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Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method

  • EL Sayed M.E. Zayed EMAIL logo and Abdul-Ghani Al-Nowehy
Published/Copyright: April 28, 2016
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Open Physics
From the journal Open Physics Volume 14 Issue 1

Abstract

In this article, we apply the generalized Kudryashov method for finding exact solutions of three nonlinear partial differential equations (PDEs), namely: the Biswas-Milovic equation with dual-power law nonlinearity; the Zakharov--Kuznetsov equation (ZK(m,n,k)); and the K(m,n) equation with the generalized evolution term. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, and hyperbolic function solutions. Physical explanations for certain solutions of the three nonlinear PDEs are obtained.

Key words: Nonlinear PDEs; Generalized Kudryashov method; Exact solutions; Symmetrical hyperbolic Fibonacci function; The Biswas-Milovic equation; The Zakharov–Kuznetsov equation; The K(m,n) equation
MSC 2010: 02.30 Jr; 02.30.Ik; 05.45.Yv

1 Introduction

The research area of nonlinear partial differential equations (PDEs) has been very active for the past few decades. There are many types of nonlinear PDEs that appear in various areas of the physical and mathematical sciences. Much effort has been expended on constructing exact solutions to these nonlinear PDEs, motivated by their important role in the study of nonlinear physical phenomena. Nonlinear phenomena appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and geo-chemistry. In recent years, a number of powerful and efficient methods for finding analytic solutions to nonlinear equations have drawn a lot of interest by a diverse group of scientists. These include, for example: Hirota’s bilinear transformation method [1, 2]; the tanh-function method [3, 4]; the (G′/G)-expansion method [5–10]; the Exp-function method [11–14]; the multiple exp-function method [15–17]; the symmetry method [18, 19]; the modified simple equation method [20–22]; the improved (G′/G)-expansion method [23]; a multiple extended trial equation method [24]; the Jacobi elliptic function expansion method [25, 26]; the Bäcklund transform method [27, 28]; the generalized Riccati equation method [29]; the modified extended Fan sub equation method [30]; the auxiliary equation method [31, 32]; the first integral method [33, 34]; the modified Kudryashov method [35–42], and the soliton ansatz method [43–70].

The objective of this paper is to apply the generalized Kudryashov method [42] to find the exact solutions of the Biswas-Milovic equation with dual-power law nonlinearity [71, 72], the ZK(m,n,k) [73, 74], and the K(m,n) equation with the generalized evolution term [75].

This paper is organized as follows: in Sec. 2, the description of the generalized Kudryashov method is given. In Sec. 3, we use this method to solve the three aforementioned nonlinear PDEs. In Sec. 4, physical explanations of certain results are presented, and conclusions are discussed in Sec. 5.

2. Description of the generalized Kudryashov method

Starting with a nonlinear PDE in the following form:

(2.1)F(u,ut,ux,utt,uxt,uxx,...)=0,

where u = u(x, t) is an unknown function, F is a polynomial in u = u (x, t) and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved.

The main steps of the generalized Kudryashov method are described as follows:

Step 1. First, we use the wave transformation:

(2.2)u(x,t)=U(ζ),  ζ=kx±λt,

where k and λ are arbitrary constants with k, λ ≠ 0, in order to reduce equation (2.1) into a nonlinear ordinary differential equation (ODE) with respect to the variable ζ of the form

(2.3)H(U,U′,U′′,U′′′,...)=0,

where H is a polynomial in U(ζ) and its total derivatives U′, U″, U‴, ... such that U′=dUdζ, U′′=d2Udζ2 and so on.

Step 2. We assume that the formal solution of the ODE (2.3) can be written in the following rational form:

(2.4)U(ζ)=∑i=0naiQi(ζ)∑j=0mbjQj(ζ)=AQ(ζ)BQ(ζ),

where Q=11±aζ,A[Q(ζ)]=∑i=0naiQi(ζ) and B[Q(ζ)]=∑j=0mbjQj(ζ).. The function Q is the solution of the equation

(2.5)Q′=Q(Q−1)ln(a),  0<a≠1.

