξ,m)-b1ns(ξ,m),(45)U25=1-mb1cd(ξ,m)-b1dc(ξ,m),
where ξ=x-2(1+m2+6m)t, α=2(1+m2+6m), β=-4.Whenm→1, from (44) the exact solution of (8) is
(46)U26=1-b1tanh(ξ)-b1coth(ξ),
where ξ=x-16t, α=16, β=-4.


## 4. Exact Solutions of the Kudryashov-Sinelshchikov Equation in the Case ofν≠0, δ≠0

In this section, we solve the Kudryashov-Sinelshchikov equation in case ofν≠0, δ≠0 by the extended version of F-expansion method. Using transformation (7), we can write the Kudryashov-Sinelshchikov equation in the following form:
(47)Ut+αUUx+Uxxx-(UUxx)x-βUxUxx-νUxx-μ(UUx)x=0,
where α=γ/ε, β=κ/ε, μ=δ/ε.We let(48)U(x,t)=1-ϕ(ξ),ξ=x-ct.Under this transformation, (47) can be reduced to the following ordinary differential equation (ODE):
(49)cϕ′-α(1-ϕ)ϕ′-ϕ′′′+((1-ϕ)ϕ′′)′-βϕ′ϕ′′+νϕ′′-μ((ϕ-1)ϕ′)′=0.By integrating (49) once with respect to ξ, we have
(50)12αϕ2+(c-α)ϕ-ϕϕ′′-12β(ϕ′)2+νϕ′-μ(ϕ-1)ϕ′+C=0,
where C is integration constant.Based on the extended version ofF-expansion method, we take the solution of ODE (50) as follows:
(51)ϕ=a0+a1F(ξ)+a2F2(ξ)+b1F′(ξ)+b2F(ξ)F′(ξ),
where a0,a1,b1,a2, and b2 are constants to be determined, and F(ξ) satisfies the elliptic Equation (5).Substituting (51) and (5) into (50), the left-hand side of (50) becomes a polynomial in Fj(ξ)F′k(ξ)(j=0,±1,±2,…, k=0,1). Setting their coefficients to zero yields a system of algebraic equations in a0,a1,b1,a2, and b2 By solving the overdetermined algebraic equations by Maple, we can obtain the following solution:
(52)a0=a1=b1=0,a2=Qb2,c=-43(5QRb2+4Q),μ=-23Q,ν=23(Q+2Rb2),α=-16Q3,β=-83,C=-83R2b22.Substituting (52) into (51) with (48), we have the following formal solution of (47):
(53)U=1-Qb2F2(ξ)-b2F(ξ)F′(ξ),
where ξ=x+(4/3)(5QRb2+4Q)t, μ=-(2/3)Q, ν=(2/3)(Q+2Rb2),  α=-16Q/3,  β=-8/3.By combining (53) with Table 1, some exact solutions of (47) are obtained.WhenP=m2,Q=-(1+m2),R=1, the solutions of elliptic equation (5) are F(ξ)=sn(ξ,m) and F(ξ)=cd(ξ,m) from Table 1, so the exact solutions of (47) are
(54)U1=1-b2(i1+m2sn2(ξ,m)1-b2+sn(ξ,m)cn(ξ,m)dn(ξ,m){1+m2}),(55)U2=1-b2(i1+m2cd2(ξ,m)&1-b2-(1-m2)cd(ξ,m)sd(ξ,m)nd(ξ,m){1+m2}),
where ξ=x-(4/3)(4(1+m2)-5b2i1+m2)t, μ=-(2/3)i1+m2,  ν=(2/3)(i1+m2+2b2),  α=(16/3)(1+m2), β=-8/3.Whenm→1, from (54), the exact solution of (47) is
(56)U3=1-b2(i2tanh2(ξ)+tanh(ξ)sech2(ξ)),
where ξ=x-(4/3)(8-5b2i2)t, μ=-(2/3)i2, ν=(2/3)(i2+2b2), α=32/3, β=-8/3.Whenm→0, from (54) and (55), the exact solutions of (47) are
(57)U4=1-b2(isin2(ξ)+sin(ξ)cos(ξ)),U5=1-b2(icos2(ξ)-cos(ξ)sin(ξ)),
where ξ=x-(4/3)(4-5b2i)t, μ=-(2/3)i, ν=(2/3)(i+2b2)α=16/3, β=-8/3.WhenP=-m2,Q=2m2-1, and R=1-m2, the solutions of elliptic Equation (5) are F(ξ)=cn(ξ,m) from Table 1, so the exact solution of (47) is
(58)U6=1-b2(2m2-1cn2(ξ,m)&1-b2-cn(ξ,m)dn(ξ,m)sn(ξ,m){2m2-1}),
where ξ=x+(4/3)(5b22m2-1(1-m2)+4(2m2-1))t, μ=-(2/3)2m2-1, ν=(2/3)(2m2-1+2b2(1-m2)), α=(16/3)(1-2m2), β=-8/3.Whenm→1, from (58) the exact solution of (47) is
(59)U7=1-b2(sech2(ξ,m)-sech2(ξ)tanh(ξ)),
where ξ=x+(16/3)t, μ=-2/3, ν=2/3, α=-16/3, β=-8/3.WhenP=1-m2, Q=2-m2, and R=1, the solutions of elliptic Equation (5) are F(ξ)=sc(ξ,m) from Table 1, so the exact solution of (47) is
(60)U8=1-b2(2-m2sc2(ξ,m)1-b2+dc(ξ,m)sc(ξ,m)nc(ξ,m){2-m2}),
where ξ=x+(4/3)(5b22-m2+4(2-m2))t, μ=-(2/3)2-m2, ν=(2/3)(2-m2+2b2),α=(16/3)(m2-2), β=-8/3.Whenm→1, from (60), the exact solution of (47) is
(61)U9=1-b2(sinh2(ξ)+sinh(ξ)cosh(ξ)),
where ξ=x+(4/3)(5b2+4)t, μ=-(2/3), ν=(2/3)(1+2b2),  α=-16/3,β=-8/3.Whenm→0, from (60), the exact solution of (47) is
(62)U10=1-b2(2tan2(ξ)+sec2(ξ)tan(ξ)),
where ξ=x+(4/3)(52b2+8)t, μ=-(2/3)2, ν=(2/3)(2+2b2),α=-32/3,  β=-8/3.


## 5. Conclusions

TheF-expansion method and its extended version are very effective in solving various NLEEs. For some NLEEs, the F-expansion method can give nontrivial solutions, for some other NLEEs, the extended version of F-expansion method can give nontrivial solutions, and for some particular NLEEs (especially the complete integrable systems), both F-expansion method and its extended version are feasible for constructing exact solutions.In summary, lots of new exact Jacobian elliptic function solutions and soliton solutions of the Kudryashov-Sinelshchikov equation are proposed by theF-expansion method and its extended version. The results of [21, 22] have been enriched. These exact solutions have been verified by symbolic computation system—Maple. Moreover, the solutions listed in this paper may be of important significance for the explanation of some relevant physical problems. We would like to study the Kudryashov-Sinelshchikov equation further.



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*Source: 895760-2013-04-28.xml*