# The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics

Authors: E. M. E. Zayed; K. A. E. Alurrfi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.


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## Body

## 1. Introduction

In the recent years, investigations of exact solutions to nonlinear partial differential equations (NPDEs) play an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and geochemistry. To obtain traveling wave solutions, many powerful methods have been presented, such as the inverse scattering method [1], the tanh-function method [2–8], the Hirota bilinear transform method [9], the truncated Painleve expansion method [10–13], the Backlund transform method [14, 15], the Exp-function method [16–20], the Jacobi elliptic function expansion method [21–23], the generalized Riccati equations method [24–26], the (G′/G)-expansion method [27–33], and the (G′/G,1/G)-expansion method [34–36]. Conte and Musette [37] presented an indirect method to seek more solitary wave solutions of some NPDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [38]. Using this method, many solitary wave solutions of many NPDEs are found [38, 39]. Recently, Yan [40] developed further Conte and Musette’s method by introducing more generalized projective Riccati equations.In this paper, we will use the generalized projective Riccati equations method to construct exact solutions for the following three nonlinear evolution equations with higher-order nonlinear terms:(i) the nonlinear Pochhammer-Chree equation [41]:
(1)utt-uxxtt-αu+βun+1+γu2n+1xx=0,lllllllllllllllllllllllllllllllllllllllllllllllllllln≥1,
where α, β, and γ are constants and γ<0,(ii) the nonlinear Burgers equation [42]:
(2)ut+aunx+buxx=0,n>1,
where a and b are constants;(iii) the nonlinear generalized Zakharov-Kuznetsov equation [43]:
(3)ut+(Aup+Bu2p)ux+C(uxxx+uxyy)=0,p>0,
where A, B, and C are nonzero real constants.Zuo [32] has applied the extended (G′/G)-expansion method and determined the exact solutions of (1), and Hayek [33] has found the exact solutions of (2) using another form of the extended (G′/G)-expansion method, while Zhang [44] has discussed (3) using an algebraic method to find some of its exact solutions. The rest of this paper is organized as follows. In Section 2, we give the description of the generalized projective Riccati equations method. In Section 3, we apply this method to solve (1)–(3). In Section 4, physical explanations of some obtained results are obtained. In Section 5, some conclusions are given.


## 2. Description of the Generalized Projective Riccati Equations Method

Consider we have the following NPDE:(4)Fu,ut,ux,utt,uxt,uxx,…=0,
where F is a polynomial in u(x,t) and its partial derivatives, in which the highest-order derivatives and nonlinear terms are involved. In the following, we give the main steps of this method.Step 1.
We use the wave transformation(5)u(x,t)=u(ξ),ξ=x-ct,
where c is a constant, to reduce (4) to the following ODE:
(6)P(u,u′,u′′,…)=0,
where P is a polynomial in u(ξ) and its total derivatives, such that ′=d/dξ.Step 2.
We assume that (6) has the formal solution
(7)uξ=A0+∑i=1mσi-1ξAiσξ+Biτξ,
where A0, Ai, and Bi are constants to be determined later. The functions σ(ξ) and τ(ξ) satisfy the ODEs
(8)σ′(ξ)=εσξτξ,τ′(ξ)=R+ετ2(ξ)-μσξ,ε=±1,
where
(9)τ2(ξ)=-εR-2μσ(ξ)+μ2-1Rσ2(ξ),
where R and μ are nonzero constants.
IfR=μ=0, (6) has the formal solution
(10)uξ=∑i=0mAiτiξ,
where τ(ξ) satisfies the ODE
(11)τ′(ξ)=τ2(ξ).Step 3.
We determine the positive integerm in (7) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in (6). In some nonlinear equations the balance number m is not a positive integer. In this case, we make the following transformations.
(a) Whenm=q/p, where q/p is a fraction in the lowest terms, we let
(12)u(ξ)=vq/p(ξ),
and then we substitute (12) into (6) to get a new equation in the new function v(ξ) with a positive integer balance number.
(b) Whenm is a negative number, we let
(13)u(ξ)=vm(ξ),and then we substitute(13) into (6) to get a new equation in the new function v(ξ) with a positive integer balance number.Step 4.
Substitute (7) along with (8)-(9) into (6) or (10) along with (11) into (6). Collect all terms of the same order of σj(ξ)τi(ξ)(j=0,1,…,i=0,1) (or τj(ξ), j=0,1,…). Setting each coefficient to zero yields a set of algebraic equations which can be solved to find the values of A0, Ai, Bi, c, μ, and R.Step 5.
It is well known [24] that (8) admits the following solutions.
(i) Ifε=-1, R≠0,
(14)σ1(ξ)=RsechRξμsech⁡Rξ+1,τ1(ξ)=RtanhRξμsech⁡Rξ+1,σ2(ξ)=RcschRξμcschRξ+1,τ2(ξ)=Rcoth⁡RξμcschRξ+1.
(ii) Ifε=1, R≠0,
(15)σ3(ξ)=RsecRξμsecRξ+1,τ3(ξ)=RtanRξμsecRξ+1,σ4(ξ)=RcscRξμcscRξ+1,τ4(ξ)=-RcotRξμcscRξ+1.
(iii) IfR=μ=0,
(16)σ5ξ=Cξ,τ5(ξ)=1εξ,
where C is nonzero constant.Step 6.
Substituting the values ofA0, Ai, Bi, c, μ, and R as well as the solutions (14)–(16) into (7), we obtain the exact solutions of (4).
We close this section with the remark that without loss of generality we takeε=-1 (similarly the case ε=1 can be done which is omitted here for simplicity).