Taking into consideration (2.4), we obtain

(2.6)U′(ζ)=Q(Q−1)[A′B−AB′B2]ln(a),
(2.7)U′′(ζ)=QQ−1(2Q−1)A′B−AB′B2ln2(a)+Q2Q−12B(A′′B−AB′′)−2A′B′B+2A(B′)2B3ln2(a),
(2.8)U′′′(ζ)=Q3Q−13ln3(a)×(A′′′B−AB′′′−3A′′B′−3A′B′′)B+6B′(A′B′+AB′′)B3−6A(B′)3B4+3Q2(Q−1)2(2Q−1)×(A′′B−AB′′−2A′B′)B+2A(B′)2B3In3(a)+Q(Q−1)(6Q2−6Q+1)A′B−AB′B2In3(a),

and that similar solutions apply for higher order differentiation terms.

Step 3. Under the terms of the given method, we suppose that the solution of Eq. (2.3) can be written in the following form:

(2.9)U(ζ)=a0+a1Q+a2Q2++anQnb0+b1Q+b2Q2++bmQm.

To calculate the values m and n in (2.9), i.e. the pole order for the general solution of Eq. (2.3), we progress as per the classical Kudryashov method by balancing the highest order nonlinear terms and the highest order derivatives of U() in Eq. (2.3). This allows us to derive a formula for m and n and determine their values.

Step 4. We substitute (2.4) into Eq. (2.3) to get a polynomial R (Q) of Q and equate all the coefficients of Qi, (i = 0, 1, 2, ...) to zero, to yield a system of algebraic equations for ai (i = 0, 1, ..., n) and bj (j = 0, 1, ..., m).

Step 5. We solve the algebraic equations obtained in Step 4 using Mathematica or Maple, to get k , and the coefficients of ai (i = 0, 1, ..., n) and bj (j = 0, 1, ..., m). In this way, we attain the exact solutions to Eq. (2.3).

The obtained solutions depended on the symmetrical hyperbolic Fibonacci functions given in [76]. The symmetrical Fibonacci sine, cosine, tangent, and cotangent functions are, respectively, defined as:

(2.10)sFs(x)=ax−a−x5,cFs(x)=ax+a−x5,tanFs(x)=ax−a−xax+a−x,cotFs(x)=ax+a−xax−a−x,
(2.11)sFs(x)=25sh(x.In(a)),cFs(x)=25ch(x. In(a)),tanFs(x)=tanh(x. In(a)),cotFs(x)=coth(x. In(a)).

3. Applications

In this section we construct the exact solutions in terms of the symmetrical hyperbolic Fibonacci functions of the following three nonlinear PDEs using the generalized Kudryashov method described in Sec. 2:

3.1 Example 1. The Biswas-Milovic equation with dual-power law nonlinearity

In this subsection, we study the Biswas-Milovic equation with dual-power law nonlinearity [71, 72]

(3.1.1)i(qm)t+a(qm)xx+b(|q|2n+k|q|4n)qm=0,

where a, b and k are constants, while m, n are positive integers.

If m = 1, then Eq. (3.1.1) becomes the nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity. In Eq. (3.1.1), the first term is the temporal evolution, while a is the coefficient of group-velocity dispersion (GVD), while b and k are the coefficients of the nonlinear terms. Let us now solve Eq. (3.1.1) using the method of Sec. 2. To this end, we use the wave transformation

(3.1.2)q(x,t)=U(ζ)exp(iθ),  θ=λx−wt,  ζ=μ(x−νt),

where λ, w, μ and v are constants.

Substituting (3.1.2) into (3.1.1), we obtain the nonlinear ODE:

(3.1.3)aμ2(Um)′′+iμ(−ν+2amλ)(Um)′+m(w−amλ2)Um+bU2n+m+bkU4n+m=0.

From Eq. (3.1.3) , we deduce that

(3.1.4)ν=2amλ,

and

(3.1.5)aμ2(Um)′′+m(w−amλ2)Um+bU2n+m+bkU4n+m=0.

Balancing (Um)″ and U4n + m in (3.1.5), then the following relation is attained:

(3.1.6)mN+2=(4n+m)N⇒N=12n.

Then we take into consideration the transformation

(3.1.7)U(ζ)=[v(ζ)]12n.

Substituting (3.1.7) into equation (3.1.5) we have the new equation

(3.1.8)2amnμ2vv′′+amμ2(m−2n)v′2+4mn2(w−amλ2)v2+4bn2v3+4bkn2v4=0.

