# Explicit Solutions to the (3+1)-Dimensional Kudryashov-Sinelshchikov Equations in Bubbly Flow Dynamics

Authors: Y. B. Chukkol; M. N. B. Mohamad; Mukhiddin Muminov
Journal: Journal of Applied Mathematics
 (2018)

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## Abstract

A modified tanh-coth method with Riccati equation is used to construct several explicit solutions of (3+1)-dimensional Kudryashov-Sinelshchikov equations in bubble gas liquid flow. The solutions include solitons and periodic solutions. The method applied can be used in further works to obtain entirely new solutions to many other nonlinear evolution equations.


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## Body

## 1. Introduction

Many complex phenomena in nature are describe in various scientific and industrial fields and especially in areas of physics such as fluid mechanics [1], optical fibres [2], plasma physics [3], and so on. Nonlinear evolution equations are used to describe these nonlinear phenomena, and that has led to the development of methods to look for exact solutions of nonlinear partial differential equations [4].Recently, Kudryashov and Sinelshchikov [5] developed a (3+1)-dimensional nonlinear evolution equation in a model of wave propagation in bubbly fluid flow. The model equation has gained a lot of attention where BÃ¤cklund transformation and conservation laws [6], bifurcation [7], and density-fluctuation [8] analysis were carried out the important evolution equation.Many methods have been developed to find the explicit solutions of nonlinear evolution equations; example of such methods are the first integral method [9], Jacobi elliptic function method [10], Hirota bilinear method [11], Wronskian determinant technique [12], F-expansion method [13], Darboux Transformations [14], Backlund transformation method [6], Miura transformation [15], homotopy perturbation method [16], and Adomian decomposition method [17]. Many algebraic methods were proposed so far, such as tanh method which was proposed by Malflie [18]. Fan [19] extended the tanh method and obtained new exact solution that cannot be obtained by using the conventional tanh method. Further extension called tanh-coth was proposed by Wazwaz [4] which provides a wider applicability for solving nonlinear evolution equations. A modification was also proposed by El-Wakil [20] and Soliman [21, 22].In this paper, we present two equations (3+1)-dimensional Kudryashov and Sinelshchikov equation written as(1)ut+αuux+γuxxxx+duyy+euzz=0,ut+αuux-βuxx+γuxxxx+duyy+euzz=0where α, β, and γ represent the nonlinearity, dissipation dispersion terms, while d and e stand for transverse variation of wave in y and z directions; we assume all the coefficient to be constant parameters. We shall use modified tanh-coth to obtain many explicit exact solutions for (3+1)-dimensional Kudryashov-Sinelshchikov equations. The travelling wave solution of a special case of (1) is given in [7]. Note that when e=0, (1) reduces to their two-dimensional counterparts(2)ut+αuux+γuxxxx+duyy=0,ut+αuux-βuxx+γuxxxx+duyy=0,which were widely studied in [16, 17, 23, 24]. Furthermore when d=e=0, (1) reduces to one-dimensional Korteweg-de-Vries equations (KdV) and one-dimensional Korteweg-de-Vries-Burgers equations (KdVB). These equations have many applications in fluid dynamics [25], plasma physics [26], cold atomic gases [27], and so on. It is well known that the KdV and KdVB equations take the form(3)ut+αuux+γuxxx=0,ut+αuux-βuxx+γuxxx=0.These two equations have been extensively investigated by many researcher in recent years [28–32].In the following section, we briefly describe the modified tanh-coth method in three variables and the application of the modified tanh-coth to (3+1)-dimensional Kudryashov and Sinelshchikov equations and their explicit solutions will be given in Section3. Conclusions will follow in Section 4.