## 3. Applications

In this section, we will apply the proposed method described in Section2 to find the exact traveling wave solutions of the nonlinear equations (1)–(3).Example 1 (the nonlinear Pochhammer-Chree equation (1)).
In this example, we find the exact solutions of (1). To this end, we see that the traveling wave variable (5) permits us to convert (1) into the following ODE:
(17)c2u′′-c2u′′′′-αu+βun+1+γu2n+1′′=0.
Integrating (17) twice with respect to ξ and vanishing the constants of integration, we get
(18)c2-αu-c2u′′-βun+1-γu2n+1=0.
By balancing u′′ with u2n+1 in (18) we get m=1/n. According to Step 3, we use the transformation
(19)u(ξ)=v1/n(ξ),
where v(ξ) is a new function of ξ. Substituting (19) into (18), we get the new ODE
(20)(c2-α)n2v2-c2nvv′′-c2(1-n)v′2-βn2v3-γn2v4=0.
Balancing vv′′ with v4 in (20), we get m=1. Consequently, we get
(21)v(ξ)=A0+A1σ(ξ)+B1τ(ξ),
where A0, A1, and B1 are constants to be determined later.
Substituting (21) into (20) and using (8)-(9) with ε=-1, the left-hand side of (20) becomes a polynomial in σ and τ. Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:
(22)σ4:-n2γA14R2+Rc2nA12-c2nμ2A12-c2μ2A12hhhhlh+c2A12-6γn2μ2A12B12+6γn2A12B12+2c2nμ2B12-c2nμ4B12-c2nB12-c2μ4B12+2c2μ2B12-c2B12-γn2μ4B14+2γn2μ2B14-γn2B14=0,σ3:2c2μ3B12-2c2μB12+2c2nA0A1+4n2γμ3B14+2Rc2μA12-Rn2βA13-3c2nμB12-4n2γμB14+3n2βA1B12+3c2nμ3B12+12n2γA0A1B12-3n2βμ2A1B12+Rc2nμA12-4Rn2γA0A13-2c2nμ2A0A1+12Rn2γμA12B12-12n2γμ2A0A1B12=0,σ3τ:-2c2nμ2A1B1+2c2nA1B1-2c2μ2A1B1+2c2A1B1-4γn2μ2A1B13-4Rγn2A13B1+4γn2A1B13=0,σ2:R2-c2A12-6γn2A12B12+R3c2nμA0A1+2c2nB12-c2μ2B12hhhhhh-6γn2μ2B14+24γn2μA0A1B12hhhhhh+6βn2μA1B12-6γn2A02A12-3βn2A0A12hhhhhh-αn2A12+2γn2B14+c2n2A12-3c2nμ2B12+c2n2μ2B12-c2n2B12-6γn2μ2A02B12-3βn2μ2A0B12-αn2μ2B12+6γn2A02B12+3βn2A0B12+αn2B12=0,σ2τ:n2βB13+2c2nA0B1-n2βμ2B13+4n2γA0B13+2Rc2μA1B1-4n2γμ2A0B13-3Rn2βA12B1-2c2nμ2A0B1-12Rn2γA0A12B1+2Rc2nμA1B1+8Rn2γμA1B13=0,σ:2c2n2A0A1-2n2αA0A1+2n2αμB12-3n2βA02A1-4n2γA03A1-2c2n2μB12-Rc2nA0A1+12n2γμA02B12+Rc2nμB12+4Rn2γμB14-3Rn2βA1B12+6n2βμA0B12-12Rn2γA0A1B12=0,στ:2c2n2A1B1-2n2αA1B1+2n2βμB13-12n2γA02A1B1-Rc2nA1B1+c2nμA0B1-6n2βA0A1B1-4Rn2γA1B13+8n2γμA0B13=0,τ:2c2n2A0B1-4γn2A03B1-3βn2A02B1-4Rγn2A0B13-2αn2A0B1-Rβn2B13=0,σ0:c2n2A02-n2αA02-n2βA03-n2γA04-Rn2αB12+Rc2n2B12-R2n2γB14-6Rn2γA02B12-3Rn2βA0B12=0.
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.Case 1.
We have(23)A0=0,A1=-β(n+1)(μ2-1)μγR(n+2),B1=0,c=±βnμn+2-n+1μ2-1γR,μ=μ,α=(R-n2)(n+1)(μ2-1)μ2γRn+22.
From (14), (19), (21), and (23), we deduce the following exact solutions:
(24)u11=-β(n+1)(μ2-1)μγ(n+2)sech⁡Rξμsech⁡Rξ+11/n,(25)u12=-β(n+1)(μ2-1)μγ(n+2)cschRξμcschRξ+11/n,
where ξ=x±βn/μ(n+2)-(n+1)(μ2-1)/γRt.Case 2.
We have(26)A0=-βn+12γn+2,A1=0,B1=±βn+12γn+2R,c=±βn2(n+2)-(n+1)γR,μ=0,α=β24R-n2n+14γRn+22.
In this case, we deduce the following exact solutions:
(27)u21=β(n+1)2γ(n+2)-1±tanhRξ1/n,(28)u22=β(n+1)2γ(n+2)-1±coth⁡Rξ1/n,
where ξ=x±βn/2(n+2)-(n+1)/γRt.Case 3.
We have(29)A0=-β(n+1)2γ(n+2),A1=±β(n+1)μ2-12γR(n+2),B1=±β(n+1)2γ(n+2)R,μ=μ,c=±βn(n+2)-(n+1)γR,α=β2(R-n2)(n+1)γRn+22.
In this case, we deduce the following exact solutions:
(30)u31=-1±μ2-1sech⁡⁡Rξ+tanh⁡Rξμsech⁡⁡Rξ+1β(n+1)2γ(n+2)×-1±μ2-1sech⁡⁡Rξ+tanh⁡Rξμsech⁡⁡Rξ+11/n,(31)u32=×-1±μ2-1csch⁡Rξ+coth⁡⁡Rξμcsch⁡Rξ+1β(n+1)2γ(n+2)×-1±μ2-1csch⁡Rξ+coth⁡⁡Rξμcsch⁡Rξ+11/n,
where ξ=x±βn/(n+2)-(n+1)/γRt.Example 2 (the nonlinear Burgers equation (2)).
In this example, we study the Burgers equation with power-law nonlinearity (2). To this end, we see that the traveling wave variable (4) permits us to convert (2) into the following ODE:
(32)-cu′+aun′+bu′′=0.
Integrating (32) once with respect ξ and setting the constant of integration to be zero yield
(33)-cu+aun+bu′=0.
By balancing u′ with un in (33) we get m=1/(n-1). According to Step 3, we use the transformation
(34)u(ξ)=v1/(n-1)(ξ),
where v(ξ) is a new function of ξ. Substituting (34) into (33), we get the new ODE
(35)-c(n-1)v+a(n-1)v2+bv′=0.
Balancing v′ with v2 in (35), we get m=1. Consequently, we get
