# Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using theG′/G2-Expansion Method

Authors: Sekson Sirisubtawee; Sanoe Koonprasert
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We apply theG′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.


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## Body

## 1. Introduction

Various phenomena such as shallow water waves and multicellular biological dynamics arising in the nonlinear physical sciences [1, 2], engineering [3, 4], and biology [5] can be modeled by a class of integrable nonlinear evolution equations which can be expressed in terms of nonlinear partial differential equations (NPDEs) of integer orders. Consequently, study of traveling wave solutions of NPDEs plays a significant role in the investigation of behaviors of nonlinear phenomena. Due to the efficiency, reliability, and easy use of symbolic software packages such as Maple or Mathematica, many powerful methods have been constructed and developed to analytically solve NPDEs with their aid. Over the last few decades, exact solutions, analytical approximate solutions, and numerical solutions of many NPDEs have been successfully obtained. The methods for obtaining exact explicit solutions of NPDEs are, for example, the (G′/G)-expansion method [6–8], the (G′/G,1/G)-expansion method [9–11], the novel (G′/G)-expansion method [12], the tanh-function method [13], the exp-function method [14, 15], theF-expansion method [16], Hirota’s direct method [17, 18], Kudryashov method [19, 20], and the extended auxiliary equation method [21]. Examples of the methods for obtaining analytical approximate solutions to NPDEs are the variational iteration method [22, 23] (VIM), the Adomian decomposition method [24, 25] (ADM), the homotopy perturbation method [26, 27] (HPM), and the reduced differential transform method [28]. In addition, the examples of useful methods for solving NPDEs numerically are the generalized finite difference method [29], the finite volume method [30], the finite element method [31], the spectral collocation method [32], and the Galerkin finite element method [33]. However, we prefer, if possible, to obtain exact solutions of NPDEs.In the recent decades, applications of theG′/G2-expansion method for solving NPDEs have been proposed in various areas of applied sciences and engineering. For example, Chen [34] gave the application of the G′/G2-expansion method for seeking exact solutions of the coupled nonlinear Klein-Gordon equation. Wen-An et al. [35] demonstrated the use of the w/g-expansion method for finding traveling wave solutions of a nonlinear evolution equation. Zayed and Arnous [36] investigated the use of the modified w/g-expansion method for finding traveling wave solutions of nonlinear evolution equations. The modified method can be thought as the generalization of the well-known G′/G-expansion method introduced in [37] with the special functions w and g including the case of w=G′ and g=G2. In particular, Wen-An et al. [35] applied the modified method, i.e., the G′/G2-expansion method, to find the traveling wave solutions of the Vakhnenko equation. Zhouzheng [38] applied the G′/G2-expansion method to obtain the exact solutions of the modified Benjamin-Bona-Mahony (MBBM) and Ostrovsky-Benjamin-Bona-Mahony (OBBM) equations. They found that the explicit exact solutions of the equations obtained by the method are in terms of some trigonometric, hyperbolic, and rational functions. Gepreel [39] employed the extended rational G′/G2-expansion method to obtain traveling wave solutions of the first equation of two integral members of nonlinear Kadomtsev-Petviashvili (KP) hierarchy equations in mathematical physics. Mohyud-Din and Bibi [40] used the G′/G2-expansion method along with the fractional complex transform to analytically solve the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) and the space-time fractional coupled Burgers equations for innovative exact solutions. Their exact solutions include trigonometric, hyperbolic, and rational function solutions, while Zhang et al. [41] proposed the use of the G′/G2-expansion method for solving the Schrödinger equation with third-order dispersion.The rest of this article is organized as follows. In Section2, the brief description of the G′/G2-expansion method is given. In Section 3, we apply the method to some real world problems modeled by NPDEs in order to obtain their exact solutions. Finally, the conclusions are drawn in Section 4.