## 2. Description of the Method

Let us consider a three-dimensional nonlinear partial differential equation in the form(4)ut=Fu,ux,uy,uz,uxx,uyy,uzz,….Upon using the transformation u(x,y,z,t)=u(ξ), where ξ=μ(x+y+z-ct), (4) can be reduced to an ordinary differential equation (ODE)(5)-cμu′ξ=Fuξ,μu′ξ,μ2u′′ξ,μ3u′′′ξ,…,where the independent variable composed of the new variable ξ=μ(x+y+z-ct), μ is the wave number, c is wave speed, x,y,z are the spatial coordinates, and t is time. The ODE (5) is then integrated as long as all terms contain derivatives, where integration constants are considered zero. The simplified ODE is then solved via the tanh-coth method [28], which admits the use of finite expansion(6)uY=∑i=0maiYi+∑i=1mbiY-iwith the Riccati equation(7)Y′=P+QY+RY2.By changing of variable(8)ddξ=P+QY+RY2ddYd2dξ2=P+QY+RY2Q+2RYddY+P+QY+RY2d2dY2,where P, Q, and R shall be given, while ai, bi are constants to be determined later. The positive integer m can be determined by considering the homogeneous balance [33] between the highest order derivatives and the most nonlinear terms appearing in (5). If m is not an integer, then a transformation formula should be used to overcome the difficulty. Substituting (6) into (5) and make use of (7) and (8) yield an equation in terms of Yi. Equating the coefficients of all the power Yi to zero, we obtain a set of algebraic equations for P,Q,R,ai,bi,c, and μ.For the aforementioned method, expansion (8) reduces to the standard tanh-coth method. In this work, we shall use the following solutions of Riccati equation:P=Q=1 and R=0, then Y=exp(ξ)-1,P=1/2,Q=0,R=-1/2, then Y=coth⁡ξ±csch⁡ξ.


## 3. Application of the Method

In this section, we will apply the method to the (3+1)-dimensional Kudryashov-Sinelshchikov equation for both dispersive and dissipative cases.