(36)v(ξ)=A0+A1σ(ξ)+B1τ(ξ),
where A0, A1, and B1 are constants to be determined later.
Substituting (36) into (35) and using (8)-(9) with ε=-1, the left-hand side of (35) becomes a polynomial in σ and τ. Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:
(37)σ2:bB1+aB12-aμ2B12-RaA12-anB12-bμ2B1+anμ2B12+RanA12=0,σ:bμB1-(n-1)cA1+2(n-1)aA0A1-2n-1aμB12=0,στ:2(n-1)aA1B1-bA1=0,τ:2(n-1)aA0B1-(n-1)cB1=0,σ0:(n-1)aA02-(n-1)cA0+(n-1)RaB12=0.
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.Case 1.
We have(38)A0=±bR2a(n-1),A1=±b2a(n-1)μ2-1R,llllllllllllllB1=b2a(n-1),c=±bRn-1.
From (14), (34), (36), and (38), we deduce the following exact solutions:
(39)u11=±1+tanh⁡Rξ±μ2-1sech⁡⁡Rξμsech⁡⁡Rξ+1bR2a(n-1)×±1+tanh⁡Rξ±μ2-1sech⁡⁡Rξμsech⁡⁡Rξ+11/(n-1),(40)u12=±1+coth⁡⁡Rξ±μ2-1csch⁡Rξμcsch⁡Rξ+1bR2an-1×±1+coth⁡⁡Rξ±μ2-1csch⁡Rξμcsch⁡Rξ+11/n-1,
where ξ=x±bR/(n-1)t.Case 2.
We have(41)lllllllllllA0=±bRan-1,A1=0,B1=ban-1,c=±2bRn-1,μ=0.
In this case, we deduce the following exact solutions:
(42)u21=bRa(n-1)tanh(Rξ)±11/(n-1),(43)u22=bRa(n-1)coth⁡(Rξ)±11/(n-1),
where ξ=x±2bR/n-1t.Example 3 (the nonlinear generalized Zakharov-Kuznetsov equation (3)).
In this example, we study the generalized Zakharov-Kuznetsov equation with power-law nonlinearity (3). To this end, we use the traveling wave variable
(44)u(x,y,t)=u(ξ),ξ=x+ky+st,
where k and s are nonzero constants, to reduce (3) to the following ODE:
(45)su′+(Aup+Bu2p)u′+C(1+k2)u′′′=0.
By balancing u′′′ with u2pu′ in (45) we get m=1/p. According to Step 3, we use the transformation
(46)u(ξ)=v1/p(ξ),
where v(ξ) is a new function of ξ. Substituting (46) into (45), we get the new ODE
(47)Cp2(1+k2)v2v′′′+3C(-p2+p-k2p2+k2p)vv′v′′+C(1-3p-3k2p+2p2+k2+2k2p2)v′3+(Av+s+Bv2)p2v2v′=0.
Balancing v2v′′′ with v4v′ in (47), we get m=1. Consequently, we get
(48)v(ξ)=A0+A1σ(ξ)+B1τ(ξ),
where A0, A1, and B1 are constants to be determined later.