## 2. Algorithm of theG′/G2-Expansion Method

In this section, we provide the description of theG′/G2-expansion method which is discussed in [34, 40]. Consider a nonlinear evolution partial differential equation (NEPDE) in two independent variables x and t as follows:(1)Pu,ut,ux,utt,uxx,uxt,…=0,where u=u(x,t) is an unknown function of independent variables x,t and P is a polynomial of u and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.The main steps of the method to obtain exact solutions of NPDEs can be given as follows [42, 43].Step 1.
Convert a nonlinear partial differential equation in (1) into an ordinary differential equation (ODE) using the traveling wave transformation in a variable ξ as follows:(2)ux,t=Uξ,ξ=kx-ct,where k and c are nonzero arbitrary constants. It can be noted that another transform ξ=k(x-ct) can be sometimes used for some certain problems. With the transformation in (2) and integrations with respect to ξ as many as possible, (1) is reduced to an ODE in U=U(ξ) as follows:(3)QU,U′,U′′,U′′′,…=0,where Q is a polynomial of U(ξ) and its various derivatives. The prime notation (′) denotes the derivative with respect to ξ.Step 2.
Suppose that the formal solution of the ODE in (3) can be expressed in powers of G′/G2 as follows:(4)Uξ=a0+∑j=1NajG′G2j+bjG′G2-j,where G=G(ξ) satisfies the following nonlinear ODE:(5)G′G2′=μ+λG′G22,in which μ≠1 and λ≠0 are integers. The unknown constants aN or bN may be zero, but both of them cannot be zero simultaneously. The coefficients a0,aj,bj(j=1,2,…,N) are unknown constants to be determined at a later step.Step 3.
The value of the positive integerN can be determined using the homogeneous balance principle, i.e., by balancing between the highest order derivatives and the nonlinear terms occurring in (3). More precisely, if the degree of U(ξ) is deg[U(ξ)]=N, then the degree of the other terms will be expressed as follows:(6)degdqUξdξq=N+q,degUξpdqUξdξqs=Np+sN+q.Step 4.
Substituting (4) along with (5) into (3), we obtain a polynomial in G′/G2. Collecting all coefficients of like-power of G′/G2k (k=0,±1,±2,…,±M, where M is some positive integer) and setting all of the obtained coefficients to zero, we acquire a system of nonlinear algebraic equations for the unknown constants a0,aj,bj(j=1,2,…,N),k, and c. Assume that the resulting algebraic equations can be solved for the unknown constants using symbolic software packages such as Maple.Step 5.
The general solutions of (5) can be categorized into the following three cases when C,D are arbitrary nonzero constants.Ifμλ>0, then we obtain the general solution(7)G′G2=μλCcosμλξ+DsinμλξDcosμλξ-Csinμλξ.Ifμλ<0, then we obtain the general solution(8)G′G2=12λ2μλ-4Cμλe2ξμλCe2ξμλ-D,which is equivalent to(9)G′G2=-μλλCsinh2μλξ+Ccosh2μλξ+DCsinh2μλξ+Ccosh2μλξ-D.Ifμ=0 and λ≠0, then we obtain the general solution(10)G′G2=-CλCξ+D.The explicit exact solutions of (1) can be obtained by inserting the values of a0,aj,bj(j=1,2,…,N),k,c and the solutions in (7)-(10) into (4) with the transformation in (2).


## 3. Applications of theG′/G2-Expansion Method

In this section, we will demonstrate the use of theG′/G2-expansion method on three of the interesting problems in mathematical physics.