### 3.1. The (3+1)-Dimensional Kudryashov-Sinelshchikov Equation with Dispersion

The generalised equation is given as(9)ut+αuux+γuxxxx+duyy+euzz=0.By using the transformation(10)ux,y,z,t=uξ,ξ=μx+y+z-ct,(9) is reduced to an ODE of the form(11)μ-cμu′+αμuu′+γμ3u′′′′+dμ2u′′+eμ2u′′=0.Integrating (11) twice and assuming the integration constant to be zero, we obtain(12)-c+d+eu+α2u2+γμ2u′′=0.Balancing terms u2 and u′′ in (12), we get m=2, which enables us to make the following ansatz:(13)uξ=a0+a1Y+a2Y2+b1Y+b2Y2.Substituting (13) into (12) and making use of (7) and (8), we obtain a system of equation in terms of ai(i=0,1,2), bi(i=1,2), α,γ,d,ec,P,Q,R, and μ.(14)12γR2μ2a2+αa22=0,4γa1R2μ2+20γQa2Rμ2+2αa1a2=0,8γa2Q2μ2+6γRQμ2a1+16γPRa2μ2+αa12+2da2+2ea2-2ca2+2αa0a2=0,2γa1Q2μ2+12γPa2Qμ2+4γPRa1μ2+2da1+2ea1-2ca1+2αa0a1+2αa2b1=0,2da0+2ea0-2ca0+αa02+2αa1b1+2αa2b2+4γP2μ2a2+4γR2μ2b2+2γPQμ2a1+2γQRμ2b1=0,2γb1Q2μ2+12γRb2Qμ2+4γPRb1μ2+2db1+2eb1-2cb1+2αa0b1+2αa1b2=0,8γb2Q2μ2+6γPQμ2b1+16γPRb2μ2+αb12+2db2+2eb2-2cb2+2αa0b2=0,4γb1P2μ2+20γQb2Pμ2+2αb1b2=0,12γP2μ2b2+αb22=0.Case I. If P=1,Q=1,R=0 are substituted into the system of (14) and solving the equations, we have the following solutions:(15a)a0=2c-d-eα,a1=a2=0,b1=b2=6a0,μ=-γc-d-eγ,(15b)a0=a1=a2=0,b1=b2=-12c-d-eα,μ=γc-d-eγ.Substituting (15a) and (15b) into (13) using Y=exp(ξ)-1,γ>0, with Ω=x+y+z-ct, we get the solutions in simple form as(16)u1x,y,z,t=2c-d-eα+3c-d-eαsinh2-γc-d-e/γΩ/2,c<d+e(17)u2x,y,z,t=-3c-d-eαsinh2γc-d-e/γΩ/2,c>d+e.Case II. If we set P=1/2,Q=0,R=-1/2 into the system of (14) and solve the equations, we have the following solutions:(18a)a0=-c-d-eα,a1=0,a2=-3a0,b1=b2=0,μ=-γc-d-eγ,(18b)a0=3c-d-eα,a1=0,a2=-3a0,b1=b2=0,μ=γc-d-eγ,(18c)a0=-c-d-eα,a1=a2=0,b1=0,b2=-3a0,μ=-γc-d-eγ,(18d)a0=3c-d-eα,a1=a2=0,b1=0,b2=-a0,μ=γc-d-eγ,(18e)a0=c-d-e2α,a1=b1=0,a2=b2=3c-d-e4α,μ=-γc-d-e2γ,(18f)a0=3c-d-e2α,a1=b1=0,a2=b2=-3c-d-e4α,μ=γc-d-e2γ.SubstitutingY=coth⁡ξ±csch⁡ξ,γ>0,Ω=x+y+z-ct into (13), making use of (18a), we obtain a soliton solution(19)u3x,y,z,t=2c-d-eα-3c-d-eαcosh2-γc-d-e/γΩ/2,c<d+eand periodic solutions(20)u4x,y,z,t=2c-d-eα-3c-d-eαcos2γc-d-e/γΩ/2,c>d+e(21)u5x,y,z,t=2c-d-eα-6c-d-eαcosγc-d-e/γΩ/2+sinγc-d-e/γΩ/22,where c>d+e. Using (18b), we have a soliton solution, written in simple form as(22)u6x,y,z,t=3c-d-eαcosh2γc-d-e/γΩ/2,c>d+eand periodic solutions(23)u7x,y,z,t=3c-d-eαcos2-γc-d-e/γΩ/2,c<d+e(24)u8x,y,z,t=6c-d-eαcos-γc-d-e/γΩ/2+sin-γc-d-e/γΩ/22,c<d+e.Equation (18c) gives a soliton solution(25)u9x,y,z,t=2c-d-eα+3c-d-eαsinh2-γc-d-e/γΩ/2,c<d+eand periodic solutions(26)u10x,y,z,t=2c-d-eα-3c-d-eαsin2γc-d-e/γΩ/2,c>d+e(27)u11x,y,z,t=2c-d-eα-6c-d-eαcosγc-d-e/γΩ/2-sinγc-d-e/γΩ/22,where c>d+e. Equation (18d) gives a soliton solution(28)u12x,y,z,t=-3c-d-eαsinh2γc-d-e/γΩ/2,c>d+eand periodic solutions(29)u13x,y,z,t=3c-d-eαsin2-γc-d-e/γΩ/2,c<d+e(30)u14x,y,z,t=6c-d-eαcos-γc-d-e/γΩ/2+sin-γc-d-e/γΩ/22,c<d+e.Equation (18e) gives soliton solutions(31)u15x,y,z,t=2c-d-eα+3c-d-eαsinh2-γc-d-e/γΩ/2,c<d+e(32)u16x,y,z,t=2c-d-eα-3c-d-eαcosh2-γc-d-e/γΩ/2,c<d+eand periodic solutions(33)u17x,y,z,t=2c-d-eα-3c-d-eαsin2γc-d-e/γΩ/2,c>d+e(34)u18x,y,z,t=2c-d-eα-3c-d-eαcos2γc-d-e/γΩ/2,c>d+e.Finally (18f) gives soliton solutions(35)u19x,y,z,t=-3c-d-eαsinh2γc-d-e/γΩ/2,c>d+e(36)u20x,y,z,t=3c-d-eαcosh2γc-d-e/γΩ/2,c>d+eand periodic solutions(37)u21x,y,z,t=3c-d-eαsin2-γc-d-e/γΩ/2,c<d+e(38)u22x,y,z,t=3c-d-eαcos2-γc-d-e/γΩ/2,c<d+e.The graphical representation of the solitary wave profile to the solutionu3(x,y,z,t) of (9) at t=0 is given in Figure 1, while for t=0.5 is given in Figure 2.Figure 1
Solitary wave profile ofu3, α=1, d=0.5, e=0.5, c=0.6, x=0, and t=0.Figure 2
Solitary wave profile ofu3, α=1, d=0.5, e=0.5, c=0.6, x=0, and t=0.5.