Balancing the vv″ and v4 in (3.1.8), then the following relation is obtained:

(3.1.9)(N−M)+(N−M)+2=4(N−M)⇒N=M+1.

If we choose M = 1 and N = 2, then the formal solution of Eq. (3.1.8) has the form:

(3.1.10)v(ζ)=a0+a1Q+a2Q2b0+b1Q,

and consequently

(3.1.11)v′(ζ)=Q(Q−1)(b0+b1Q)2[(a1+2a2Q)(b0+b1Q)−b1(a0+a1Q+a2Q2)]ln(a),
(3.1.12)v′′(ζ)=Q(Q−1)(2Q−1)(b0+b1Q)2[(a1+2a2Q)(b0+b1Q)−b1(a0+a1Q+a2Q2)]ln2(a)+Q2(Q−1)2ln2(a)(b0+b1Q)3×[2a2(b0+b1Q)2−2b1(a1+2a2Q)(b0+b1Q)+2b12(a0+a1Q+a2Q2)].

Substituting (3.1.10), (3.1.11) and (3.1.12) into (3.1.8), collecting the coefficients of each power of Qi, (i = 0, 1, ..., 8) and setting each of the coefficients to zero, we obtain the following system of algebraic equations:

Q8:am2a22b12μ2ln2(a)+2amna22b12μ2ln2(a)  +4bkn2a24=0,
Q7:−2amna22b12μ2ln2(a)+16bkn2a1a23 +4am2a22b0b1μ2ln2(a)+4amna22b0b1μ2ln2(a) +4amna2b12a1μ2ln2(a)−2am2a22b12μ2ln2(a) +4bn2a23b1=0,
Q6:12bn2a1a22b1+4amna22b02μ2ln2(a) −2amna22b0b1μ2ln2(a)+2am2a1b0a2b1μ2ln2(a) −4n2m2aλ2a22b12−2am2a2b12a0μ2ln2(a) −8am2a22b0b1μ2ln2(a)+24bkn2a12a22 +8amna1b0b1a2μ2ln2(a)+4bn2a23b0 −6amna2b12a1μ2ln2(a)+4am2a22b02μ2ln2(a) +am2a22b12μ2ln2(a)+16bkn2a0a23+4n2mwa22b12 +8amna2b12a0μ2ln2(a)=0,
Q5:12bn2a12a2b1+48bkn2a0a1a22 +16amna2b0b1a0μ2ln2(a)+8n2mwa22b0b1 +12bn2a0a22b1+4am2a22b0b1μ2ln2(a) −8amna1b0b1a2μ2ln2(a)−2amna22b0b1μ2ln2(a) +12bn2a1a22b0−4am2a1b0a2b1μ2ln2(a) −16amna2b12a0μln2(a)+4am2a1b02a2μ2ln2(a) −4am2a2b0b1a0μ2ln2(a)+8amna2b02a1μ2ln2(a)  +16bkn2a13a2+8n2mwa1a2b12 +2amna2b12a1μ2ln2(a)+4am2a2b12a0μ2ln2(a) −8n2m2aλ2a22b0b1−8n2m2aλ2a1a2b12 −4amna22b02μ2ln2(a)−8am2a22b02μ2ln2(a)=0,
Q4:−8n2m2aλ2a0a2b12−2am2a2b12a0+16n2mwa1a2b0b1+2am2a1b0a2b1μ2ln2(a)+4bkn2a14+8amna2b12a0μ2ln2(a)−4n2m2aλ2a12b12−4n2m2aλ2a22b02+8n2mwa0a2b12+24bn2a0a1a2b1+48bkn2a0a12a2+am2a12b02μ2ln2(a)+4bn2a13b1+2amna12b02μ2ln2(a)+8am2a2b0b1a0μ2ln2(a)−2amnb12a02μ2ln2(a)+4n2mwa12b12+4n2mwa22b02+12bn2a0a22b0−16n2m2aλ2a1a2b0b1+12bn2a12a2b0+4am2a22b02μ2ln2(a)−8am2a1b02a2μ2ln2(a)−28amna2b0b1a0μ2ln2(a)−10amna2b02a1μ2ln2(a)+24bkn2a02a22+am2b12a02μ2ln2(a)+2amna12b0b1μ2ln2(a)+12amna2b02a0μ2ln2(a)−2amnb12a0a1μ2ln2(a)−2am2a1b0b1a0μ2ln2(a)=0,
Q3:−2am2a12b02μ2ln2(a)−2amna12b02μ2ln2(a)+2amnb12a02μ2ln2(a)−16n2m2aλ2a0a2b0b1+8n2mwa0a1b12+8n2mwa12b0b1+8n2mwa1a2b02+24bn2a0a1a2b0+48bkn2a0a1a2+4bn2a13b0+12bn2a02a2b1+12amna2b0b1a0μ2ln2(a)+12bn2a0a12b1+16bkn2a0a13+4amna1b02a0μ2ln2(a)−4amnb1a02b0μ2ln2(a)+4am2a1b02a2μ2ln2(a)+2amna2b02a1μ2ln2(a)−2am2b12a02μ2ln2(a)−8n2m2aλ2a0a1b12−8n2m2aλ2a12b0b1−8n2m2aλ2a1a2b02+16n2mwa0a2b0b1−2amna12b0b1μ2ln2(a)−20amna2b02a0μ2ln2(a)−4am2a2b0b1a0μ2ln2(a)+2amnb12a0a1μ2ln2(a)+4am2a1b0b1a1μ2ln2(a)=0,
Q2:12bn2a02a2b0−4n2m2aλ2a12b02+am2b12a02μ2ln2(a)−8n2m2aλ2a0a2b02−4n2m2aλ2a02b12+am2a12b02μ2ln2(a)+12bn2a02a1b1+6amnb1a02b0μ2ln2(a)+4n2mwa12b02+16bkn2a03a2+8amna2b02a0μ2ln2(a)+12bn2a0a12b0+24bkn2a02a12+4n2mwa02b12−16n2m2aλ2a0a1b0b1+8n2mwa0a2b02−2am2a1b0b1a0μ2ln2(a)+16n2mwa0a1b0b1−6amna1b02a0μ2ln2(a)=0,
Q1:−8n2m2aλ2a0a1b02−2amnb1a02b0μ2ln2(a)−8n2m2aλ2a02b0b1+8n2mwa0a1b02+4bn2a03b1+16bkn2a03a1+12bn2a02a1b0+2amna1b02a0μ2ln2(a)+8n2mwa02b0b1=0,
(3.1.13)Q0:−4n2m2aλ2a02b02+4bkn2a04+4bn2a03b0+4n2mwa02b02=0.