### 3.1. The Benney-Luke Equation

In this section, we will provide a use of theG′/G2-expansion method for seeking exact solitary wave solutions of the Benney-Luke equation, which is used to approximate the full water wave equations and appropriately described two-way water wave propagation with surface tension. The equation can be written in the following form [44, 45]:(11)utt-uxx+αuxxxx-βuxxtt+utuxx+2uxuxt=0,where α,β are the positive integers such that their difference is in terms of the inverse bond number capturing the effects of gravity forces and surface tension.Using the traveling wave transformationξ=kx-ct, (11) is converted into the following ODE in the variable U=U(ξ):(12)c2-k2U′′+αk4-βk2c2U4-3ck2U′U′′=0.Integrating (12) with respect to ξ once and then choosing the constant of integration to be zero, we obtain the following ODE:(13)2c2-k2U′+2αk4-βk2c2U′′′-3ck2U′2=0,for which the homogeneous balance principle is applied. Following Step 3 of the mentioned method, the highest order derivative U′′′ and the nonlinear term of the highest order (U′)2 are balanced via using formula (6) as follows:(14)degU′′′=N+3=degU′2=2N+1,which leads to N=1. Hence, the form of exact solutions of the ODE in (13) using the G′/G2-expansion method can be expressed as(15)Uξ=a0+a1G′G2+b1G′G2-1,where a0,a1,b1 are unknown constants. Substituting (15) into (13) along with (5), then collecting all the coefficients with the same power of G′/G2i,(i=0,±1,±2,…), and finally setting these resulting coefficients to be zero, we consequently obtain the following system of algebraic equations in a0,a1,b1,k,c,μ,λ,α,β:(16)G′G2-4:-12αk4μ3b1+12βc2k2μ3b1-3ck2μ2b12=0,G′G2-2:-16αk4λμ2b1+16βc2k2λμ2b1-6ck2λμb12+6ck2μ2a1b1-2c2μb1+2k2μb1=0,G′G20:-4αk4λ2μb1+4αk4λμ2a1+4βc2k2λ2μb1-4βc2k2λμ2a1-3ck2λ2b12+12ck2λμa1b1-3ck2μ2a12-2c2λb1+2c2μa1+2k2λb1-2k2μa1=0,G′G22:16αk4λ2μa1-16βc2k2λ2μa1+6ck2λ2a1b1-6ck2λμa12+2c2λa1-2k2λa1=0,G′G24:12αk4λ3a1-12βc2k2λ3a1-3ck2λ2a12=0.Solving the obtained algebraic system (16) by use of Maple, we get the following three cases.Case 1.
(17) a 0 = a 0 , a 1 = 0 , b 1 = ∓ 4 k α - β μ 4 α k 2 λ μ + 1 4 β k 2 λ μ + 1 , k = k , c = ± 4 α k 2 λ μ + 1 k 4 β k 2 λ μ + 1 ,where a0,k,α,β,μ,λ are arbitrary constants.Case 2.
(18) a 0 = a 0 , a 1 = ± 4 k α - β λ 4 α k 2 λ μ + 1 4 β k 2 λ μ + 1 , b 1 = 0 , k = k , c = ± 4 α k 2 λ μ + 1 k 4 β k 2 λ μ + 1 ,where a0,k,α,β,μ,λ are arbitrary constants.Case 3.
(19) a 0 = a 0 , a 1 = ± 4 k α - β λ 16 α k 2 λ μ + 1 16 β k 2 λ μ + 1 , b 1 = ∓ 4 k α - β μ 16 α k 2 λ μ + 1 16 β k 2 λ μ + 1 , k = k , c = ± 16 α k 2 λ μ + 1 k 16 β k 2 λ μ + 1 ,where a0,k,α,β,μ,λ are arbitrary constants.
When we substitute the above three cases of the obtained parameters along with the functionsG′/G2 specified in (7)-(10) into the solution form (15), we can write three results of solutions of (11) as follows.Result 1.
From Case 1 in (17), we have ξ=kx∓4αk2λμ+1k/4βk2λμ+1t and the following exact solutions.
Whenμλ>0, the trigonometric function solution corresponding to the parameter values in Case 1 can be written as(20)u11x,t=a0∓4kα-βμλ4αk2λμ+14βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ-1.
Whenμλ<0, the exponential function solution associated with the parameter values in Case 1 can be expressed as(21)u21x,t=a0∓8kα-βμλ4αk2λμ+14βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D-1.
Whenμ=0,λ≠0, the exact solution corresponding to the parameter values in Case 1 is u31(x,t)=a0, which is the constant solution.Result 2.
From Case 2 in (18), we have ξ=kx∓4αk2λμ+1k/4βk2λμ+1t and the following exact solutions.