### 3.2. The (3+1)-Dimensional Kudryashov-Sinelshchikov Equation with Dispersion and Dissipation

The equation is given as(39)ut+αuux-βuxx+γuxxxx+duyy+euzz=0.By using the transformation(40)ux,y,z,t=uξ,ξ=μx+y+z-ct,(39) is reduced to an ODE of the form(41)μ-cμu′+αμuu′-βμ2u′′+γμ3u′′′′+dμ2u′′+eμ2u′′=0.Integrating (41) twice and assuming the integration constant to be zero, we obtain(42)-c+d+eu+α2u2-βμu′+γμ2u′′=0.Balancing u2 and u′′ in (42), we get m=2, which enables us to make the following ansatz:(43)uξ=a0+a1Y+a2Y2+b1Y+b2Y2.Substituting (43) into (42) and making use of (7) and (8), we obtained a system of equations in ai(i=0,1,2),bi(i=1,2),c,P,Q,R, α,β,γ, and μ.(44)12γR2μ2a2+αa22=0,4γa1R2μ2+20γQa2Rμ2-4βa2Rμ+2αa1a2=0,8γa2Q2μ2+6γRQμ2a1-4βa2Qμ+16γPRa2μ2-2βRμa1+αa12+2da2+2ea2-2ca2+2αa0a2=0,2γa1Q2μ2+12γPa2Qμ2-2βa1Qμ+4γARa1μ2-4βPa2μ+2da1+2ea1-2ca1+2αa0a1+2αa2b1=0,2da0+2ea0-2wa0+αa02+2αa1b1+2αa2b2+4γP2μ2a2+4γR2μ2b2-2βPμa1+2βRμb1+2γPQμ2a1+2γQRμ2b1=0,2γb1Q2μ2+12γRb2Qμ2+2βb1Qμ+4γPRb1μ2+4βRb2μ+2db1+2eb1-2cb1+2αa0b1+2αa1b2=0,8γb2Q2μ2+6γPQμ2b1+4βb2Qμ+16γPQb2μ2+2βPμb1+αb12+2db2+2eb2-2cb2+2αa0b2=0,4γb1P2μ2+20γQb2Pμ2+4βb2Pμ+2αb1b2=0,12γP2μ2b2+αb22=0.Case I. If P=1,Q=1,R=0 are used in the system (44) and solving the equations, we have obtained the following solutions:(45a)a0=a1=a2=0,b1=-24β225γα,b2=12b1,μ=β5γ,c=6β2+25γd+25γe25γ,(45b)a0=b2=-12β225γα,a1=a2=0,b1=2b2,μ=β5γ,c=25γd+25γe-6β225γ.Substituting (45a) and (45b) into (43) and using Y=exp(ξ)-1, we get(46)u1x,y,z,t=12β225γα1-exp⁡-ξ-1-2,ξ=β5γx+y+z-25γd+25γe+6β225γt,and(47)u2x,y,z,t=-12β225γαexp⁡-ξ-1-2,ξ=β5γx+y+z-25γd+25γe-6β225γt.Case II. If we set P=1/2, Q=0, and R=-1/2 into system (44) and solve the equation, we have(48a)a0=a2=-3β225γα,a1=2a0=,b1=b2=0,μ=β5γ,c=25γd+25γe-6β225γ,(48b)a0=9β225γα,a1=-2a03,a2=-a03,b1=b2=0,μ=β5γ,c=25γd+25γe+6β225γ,(48c)a0=b2=-3β225γα,a1=a2=0,b1=2b2,μ=β5γ,c=25γd+25γe-6β225γ,(48d)a0=9β225γα,a1=a2=0,b1=-2a03,b2=-a03,μ=β5γ,c=25γd+25γe+6β225γ,(48e)a0=-9β250γα,a1=b1=2a03,a2=b2=a06,μ=q10γ,c=25γd+25γe-6β225γ,(48f)a0=3β210γα,a1=b1=-2a05,a2=b2=-a010,μ=β10γ,c=25γd+25γe+6q225γ.It can be shown that ifY=coth⁡ξ±csch⁡ξ, the set of values in (48a), (48c), and (48e) give the same solution as(49)u3,4x,y,z,t=-3β225γα-6β225γαcoth⁡ξ±csch⁡ξ-3β225γαcoth⁡ξ±csch⁡ξ2=-12β225γαexp⁡-ξ∓1-2,ξ=β5γx+y+z-25γd+25γe-6β225γt.The set values in (48b), (48d), and (48f) give exactly the same solution as(50)u5,6x,y,z,t=9β225γα-6β225γαcoth⁡ξ±csch⁡ξ-3β225γαcoth⁡ξ±csch⁡ξ2=12β225γα1-exp⁡-ξ∓1-2,ξ=β5γx+y+z-25γd+25γe+6β225γt.We useu5,6(x,y,z,t) to represent the solution to (39) graphically; Figure 3 gives a kink wave profile at t=0, while at t=0.5 it is given in Figure 4.Figure 3
Kink wave profile ofu5,6, α=1, γ=1, d=0.5, e=0.5, β=4, x=0, and t=0.Figure 4
Kink wave profile ofu5,6, α=1, γ=1, d=0.5, e=0.5, β=4, x=0, and t=0.5.