Substituting (48) into (47) and using (8)-(9) with ε=-1, the left-hand side of (47) becomes a polynomial in σ and τ. Setting the coefficients of this polynomial to be zero yields a system of algebraic equations in A0, A1, B1, μ, and c, which can be solved using the Maple or Mathematica; we get the following results.Case 1.
We have(49)A0=0,A1=-A2p+1μ2-1μBR(p+2),B1=0,μ=μ,s=A22p+1μ2-1μ2Bp+22p+1,k=±-A2p2(2p+1)(μ2-1)μ2RBCp+22(p+1)-1.
From (14), (46), (48), and (49), we deduce the following exact solutions:
(50)u11=-A(2p+1)(μ2-1)μB(p+2)sech⁡Rξμsech⁡Rξ+11/p,(51)u12=-A(2p+1)(μ2-1)μB(p+2)cschRξμcschRξ+11/p,
where ξ=x+ky+st, k=±-A2p2(2p+1)(μ2-1)/μ2RBC(p+2)2(p+1)-1, s=(A2(2p+1)(μ2-1)/μ2B(p+2)2(p+1)).Case 2.
We have(52)A0=-A(2p+1)2B(p+2),A1=0,B1=±A2p+12Bp+2R,μ=1,s=A2(2p+1)Bp+22(p+1)k=±-A2p2(2p+1)RBCp+22(p+1)-1.
In this case, we have the exact solutions
(53)u21=A(2p+1)2B(p+2)-1±tanhRξsech⁡Rξ+11/p,(54)u22=A(2p+1)2B(p+2)-1±coth⁡RξcschRξ+11/p,
where ξ=x+ky+st, k=±-A2p2(2p+1)/RBC(p+2)2(p+1)-1, s=(A2(2p+1)/B(p+2)2(p+1)).Case 3.
We have(55)A0=-A2p+12Bp+2,A1=0,B1=±A(2p+1)2B(p+2)R,μ=-1,s=A22p+1Bp+22p+1,k=±-A2p2(2p+1)RBCp+22(p+1)-1.
In this case, we have the exact solutions
(56)u31=A(2p+1)2B(p+2)-1±tanhRξ1-sech⁡Rξ1/p,(57)u32=A(2p+1)2B(p+2)-1±coth⁡Rξ1-cschRξ1/p,
where ξ=x+ky+st, k=±-A2p2(2p+1)/RBC(p+2)2(p+1)-1, s=(A2(2p+1)/B(p+2)2(p+1)).Case 4.
We have(58)A0=-A2p+12Bp+2,A1=0,B1=±A(2p+1)2B(p+2)R,μ=0,s=A2(2p+1)Bp+22(p+1)k=±-A2p2(2p+1)4RBCp+22(p+1)-1.
In this case, we have the exact solutions
(59)u41=A(2p+1)2B(p+2)-1±tanhRξ1/p,(60)u42=A(2p+1)2B(p+2)-1±coth⁡Rξ1/p,
where ξ=x+ky+st, k=±-A2p2(2p+1)/4RBC(p+2)2(p+1)-1, s=(A2(2p+1)/B(p+2)2(p+1)).Case 5.
We have(61)A0=-A(2p+1)2B(p+2),A1=±A(2p+1)μ2-12BR(p+2),B1=±A2p+12Bp+2R,μ=μ,s=A22p+1Bp+22p+1,k=±-A2p2(2p+1)RBCp+22(p+1)-1.
In this case, we have the exact solutions
(62)u51=A(2p+1)2B(p+2)-1±μ2-1sech⁡⁡Rξ+tanh⁡Rξμsech⁡⁡Rξ+1×-1±μ2-1sech⁡⁡Rξ+tanh⁡Rξμsech⁡⁡Rξ+11/p,(63)u52=A(2p+1)2B(p+2)-1±μ2-1csch⁡Rξ+coth⁡⁡Rξμcsch⁡Rξ+1×-1±μ2-1csch⁡Rξ+coth⁡⁡Rξμcsch⁡Rξ+11/p,
where ξ=x+ky+st, k=±-A2p2(2p+1)/RBC(p+2)2(p+1)-1, s=A2(2p+1)/B(p+2)2(p+1).