Whenμλ>0, the trigonometric function solution corresponding to the parameter values in Case 2 can be written as(22)u12x,t=a0±4kα-βμλ4αk2λμ+14βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ.
Whenμλ<0, the exponential function solution associated with the parameter values in Case 2 can be expressed as(23)u22x,t=a0±2kα-β4αk2λμ+14βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D.
Whenμ=0,λ≠0, the rational function solution corresponding to the parameter values in Case 2 can be expressed as(24)u32x,t=a0∓4kα-βCCξ+D,for which the traveling wave transformation for this case is ξ=k(x∓t).Result 3.
From Case 3 in (19), we have ξ=kx∓16αk2λμ+1k/16βk2λμ+1t and the following exact solutions.
Whenμλ>0, the trigonometric function solution corresponding to the parameter values in Case 3 can be written as(25)u13x,t=a0±4kα-βμλ16αk2λμ+116βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ∓4kα-βμλ16αk2λμ+116βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ-1.
Whenμλ<0, the exponential function solution associated with the parameter values in Case 3 can be expressed as(26)u23x,t=a0±2kα-β16αk2λμ+116βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D∓8kα-βμλ16αk2λμ+116βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D-1.
Whenμ=0,λ≠0, the rational function solution corresponding to the parameter values in Case 3 can be expressed as(27)u33x,t=a0∓4kα-βCCξ+D,for which the traveling wave transformation for this case is ξ=k(x∓t).Akter and Akbar [44] utilized the modified simple equation method to obtain exact solutions of (11) which were written as fractions of exponential functions. They can be transformed into the tanh and coth functions for which the arbitrary constants are selected appropriately as shown in their paper. Islam et al. [45] demonstrated the application of the improved F-expansion method with Riccati equation to obtain exact traveling wave solutions of (11). Their results were expressed in terms of the following functions:(i)
tanh, coth, their reciprocals, and their summations(ii)
tan, cot, their reciprocals, and their summations(iii)
rational functions.However, our obtained results in (20)-(27) are more generalized than the ones described above; i.e., selecting the appropriate constants C and D in our solutions can lead to the solutions obtained by other existing methods mentioned previously. Here we present the plots of the exact solution u21(x,t) in (21) with the positive formula using the following parameter values: α=2,β=3,k=1,a0=1,μ=0.5,λ=-1,C=D=1. The three-dimensional and two-dimensional plots of this solution are portrayed in Figure 1 demonstrating the solitary wave solution of kink type.Figure 1
The solitary wave solutions of kink type foru21(x,t) in (21) when α=2, β=3, k=1, a0=1, μ=0.5, λ=-1, and C=D=1.
(a)
3D graph (b)
2D graph whent=0


### 3.2. Equation of Nanoionic Currents along Microtubules

In this section, we will show an application of theG′/G2-expansion method in nanobiosciences. One of the important models in such fields is the nonlinear transmission line model for nanoionic currents along microtubules (MTs) segmented into identical elementary rings (ERs). The model is playing an important role in cellular signalling and the elaborated details of derivation of the equation can be found in [46, 47]. The equation of nanoionic currents along MTs is described as follows [47]:(28)l23uxxx+Z3/2lwG0-2δC0uut+2ux+ZC0lut+1lRZ-1-G0Zu=0,where R=0.34×109  Ω is