## 3.1. The (3+1)-Dimensional Kudryashov-Sinelshchikov Equation with Dispersion

The generalised equation is given as(9)ut+αuux+γuxxxx+duyy+euzz=0.By using the transformation(10)ux,y,z,t=uξ,ξ=μx+y+z-ct,(9) is reduced to an ODE of the form(11)μ-cμu′+αμuu′+γμ3u′′′′+dμ2u′′+eμ2u′′=0.Integrating (11) twice and assuming the integration constant to be zero, we obtain(12)-c+d+eu+α2u2+γμ2u′′=0.Balancing terms u2 and u′′ in (12), we get m=2, which enables us to make the following ansatz:(13)uξ=a0+a1Y+a2Y2+b1Y+b2Y2.Substituting (13) into (12) and making use of (7) and (8), we obtain a system of equation in terms of ai(i=0,1,2), bi(i=1,2), α,γ,d,ec,P,Q,R, and μ.(14)12γR2μ2a2+αa22=0,4γa1R2μ2+20γQa2Rμ2+2αa1a2=0,8γa2Q2μ2+6γRQμ2a1+16γPRa2μ2+αa12+2da2+2ea2-2ca2+2αa0a2=0,2γa1Q2μ2+12γPa2Qμ2+4γPRa1μ2+2da1+2ea1-2ca1+2αa0a1+2αa2b1=0,2da0+2ea0-2ca0+αa02+2αa1b1+2αa2b2+4γP2μ2a2+4γR2μ2b2+2γPQμ2a1+2γQRμ2b1=0,2γb1Q2μ2+12γRb2Qμ2+4γPRb1μ2+2db1+2eb1-2cb1+2αa0b1+2αa1b2=0,8γb2Q2μ2+6γPQμ2b1+16γPRb2μ2+αb12+2db2+2eb2-2cb2+2αa0b2=0,4γb1P2μ2+20γQb2Pμ2+2αb1b2=0,12γP2μ2b2+αb22=0.Case I. If P=1,Q=1,R=0 are substituted into the system of (14) and solving the equations, we have the following solutions:(15a)a0=2c-d-eα,a1=a2=0,b1=b2=6a0,μ=-γc-d-eγ,(15b)a0=a1=a2=0,b1=b2=-12c-d-eα,μ=γc-d-eγ.Substituting (15a) and (15b) into (13) using Y=exp(ξ)-1,γ>0, with Ω=x+y+z-ct, we get the solutions in simple form as(16)u1x,y,z,t=2c-d-eα+3c-d-eαsinh2-γc-d-e/γΩ/2,c<d+e(17)u2x,y,z,t=-3c-d-eαsinh2γc-d-e/γΩ/2,c>d+e.Case II. If we set P=1/2,Q=0,R=-1/2 into the system of (14) and solve the equations, we have the following solutions:(18a)a0=-c-d-eα,a1=0,a2=-3a0,b1=b2=0,μ=-γc-d-eγ,(18b)a0=3c-d-eα,a1=0,a2=-3a0,b1=b2=0,μ=γc-d-eγ,(18c)a0=-c-d-eα,a1=a2=0,b1=0,b2=-3a0,μ=-γc-d-eγ,(18d)a0=3c-d-eα,a1=a2=0,b1=0,b2=-a0,μ=γc-d-eγ,(18e)a0=c-d-e2α,a1=b1=0,a2=b2=3c-d-e4α,μ=-γc-d-e2γ,(18f)a0=3c-d-e2α,a1=b1=0,a2=b2=-3c-d-e4α,μ=γc-d-e2γ.SubstitutingY=coth⁡ξ±csch⁡ξ,γ>0,Ω=x+y+z-ct