Solving the system of algebraic equations (3.1.13) by Maple or Mathematica, we obtain the following sets:

Set 1:

(3.1.14)μ=1ln(a)−bn2(2n+m)amk(n+m)2,w=amλ2+b(2n+m)4k(n+m)2,a0=0,a1=−b0(2n+m)2k(n+m),a2=−b1(2n+m)2k(n+m),b0=b0,b1=b1,λ=λ,m=m,n=n,k=k,a=a,b=b.

Substituting (3.1.14) into (3.1.10), we get the following solution:

(3.1.15)v(ζ)=−(2n+m)2k(n+m)1(1±aζ).

From (3.1.7) and (3.1.15) we have

(3.1.16)U(ζ)=[−(2n+m)2k(n+m)1(1±aζ)]12n.

With the help of (2.10) and (2.11) the exact solution of Eq. (3.1.1) has the form:

(3.1.17)q(x,t)={−(2n+m)4k(n+m)×[1−tanFs(12ln(a)−bn2(2n+m)amk(n+m)2(x−2amλt))]}12n×exp(iθ),
(3.1.18)​=​{​−(2n+m)4k(n+m)[1−tanh(12−bn2(2n+m)amk(n+m)2(x−2amλt))]​}12n×exp(iθ),

or

(3.1.19)q(x,t)={−(2n+m)4k(n+m)×[1−cotFs(12ln(a)−bn2(2n+m)amk(n+m)2(x−2amλt))]}12n×exp(iθ),
(3.1.20)={−(2n+m)4k(n+m)×[1−coth(12−bn2(2n+m)amk(n+m)2(x−2amλt))]}12n×exp(iθ),

where θ=λx−(amλ2+b(2n+m)4k(n+m)2)t, provided that ab > 0 and k < 0.

set 2:

(3.1.21)μ=1ln(a)−bn2(2n+m)amk(n+m)2,w=amλ2+b(2n+m)4k(n+m)2,a0=0,a1=−b1(2n+m)2k(n+m),a2=b1(2n+m)2k(n+m),b0=0,b1=b1,  λ=λ,m=m,n=n,k=k,a=a,b=b.