the resistance of the ER with length l=8×10-9 m,C0=1.8×10-15 F is the total maximal capacitance of the ER, G0=1.1×10-13 Si is conductance of pertaining nanopores (NPs), and Z=5.56×1010  Ω is the characteristic impedance of the system. Parameters ω and δ represent conductance of NPs in ER and nonlinearity of ER capacitor, respectively.Using the dimensionless wave variableξ=1/lx-c/τt, where c is the dimensionless velocity of wave and τ=RC0=0.6×10-6 s is the characteristic time of charging ER capacitor, we obtain the traveling wave transformation u(x,t)=U(ξ) and then (28) is converted to the following new ODE:(29)U′′′+αcUU′+6-βcU′+γU=0,where α=3Z3/2/τ(2δC0-wG0), β=3ZC0/τ, and γ=3(RZ-1-G0Z).Balancing between the highest order derivativeU′′′ and the nonlinear term of the highest order UU′ by using formula (6), we obtain(30)deg⁡U′′′=N+3=deg⁡UU′=2N+1⟹N=2.Thus, the form of exact solutions of (29) using the G′/G2-expansion method can be expressed as(31)Uξ=a0+a1G′G2+a2G′G22+b1G′G2-1+b2G′G2-2,where a0,a1,a2,b1,b2 are unknown constants. Substituting (31) into (29) along with (5), then collecting all the coefficients with the same power of G′/G2i,(i=0,±1,±2,…), and finally setting these resulting coefficients to be zero, we consequently obtain the following system of algebraic equations in a0,a1,a2,b1,b2,c,μ,λ,α,β,γ:(32)G′G2-5:-2αcμb22-24μ3b2=0,G′G2-4:-3αcμb1b2-6μ3b1=0,G′G2-3:-2αcλb22-2αcμa0b2-αcμb12+2βcμb2-40λμ2b2-12μb2=0,G′G2-2:-3αcλb1b2-αcμa0b1-αcμa1b2+βcμb1-8λμ2b1+γb2-6μb1=0,G′G2-1:-2αcλa0b2-αcλb12+2βcλb2-16λ2μb2+γb1-12λb2=0,G′G20:-αcλa0b1-αcλa1b2+αcμa0a1+αcμa2b1+βcλb1-βcμa1-2λ2μb1+2λμ2a1+γa0-6λb1+6μa1=0,G′G21:2αcμa0a2+αcμa12-2βcμa2+16λμ2a2+γa1+12μa2=0,G′G22:αcλa0a1+αcλa2b1+3αcμa1a2-βcλa1+8λ2μa1+γa2+6λa1=0,G′G23:2αcλa0a2+αcλa12+2αcμa22-2βcλa2+40λ2μa2+12λa2=0,G′G24:3αcλa1a2+6λ3a1=0,G′G25:2αcλa22+24λ3a2=0.Using Maple to solve algebraic system (32), we obtain the following three cases.Case 1.
(33) a 0 = β c - 8 λ μ - 6 α c , a 1 = 0 , a 2 = 0 , b 1 = 0 , b 2 = - 12 μ 2 α c , c = c ,with c≠0,μ,λ which are arbitrary constants.Case 2.
(34) a 0 = β c - 8 λ μ - 6 α c , a 1 = 0 , a 2 = - 12 λ 2 α c , b 1 = 0 , b 2 = 0 , c = c ,with c≠0,μ,λ which are arbitrary constants.Case 3.
(35) a 0 = β c - 8 λ μ - 6 α c , a 1 = 0 , a 2 = - 12 λ 2 α c , b 1 = 0 , b 2 = - 12 μ 2 α c , c = c ,with c≠0,μ,λ which are arbitrary constants.Inserting the above three cases in (33)-(35) along with the functions G′/G2 described in (7)-(10) into the solution form (31), we attain three results of solutions of (28) as follows.Result 1.
Using parameter values of Case 1 in (33) and ξ=1/lx-c/τt, we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of (28) can be written as(36)u11x,t=βc-8μλ-6αc-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-2.
Whenμλ<0, the exponential function solution of (28) can be expressed as (37)u21x,t=βc-8μλ-6αc-48μ2λ2αc2μλ-4Cμλe2ξμλCe2ξμλ-D-2.
Whenμ=0,λ≠0, the exact solution of (28) is the constant solution u31(x,t)=βc-6/αc.Result 2.
Using parameter values of Case 2 in (34) and ξ=1/lx-c/τt, we have the following exact solutions.Whenμλ>0, the trigonometric function solution of (28) can be written as(38)u12x,t=βc-8μλ-6αc-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ2.Whenμλ<0, the exponential function solution of (28) can be expressed as(39)u22x,t=βc-8μλ-6αc-3αc2μλ-4Cμλe2ξμλCe2ξμλ-D2.Whenμ=0,λ≠0, the rational function solution of (28) can be expressed as(40)u32x,t=βc-6αc-12C2αcCξ+D2.Result 3.