## 4. Physical Explanations of Some Obtained Solutions

In this section, we have presented some graphs of the obtained solutions constructed by taking suitable values of involved unknown parameters to visualize the underlying mechanism of the original equation. Using mathematical software Maple, three-dimensional plots of some obtained exact solutions have been shown in Figures1, 2, 3, 4, 5, 6, 7, and 8.Figure 1
The plot of solution (24) with β=R=1, μ=n=2, and γ=-1.Figure 2
The plot of solution (27) with β=R=n=1, and γ=-1.Figure 3
The plot of solution (30) with β=R=1, μ=n=2, and γ=-1.Figure 4
The plot of solution (31) with β=R=2, μ=3, n=1, and γ=-1.Figure 5
The plot of solution (39) with a=n=μ=2, b=R=1.Figure 6
The plot of solution (42) with a=b=1, R=n=2.Figure 7
The plot of solution (51) with p=1, A=R=2, B=μ=3, and k=0.Figure 8
The plot of solution (60) with p=1, A=62, R=B=3, and k=0.


### 4.1. The Nonlinear Pochhammer-Chree Equation (1)

The obtained solutions for this equation are hyperbolic. From these explicit results it is easy to say that the solution (24) is a bell shaped soliton solution; (25) is a singular bell shaped soliton solution; (27) is a kink shaped soliton solution; (28) is a singular kink shaped soliton solution; (30) is a bell-kink shaped soliton solution; and (31) is a singular bell-kink shaped soliton solution.


### 4.2. The Nonlinear Burgers Equation (2)

From the obtained solutions for the nonlinear Burgers equation (2) we observe that the solution (39) is a bell-kink shaped soliton solution; (40) is a singular bell-kink shaped soliton solution; (42) is a kink shaped soliton solution; and (43) is a singular kink shaped soliton solution.


### 4.3. The Generalized Nonlinear Zakharov-Kuznetsov Equation (3)

From the obtained solutions for the generalized nonlinear Zakharov-Kuznetsov equation (3) we can easily conclude that the solution (50) is a bell shaped soliton solution; (51) is a singular bell shaped soliton solution; (53), (56), and (62) are bell-kink shaped solitons solutions; (54), (57), and (63) are singular bell-kink shaped solitons solutions; (59) is a kink shaped soliton solution; and (60) is a singular kink shaped soliton solution.


## 4.1. The Nonlinear Pochhammer-Chree Equation (1)

The obtained solutions for this equation are hyperbolic. From these explicit results it is easy to say that the solution (24) is a bell shaped soliton solution; (25) is a singular bell shaped soliton solution; (27) is a kink shaped soliton solution; (28) is a singular kink shaped soliton solution; (30) is a bell-kink shaped soliton solution; and (31) is a singular bell-kink shaped soliton solution.


## 4.2. The Nonlinear Burgers Equation (2)

From the obtained solutions for the nonlinear Burgers equation (2) we observe that the solution (39) is a bell-kink shaped soliton solution; (40) is a singular bell-kink shaped soliton solution; (42) is a kink shaped soliton solution; and (43) is a singular kink shaped soliton solution.


## 4.3. The Generalized Nonlinear Zakharov-Kuznetsov Equation (3)

From the obtained solutions for the generalized nonlinear Zakharov-Kuznetsov equation (3) we can easily conclude that the solution (50) is a bell shaped soliton solution; (51) is a singular bell shaped soliton solution; (53), (56), and (62) are bell-kink shaped solitons solutions; (54), (57), and (63) are singular bell-kink shaped solitons solutions; (59) is a kink shaped soliton solution; and (60) is a singular kink shaped soliton solution.


## 5. Conclusions

The generalized projective Riccati equations method is used in this paper to obtain some new exact solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation, and the generalized Zakharov-Kuznetsov equation. On comparing our results in this paper with the well-known results obtained in [32, 33, 41–44] we deduce that our results are different and new and are not published elsewhere. The proposed method of this paper is effective and can be applied to many other nonlinear equations in mathematical physics.



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