Substituting (3.1.21) into (3.1.10), we get the following solution:

(3.1.22)v(ζ)=−(2n+m)2k(n+m)aζ(aζ±1).

From (3.1.7) and (3.1.122) we have

(3.1.23)U(ζ)=[−(2n+m)2k(n+m)aζ(aζ±1)]12n.

With the help of (2.10) and (2.11) the exact solution of Eq. (3.1.1) has the form:

(3.1.24)q(x,t)=[−(2n+m)4k(n+m)(1+tanhη)]12nexp(iθ),

or

(3.1.25)q(x,t)=[−(2n+m)4k(n+m)(1+cothη)]12nexp(iθ),

where η=12−bn2(2n+m)amk(n+m)2(x−2amλt), θ=λx−(amλ2+b(2n+m)4k(n+m)2)t provided that ab > 0 and k < 0.

set 3:

(3.1.26)μ=1ln(a)−bn2(2n+m)amk(n+m)2,w=amλ2+b(2n+m)4k(n+m)2,a1=−(b0+b1)(2n+m)2k(n+m),a0=0,a2=0,b0=b0,b1=b1,  λ=λ,m=m,n=n,k=k,a=a,b=b.

Consequently,we have the exact solutions of Eq. (3.1.1) in the form:

(3.1.27)q(x,t)=[−(b0+b1)(2n+m)[1−tanhη]2k(n+m)[2b0+b1(1−tanhη)]]12nexp(iθ),

or

(3.1.28)q(x,t)=[−(b0+b1)(2n+m)[1−cothη]2k(n+m)[2b0+b1(1−cothη)]]12nexp(iθ),

where η=12−bn2(2n+m)amk(n+m)2(x−2amλt), θ=λx−(amλ2+b(2n+m)4k(n+m)2)t, provided that ab > 0 and k < 0.

On comparing our result (3.1.24), with the result (18) obtained in [71], we conclude that the two results are equivalent with λ = −κ, while our results (3.1.18), (3.1.20), (3.1.25), (3.1.27) and (3.1.28) are new, and not discussed elsewhere.

3.2 Example 2. The Zakharov–Kuznetsov equation (ZK(m,n,k))

In this subsection, we apply the given method to solve the ZK(m,n,k) [73, 74]

(3.2.1)ut+λ0(um)x+λ1(un)xxx+λ2(uk)xyy=0,

where u = u (x, y, t) is a readily differentiable function, λ0, λ1 and λ2 are arbitrary constants while m, n and k are positive integers. The function governs the behaviour of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [77]. Recently, Ma et al. [73] used the auxiliary equation method to find the solutions of the ZK(2,1,1) equation with λ0=12,λ1=13,λ2=23.

This work is concerned with two cases of Eq. (3.2.1):

3.2.1 Case 1: ZK(2,1,1)

In this case Eq. (3.2.1) becomes the form:

(3.2.2)ut+λ0(u2)x+λ1uxxx+λ2uxyy=0.

To seek travelling wave solutions, we use the wave transformation

(3.2.3)u(x,y,t)=U(ζ),  ζ=x+βy+γt,

where β and γ are arbitrary constants with β, γ ≠ 0, to reduce equation (3.2.2) to the ODE:

(3.2.4)γU′+λ0(U2)′+λ1U′′′+λ2β2U′′′=0.

Integrating Eq. (3.2.4) with respect to ζ, with zero constant of integration, we get

(3.2.5)γU+λ!0U2+(λ1+λ2β2)U′′=0.