Using parameter values of Case 3 in (35) and ξ=1/lx-c/τt, we have the following exact solutions.Whenμλ>0, the trigonometric function solution of (28) can be written as(41)u13x,t=βc-8μλ-6αc-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ2-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-2.Whenμλ<0, the exponential function solution of (28) can be expressed as(42)u23x,t=βc-8μλ-6αc-3αc2μλ-4Cμλe2ξμλCe2ξμλ-D2-48μ2λ2αc2μλ-4Cμλe2ξμλCe2ξμλ-D-2.Whenμ=0,λ≠0, the rational function solution of (28) can be expressed as(43)u33x,t=βc-6αc-12C2αcCξ+D2.Satarić et al. [46] firstly proposed model (28) and obtained its analytical solution by converting (28) with the change of variables and the use of the appropriate boundary conditions into the solvable ODE in a new variable. The analytical solution was expressed in terms of a square of the exponential secant function. Later, Sekulić et al. [47] applied the modified extended tanh-function method to solve (28) for exact traveling wave solutions. The solutions were written as the square of the following functions: tan, cot, tanh, and coth. It is not difficult to verify that the mathematical structures of their exact solutions are the particular cases of those of our solutions as shown in (36), (37), (38), (39), (41), and (42). Meanwhile our rational function solutions do not appear in their work. In addition, Zayed and Alurrfi [48] utilized the (G′/G,1/G)-expansion method to analytically obtain the exact traveling wave solutions of problem (28). Their solutions included the solitary wave solutions and the periodic wave solutions when the appropriate values of the parameters were particularly selected. Comparing the mathematical structures of their specific solutions with our obtained solutions, almost all of their specific solutions can be obtained from our solutions. For instance, solutions (20), (23), and (24) in their paper are structurally equivalent to the hyperbolic form of the solution in (39) via using C=D=1 and C=1,D=-1. The trigonometric solution (28) in their paper has the same mathematical structure as our solution in (38) with C=D=1 and C=-1,D=1. Solutions (29), (32), and (33) in their article can be equivalently transformed to our solution in (38) by choosing C=0,D=1 and C=1,D=0. Finally, rational solutions (36) and (37) in their paper with ξ of degree two in the denominators are structurally equivalent to our rational solution in (40). Furthermore, the exact solutions in (41) and (42) can generate other types of solutions different from the compared ones.In [46, 47], they used the estimated dimensionless parameter σ=1.67×102 and took σc=2.5,2δ-wG0/C0=0.1 for plotting their solutions. Using the parameter values mentioned above and choosing μ=0.5,λ=-1,C=1,D=-1, we obtain the graphical representations of the exact traveling wave solution u22(x,t) in (39) as demonstrated in Figure 2 describing the soliton solution of bell-type. The obtained three-dimensional graph shown in Figure 2(a) is similar to their results.Figure 2
The soliton solution of bell-type foru22(x,t) in (39) when α=1.8×107, β=500.4, c=0.01497, μ=0.5, λ=-1, C=1, and D=-1.