into (13), making use of (18a), we obtain a soliton solution(19)u3x,y,z,t=2c-d-eα-3c-d-eαcosh2-γc-d-e/γΩ/2,c<d+eand periodic solutions(20)u4x,y,z,t=2c-d-eα-3c-d-eαcos2γc-d-e/γΩ/2,c>d+e(21)u5x,y,z,t=2c-d-eα-6c-d-eαcosγc-d-e/γΩ/2+sinγc-d-e/γΩ/22,where c>d+e. Using (18b), we have a soliton solution, written in simple form as(22)u6x,y,z,t=3c-d-eαcosh2γc-d-e/γΩ/2,c>d+eand periodic solutions(23)u7x,y,z,t=3c-d-eαcos2-γc-d-e/γΩ/2,c<d+e(24)u8x,y,z,t=6c-d-eαcos-γc-d-e/γΩ/2+sin-γc-d-e/γΩ/22,c<d+e.Equation (18c) gives a soliton solution(25)u9x,y,z,t=2c-d-eα+3c-d-eαsinh2-γc-d-e/γΩ/2,c<d+eand periodic solutions(26)u10x,y,z,t=2c-d-eα-3c-d-eαsin2γc-d-e/γΩ/2,c>d+e(27)u11x,y,z,t=2c-d-eα-6c-d-eαcosγc-d-e/γΩ/2-sinγc-d-e/γΩ/22,where c>d+e. Equation (18d) gives a soliton solution(28)u12x,y,z,t=-3c-d-eαsinh2γc-d-e/γΩ/2,c>d+eand periodic solutions(29)u13x,y,z,t=3c-d-eαsin2-γc-d-e/γΩ/2,c<d+e(30)u14x,y,z,t=6c-d-eαcos-γc-d-e/γΩ/2+sin-γc-d-e/γΩ/22,c<d+e.Equation (18e) gives soliton solutions(31)u15x,y,z,t=2c-d-eα+3c-d-eαsinh2-γc-d-e/γΩ/2,c<d+e(32)u16x,y,z,t=2c-d-eα-3c-d-eαcosh2-γc-d-e/γΩ/2,c<d+eand periodic solutions(33)u17x,y,z,t=2c-d-eα-3c-d-eαsin2γc-d-e/γΩ/2,c>d+e(34)u18x,y,z,t=2c-d-eα-3c-d-eαcos2γc-d-e/γΩ/2,c>d+e.Finally (18f) gives soliton solutions(35)u19x,y,z,t=-3c-d-eαsinh2γc-d-e/γΩ/2,c>d+e(36)u20x,y,z,t=3c-d-eαcosh2γc-d-e/γΩ/2,c>d+eand periodic solutions(37)u21x,y,z,t=3c-d-eαsin2-γc-d-e/γΩ/2,c<d+e(38)u22x,y,z,t=3c-d-eαcos2-γc-d-e/γΩ/2,c<d+e.The graphical representation of the solitary wave profile to the solutionu3(x,y,z,t) of (9) at t=0 is given in Figure 1, while for t=0.5 is given in Figure 2.Figure 1
Solitary wave profile ofu3, α=1, d=0.5, e=0.5, c=0.6, x=0, and t=0.Figure 2
Solitary wave profile ofu3, α=1, d=0.5, e=0.5, c=0.6, x=0, and t=0.5.