Balancing the U˶ and U2 in (3.2.5), then we have N = M + 2. If we choose M = 1 and N = 3, then the formal solution of Eq. (3.2.5) has the form:

(3.2.6)U(ζ)=a0+a1Q+a2Q2+a3Q3b0+b1Q,

and consequently,

(3.2.7)U′(ζ)=Q(Q−1)(b0+b1Q)2[(a1+2a2Q+3a3Q2)(b0+b1Q)−b1(a0+a1Q+a2Q2+a3Q3)]ln(a),
(3.2.8)U′′(ζ)=Q(Q−1)(2Q−1)(b0+b1Q)2[(a1+2a2Q+3a3Q2)(b0+b1Q)−b1(a0+a1Q+a2Q2+a3Q3)]ln2(a)+Q2(Q−1)2(b0+b1Q)3[(b0+b1Q)2(2a2+6a3Q)−2b1(a1+2a2Q+3a3Q2)(b0+b1Q)+2b12(a0+a1Q+a2Q2+a3Q3)]ln2(a).

Substituting (3.2.6) and (3.2.8) into (3.2.5), and equating all the coefficients of Qi, (i = 0, 1, ..., 7) to zero, we get a system of algebraic equations, which can be solved using the Maple, to get the following sets:

Set 1:

(3.2.9)a0=0,a1=6b0(λ1+λ2β2)ln2(a)λ0,a2=−6(b0−b1)(λ1+λ2β2)ln2(a)λ0,a3=−6b1(λ1+λ2β2)ln2(a)λ0,b0=b0,b1=−2b0,γ=−(λ1+λ2β2)ln2(a),β=β,λ0=λ0,λ1=λ1,λ2=λ2. 

Substituting (3.2.9) into (3.1.10), we get the following solution of Eq. (3.2.6):

(3.2.10)U(ζ)=±6(λ1+λ2β2)ln2(a)λ0aζ(aζ±1)2.

With the help of (2.10) and (2.11) the exact solution of Eq. (3.2.2) in the form:

(3.2.11)u(x,y,t)=3(λ1+λ2β2)ln2(a)2λ0sech2 η,
(3.2.12)u(x,y,t)=−3(λ1+λ2β2)ln2(a)2λ0csch2 η,

where η=12[x+βy−(λ1+λ2β2)ln2(a).t] in(a).

set 2:

(3.2.13)a0=−b0(λ1+λ2β2)ln2(a)λ0,a1=(6b0−b1)(λ1+λ2β2)ln2(a)λ0,a2=−6(λ1+λ2β2)(b0−b1)ln2(a)λ0,a3=−6b1(λ1+λ2β2)ln2(a)λ0,γ=(λ1+λ2β2)ln2(a),b0=b0,b1=b1,β=β,λ0=λ0,λ1=λ1,λ2=λ2.

Consequently, we have the exact solutions of Eq. (3.2.2) in the form:

(3.2.14)u(x,y,t)=(λ1+λ2β2)ln2(a)2λ0[1−3tanh2η],
(3.2.15)u(x,y,t)=(λ1+λ2β2)ln2(a)2λ0[1−3coth2η],

where η=12[x+βy−(λ1+λ2β2)ln2(a).t] in(a).

If we choose λ0=12, λ1=13, λ2=23, a =e in our results (3.2.11),(3.2.12),(3.2.14) and (3.2.15) we have the well-known results u1, u2 of (10) and u1, u2 of (13) obtained in [73].

3.2.2 Case 2: ZK(m,1,1), m ⩾ 3

In this case Eq. (3.2.1) becomes the form:

(3.2.16)ut+λ0(um)x+λ1uxxx+λ2uxyy=0.

Using the same wave transformation (3.2.3), to reduce equation (3.2.16) to the ODE:

(3.2.17)γU′+λ!0(Um)′+λ1U‴+λ2β2U‴=0.   

Integrating Eq. (3.2.17) with respect to ζ, with zero constant of integration, we get

(3.2.18)γU+λ!0Um+(λ1+λ2β2)U′′=0.

By balancing U″ with Um we have N=2m−1, m ≥ 3. Then we use the transformation

(3.2.19)U(ζ)=[v(ζ)]2m−1.  

Substituting (3.2.19) into equation (3.2.18) we have the new equation

(3.2.20)(m−1)2(γv2+λ0v4)+(λ1+λ2β2)×6−2m(v′)2+2(m−1)vv′′=0. 

Balancing the vv″ and v4 in (3.2.21), then we have N = M + 1. If we choose M = 1 and N = 2, then Eq. (3.2.20) has the same formal solution (3.1.10).