(a)
3D graph (b)
2D graph whent=0


### 3.3. The Generalized Hirota-Satsuma Coupled KdV System

In 1982 Satsuma and Hirota proposed the new system of equations which is called the generalized Hirota-Satsuma coupled KdV system as follows [49]:(44)Ut=14Uxxx+3UUx+3-V2+Wx,Vt=-12Vxxx-3UVx,Wt=-12Wxxx-3UWx.The above system can be obtained from the four reductions of KP hierarchy. In particular, the well-known Hirota-Satsuma coupled KdV system [50], which was derived in 1981 by Hirota and Satsuma to describe interactions of two long waves with different dispersion relations, can be obtained by setting W=0 in (44). We want to obtain traveling wave solutions for system (44) which are in the following form:(45)Ux,t=uξ,Vx,t=vξ,Wx,t=wξ,ξ=kx-ct,where k and c are nonzero arbitrary constants to be determined later. Substituting (45) into (44), we yield a system of ODEs as follows:(46)-cku′=14k3u′′′+3kuu′+3k-v2+w′,(47)-ckv′=-12k3v′′′-3kuv′,(48)-ckw′=-12k3w′′′-3kuw′.Let [51](49)u=αv2+βv+γ,w=Av+Bwhere α,β,γ,A, and B are constants to be also determined later.Substituting (49) into (47) and (48) and then integrating once, we know that (47) and (48) give the same equation as follows:(50)k2v′′=-2αv3-3βv2+2c-3γv+c1,where c1 is a constant of integration. Multiplying (50) by v′ and then integrating the resulting equation with respect to ξ, we obtain(51)k2v′2=-αv4-2βv3+2c-3γv2+2c1v+c2,where c2 is also a constant of integration.From (49)-(51), we obtain(52)k2u′′=2αk2v′2+k22αv+βv′′,=2α-αv4-2βv3+2c-3γv2+2c1v+c2+2αv+β-2αv3-3βv2+2c-3γv+c1.Integrating (46) once, we get(53)14k2u′′+32u2+cu+3-v2+w+c3=0,where c3 is a constant of integration. Substituting (49) and (52) into (53), we yield the following system:(54)3αc-3αγ+34β2-3=0,12αc1+βc+γβ+A=0,142αc2+βc1+32γ2+cγ+3B+c3=0.Let(55)c1=12α2β3+2cαβ-6αβγ,vξ=aPξ-β2α.We find from (54) that(56)α=β2-44γ-c,A=4βc-γβ2-4,B=16-γ+cβ2-4216c3cβ2-2c3cβ4-16c3γβ2+2c3γβ4+56c2γβ2-48γ2cβ2-16c2+14c2β6-3c2β4+12c2β2-16γ2c-32c2γ-8c3β2+β4γ3-2β4c3+32c3γ-32c3c+48γ3+β4γ2c.From (50), we therefore obtain(57)ak2P′′-a2c-6γ+3β22αP+2αa3P3=0.Applying the homogeneous balance principle and (6) mentioned in Step 3 to the terms P′′ and P3, we then have that(58)deg⁡P′′=N+2=deg⁡P3=3N,which leads to N=1. Hence, the form of exact solutions of the ordinary differential equation in (57) using the G′/G2-expansion method is(59)Pξ=a0+a1G′G2+b1G′G2-1,where a0,a1,b1 are unknown constants. Substituting (59) into (57) along with (5), then collecting all the coefficients with the same power of G′/G2i,(i=0,±1,±2,…), and finally setting these resulting coefficients to be zero, we consequently attain the following system of algebraic equations in a0,a1,b1,k,c,a,α,β,γ:(60)G′G2-3:2αa3b13+2ab1k2μ2=0,G′G2-2:6αa0a3b12=0,G′G2-1:2ab1μλk2-3ab1β22α-2acb1+6aγb1+6αb1a3a02+6αa1a3b12=0,G′G20:-3aβ2a02α-2aca0+6aγa0+2αa3a03+12αa3a0a1b1=0,G′G21:2k2aa1λμ-3aβ2a12α-2aca1+6aγa1+6αa3a02a1+6αa3a12b1=0,G′G22:6αa3a0a12=0,G′G23:2a3αa13+2ak2λ2a1=0.By solving the nonlinear system in (60) with Maple, we obtain the following cases.Case 1.
(61) a 0 = 0 , a 1 = 0 , b 1 = ± - 1 / α μ k a , k = k , c = 1 4 α 4 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Case 2.
(62) a 0 = 0 , a 1 = ± - 1 / α λ k a , b 1 = 0 , k = k , c = 1 4 α 4 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Case 3.
(63) a 0 = 0 , a 1 = ± - 1 / α λ k a , b 1 = ± - 1 / α μ k a , k = k , c = 1 4 α - 8 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Case 4.