## 3.2. The (3+1)-Dimensional Kudryashov-Sinelshchikov Equation with Dispersion and Dissipation

The equation is given as(39)ut+αuux-βuxx+γuxxxx+duyy+euzz=0.By using the transformation(40)ux,y,z,t=uξ,ξ=μx+y+z-ct,(39) is reduced to an ODE of the form(41)μ-cμu′+αμuu′-βμ2u′′+γμ3u′′′′+dμ2u′′+eμ2u′′=0.Integrating (41) twice and assuming the integration constant to be zero, we obtain(42)-c+d+eu+α2u2-βμu′+γμ2u′′=0.Balancing u2 and u′′ in (42), we get m=2, which enables us to make the following ansatz:(43)uξ=a0+a1Y+a2Y2+b1Y+b2Y2.Substituting (43) into (42) and making use of (7) and (8), we obtained a system of equations in ai(i=0,1,2),bi(i=1,2),c,P,Q,R, α,β,γ, and μ.(44)12γR2μ2a2+αa22=0,4γa1R2μ2+20γQa2Rμ2-4βa2Rμ+2αa1a2=0,8γa2Q2μ2+6γRQμ2a1-4βa2Qμ+16γPRa2μ2-2βRμa1+αa12+2da2+2ea2-2ca2+2αa0a2=0,2γa1Q2μ2+12γPa2Qμ2-2βa1Qμ+4γARa1μ2-4βPa2μ+2da1+2ea1-2ca1+2αa0a1+2αa2b1=0,2da0+2ea0-2wa0+αa02+2αa1b1+2αa2b2+4γP2μ2a2+4γR2μ2b2-2βPμa1+2βRμb1+2γPQμ2a1+2γQRμ2b1=0,2γb1Q2μ2+12γRb2Qμ2+2βb1Qμ+4γPRb1μ2+4βRb2μ+2db1+2eb1-2cb1+2αa0b1+2αa1b2=0,8γb2Q2μ2+6γPQμ2b1+4βb2Qμ+16γPQb2μ2+2βPμb1+αb12+2db2+2eb2-2cb2+2αa0b2=0,4γb1P2μ2+20γQb2Pμ2+4βb2Pμ+2αb1b2=0,12γP2μ2b2+αb22=0.Case I. If P=1,Q=1,R=0 are used in the system (44) and solving the equations, we have obtained the following solutions:(45a)a0=a1=a2=0,b1=-24β225γα,b2=12b1,μ=β5γ,c=6β2+25γd+25γe25γ,(45b)a0=b2=-12β225γα,a1=a2=0,b1=2b2,μ=β5γ,c=25γd+25γe-6β225γ.Substituting (45a) and (45b) into (43) and using Y=exp(ξ)-1, we get(46)u1x,y,z,t=12β225γα1-exp⁡-ξ-1-2,ξ=β5γx+y+z-25γd+25γe+6β225γt,and(47)u2x,y,z,t=-12β225γαexp⁡-ξ-1-2,ξ=β5γx+y+z-25γd+25γe-6β225γt.Case II. If we set P=1/2, Q=0, and R=-1/2 into system (44) and solve the equation, we have(48a)a0=a2=-3β225γα,a1=2a0=,b1=b2=0,μ=β5γ,c=25γd+25γe-6β225γ,(48b)a0=9β225γα,a1=-2a03,a2=-a03,b1=b2=0,μ=β5γ,c=25γd+25γe+6β225γ,(48c)a0=b2=-3β225γα,a1=a2=0,b1=2b2,μ=β5γ,c=25γd+25γe-6β225γ,(48d)a0=9β225γα,a1=a2=0,b1=-2a03,b2=-a03,μ=β5γ,c=25γd+25γe+6β225γ,(48e)a0=-9β250γα,a1=b1=2a03,a2=b2=a06,μ=q10γ,c=25γd+25γe-6β225γ,(48f)a0=3β210γα,a1=b1=-2a05,a2=b2=-a010,μ=β10γ,c=25γd+25γe+6q225γ.It can be shown that ifY=coth⁡ξ±csch⁡ξ, the set of values in (48a), (48c), and (48e) give the same solution as(49)u3,4x,y,z,t=-3β225γα-6β225γαcoth⁡ξ±csch⁡ξ-3β225γαcoth⁡ξ±csch⁡ξ2=-12β225γαexp⁡-ξ∓1-2,ξ=β5γx+y+z-25γd+25γe-6β225γt.The set values in (48b), (48d), and (48f) give exactly the same solution as(50)u5,6x,y,z,t=9β225γα-6β225γαcoth⁡ξ±csch⁡ξ-3β225γαcoth⁡ξ±csch⁡ξ2=12β225γα1-exp⁡-ξ∓1-2,ξ=β5γx+y+z-25γd+25γe+6β225γt.We useu5,6(x,y,z,t) to represent the solution to (39) graphically; Figure 3 gives a kink wave profile at t=0, while at t=0.5 it is given in Figure 4.Figure 3
Kink wave profile ofu5,6, α=1, γ=1, d=0.5, e=0.5, β=4, x=0, and t=0.Figure 4
Kink wave profile ofu5,6, α=1, γ=1, d=0.5, e=0.5, β=4, x=0, and t=0.5.


## 4. Conclusion

In this paper, the modified tanh-coth method has been applied to nonlinear evolution equations. The explicit travelling wave solutions of the generalised (3+1)-dimensional Kudryashov-Sinelshchikov equation are derived. Some of the solutions obtained are in good agreement with the known exact solutions [7] using specific parameter values. The results show that the modified tanh-coth method is an efficient approach for the seeking solutions of such type of nonlinear evolution equations.



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