Substituting (3.1.10),(3.1.11) and (3.1.12) into (3.2.20), and equating all the coefficients of Qi,(i = 0, 1, ..., 8) to zero, we get a system of algebraic equations, which can be solved using the aid of Maple or Mathematica, to get the following result:

(3.2.21)a0=0,a1=±2b0ln(a)(m−1)−2(λ1+λ2β2)(m+1)λ0,a2=∓2b0ln(a)(m−1)−2(λ1+λ2β2)(m+1)λ0,    b0=b0,b1=−2b0,γ=−4(λ1+λ2β2)ln2(a)(m−1)2,β=β,λ0=λ0,λ1=λ1,λ2=λ2.  

provided that (λ1+λ2β2)λ0<0.

Substituting (3.2.21) into (3.1.10), we get the following solution of Eq. (3.2.20):

(3.2.22)v(ζ)=±2ln(a)(m−1)−2(λ1+λ2β2)(m+1)λ0aζ(a2ζ−1).  

From (3.2.22) and (3.2.19) we have

(3.2.23)U(ζ)​=​[±2ln(a)(m−1)−2(λ1+λ2β2)(m+1)λ0aζ(a2ζ−1)]2m−1.  

With the help of (2.10) and (2.11) the exact solution of Eq. (3.2.16) in the form:

(3.2.24)u(x,y,t)​=​[±ln(a)(m−1)−2(λ1+λ2β2)(m+1)λ0cschη]2m−1​​​,    

where η=(x+βy−4(λ1+λ2β2)ln2(a)(m−1)2t)ln(a).

3.3 Example 3. The K(m,n) equation

The K(m,n) equation with the generalized evolution term [75, 78, 79] is given by

(3.3.1)(ql)t+aqmqx+b(qn)xxx=0,    

where, the first term is the generalized evolution term, the second term represents the nonlinearity, and the third term is the dispersion. Also, a , b ∈ R and are constants, while l, m and n ∈ Z+.

Eq. (3.3.1) is the generalized form of the KdV equation, where, in particular, the case l = m = n = 1 leads to the KdV equation. Eq. (3.3.1) appeared for the first time in [78] for l = 1. Thus, Eq. (3.3.1) reduces to the K(m,n) equation for l = 1. Therefore, for l = 1, K(1,1) is the KdV equation while K(2,1) is the mKdV equation. Eq. (3.3.1) has been discussed in [75] using the (G′/G)-expansion method and its exact solutions have been found.

In order to solve Eq. (3.3.1) using the method of section 2, we introduce the wave transformation

(3.3.2)q(x,t)=U(ζ),  ζ=μ(x−ct),    

where µ and c are constants, to reduce Eq. (3.3.1) to the ODE:

(3.3.3)−μc(Ul)′+μaUmU′+μ3bUn′′′=0. 

Integrating Eq. (3.3.3) with respect to ζ, with zero constant of integration, we have

(3.3.4)−μcUl+μam+1Um+1+μ3bUn′′=0. 

Let l = n, balancing (Un)′ and Um+1 in (3.3.4) yields N=2m+1−n, where m + 1 ≠ n. In order to obtain the closed form solutions, we use the transformation

(3.3.5)U(ζ)=[v(ζ)]1m+1−n,

to reduce Eq. (3.3.4) into the ODE

(3.3.6)−cm+1−n2v2+am+1−n2m+1v3+bnμ22n−m−1v′2+bnμ2m+1−n2vv′′=0.

Balancing the vv″ and v3 in (3.3.6), then we have N = M + 2. If we choose M = 1 and N = 3, then Eq. (3.3.6) has the same formal solution (3.2.6).

Substituting (3.2.6),(3.2.7) and (3.2.8) into (3.3.6), and equating all the coefficients of Qi,(i = 0, 1, ..., 10) to zero, we get a system of algebraic equations, which can be solved using the aid of Maple or Mathematica, to get the following result:

(3.3.7)a0=0,a1=2cb0(m+1)(m+1+n)na,a2=−2c(b0−b1)(m+1)(m+1+n)na,  a3=−2cb1(m+1)(m+1+n)na,μ=±cb(m+1−n)nln(a),b0=b0,b1=b1,c=c,

provided that bc>0.

Substituting (3.3.7) into (3.2.6), we get the following solution of Eq. (3.3.6):

(3.3.8)v(ζ)=±2c(m+1)(m+1+n)naaζ(aζ±1).  

From (3.2.8) and (3.2.5) we have

(3.3.9)U(ζ)=[±2c(m+1)(m+1+n)naaζ(aζ±1)]1m+1−n.  

With the help of (2.10) and (2.11) the exact solution of Eq. (3.3.1) in the form:

(3.3.10)u(x,y,t)=[c(m+1)(m+1+n)2nasech2η]1m+1−n,  

or

(3.3.11)u(x,y,t)=[−c(m+1)(m+1+n)2nacsch2η]1m+1−n,  

where η=±cb(m+1−n)2n(x−ct)

4 Physical explanations for some of our solutions

In this section, we will illustrate the application of the results established above. Exact solutions of the results describe different nonlinear waves. The established exact solutions with symmetrical hyperbolic Fibonacci functions are special kinds of solitary waves.

We now examine Figures (1–4) which illustrate a selection of the results obtained above. Specific parameter values are selected, for example: in some of the solutions (3.1.24) and (3.1.25) of the Biswas-Milovic equation with dual-power law nonlinearity with −10 < x, t < 10, solutions (3.2.11) and (3.2.12) of the ZK(2,1,1) with −10 < x, t < 10, solution (3.2.24) of the ZK(m,1,1) with −10 < x, t < 10, solutions (3.3.10) and (3.3.11) of the K(m,n) equation with the generalized evolution with −10 < x, t < 10.

Figure 1 Symmetrical Fibonacci hyperbolic function solutions of the Biswas-Milovic equation with dual-power law nonlinearity when a = b = λ = n = m = 1, K = −1.
Figure 1

Symmetrical Fibonacci hyperbolic function solutions of the Biswas-Milovic equation with dual-power law nonlinearity when a = b = λ = n = m = 1, K = −1.

Figure 2 Symmetrical Fibonacci hyperbolic function solutions of the ZK(2,1,1) when λ0 = λ1 = λ2 = β = 1, a = e, y = 0.
Figure 2

Symmetrical Fibonacci hyperbolic function solutions of the ZK(2,1,1) when λ0 = λ1 = λ2 = β = 1, a = e, y = 0.

Figure 3 Symmetrical Fibonacci hyperbolic function solutions of the ZK(m,1,1) when λ0 = −1, λ1 = λ2 = 1, β = 0, a = e.
Figure 3

Symmetrical Fibonacci hyperbolic function solutions of the ZK(m,1,1) when λ0 = −1, λ1 = λ2 = 1, β = 0, a = e.

Figure 4 Symmetrical Fibonacci hyperbolic function solutions of K(m,n) equation with the generalized evolution when a = b = c = m = n = 1.
Figure 4

Symmetrical Fibonacci hyperbolic function solutions of K(m,n) equation with the generalized evolution when a = b = c = m = n = 1.

5 Conclusions

In this paper we have shown that the symmetrical hyperbolic Fibonacci function solutions can be obtained for the general Expa- function by using generalized Kudryashov method. We have successfully extended the generalized Kudryashov method to solve three nonlinear partial differential equations. In terms of practical applications, we have obtained many new symmetrical hyperbolic Fibonacci function solutions for the Biswas-Milovic equation with dual-power law nonlinearity, the ZK(m,n,k) and the K(m,n) equation with the generalized evolution term. This demonstrates that the generalized Kudryashov method is powerful, effective and convenient for solving nonlinear PDEs. The physical explanation for certain solutions of such equations have been presented. The generalized Kudryashov method provides a powerful mathematical tool to obtain more general exact analytical solutions of many nonlinear PDEs in mathematical physics.

Acknowledgement

The authors wish to thank the referees for their comments on this paper.

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Received: 2015-10-3
Accepted: 2016-2-12
Published Online: 2016-4-28
Published in Print: 2016-1-1

© 2016 EL Sayed M. E. Zaye and Abdul-Ghani Al-Nowehy, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2016-0013
Keywords for this article
Nonlinear PDEs; Generalized Kudryashov method; Exact solutions; Symmetrical hyperbolic Fibonacci function; The Biswas-Milovic equation; The Zakharov–Kuznetsov equation; The K(m,n) equation
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  74. Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
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  146. POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
  147. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  148. Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
  149. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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