(64) a 0 = 0 , a 1 = ± - 1 / α λ k a , b 1 = ∓ - 1 / α μ k a , k = k , c = 1 4 α 16 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Inserting the above four cases shown in (61)-(64) along with the functions G′/G2 described in (7)-(10) into the solution form (59), we obtain four results of solutions of system (44) as follows.Result 1.
Using parameter values specified in Case 1 as shown in (61) and ξ=k(x-1/4α(4k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(65)v11x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,u11x,t=αv11x,t+βv11x,t+γ,w11x,t=Av11x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(66)v21x,t=±2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2α,u21x,t=αv21x,t+βv21x,t+γ,w21x,t=Av21x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as (67)v31x,t=-β2α,u31x,t=αv31x,t+βv31x,t+γ,w31x,t=Av31x,t+B.Result 2.
Using parameter values specified in Case 2 as shown in (62) and ξ=k(x-1/4α(4k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(68)v12x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-β2α,u12x,t=αv12x,t+βv12x,t+γ,w12x,t=Av12x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(69)v22x,t=±k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-β2α,u22x,t=αv22x,t+βv22x,t+γ,w22x,t=Av22x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as(70)v32x,t=∓kC-1/αCξ+D-β2α,u32x,t=αv32x,t+βv32x,t+γ,w32x,t=Av32x,t+B,for which the traveling wave transformation for this case is ξ=k(x-6γ/β2+2t).Result 3.
Using parameter values specified in Case 3 as shown in (63) and ξ=k(x-1/4α(-8k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(71)v13x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,u13x,t=αv13x,t+βv13x,t+γ,w13x,t=Av13x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(72)v23x,t=±k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D±2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2α,u23x,t=αv23x,t+βv23x,t+γ,w23x,t=Av23x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as(73)v33x,t=∓kC-1/αCξ+D-β2α,u33x,t=αv33x,t+βv33x,t+γ,w33x,t=Av33x,t+B,for which the traveling wave transformation for this case is ξ=k(x-6γ/β2+2t).Result 4.
Using parameter values specified in Case 4 as shown in (64) and ξ=k(x-1/4α(16k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(74)v14x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ∓k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,u14x,t=αv14x,t+βv14x,t+γ,w14x,t=Av14x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(75)v24x,t=±k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D∓2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2α,u24x,t=αv24x,t+βv24x,t+γ,w24x,t=Av24x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as(76)v34x,t=∓kC-1/αCξ+D-β2α,u34x,t=αv34x,t+βv34x,t+γ,w34x,t=Av34x,t+B,for which the traveling wave transformation for this case is ξ=k(x-6γ/β2+2t).Lu et al. [52] applied the new improved Riccati equation method to system (44) and then found that its exact solutions were expressed in terms of fractions of trigonometric functions and fractions of hyperbolic functions. El-Wakil and Abdou [53] found some exact solutions for system (44) by the modified extended tanh-function method. All of the obtained solutions had the forms of the following functions: tanh, coth, tan, and cot. It is not difficult to check that our exact solutions of system (44), which are shown in (65)-(76), are the functions whose mathematical forms are equivalent to those of most of the solutions found previously by choosing suitable parameters and some of them are obviously novel.In Figures3 and 4, we present the three-dimensional and two-dimensional plots of some exact solutions, which are v14(x,t) and v24(x,t), expressed in (74) and (75), respectively. They are depicted using the sets of the appropriate parameter values as described below. Employing c=-7.25,α=-0.23077, which are solved implicitly using (56), (61) and the parameter values k=0.5,β=1,γ=-4,μ=1.5,λ=1, and choosing C=D=1, we obtain the plots of the selected exact traveling wave solution(77)v14x,t=-k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,as shown in Figure 3 which represents the periodic wave solutions.Figure 3
The periodic wave solutions ofv14(x,t) in (77) when k=0.5, β=1, γ=-4, μ=1.5, λ=1, c=-7.25, α=-0.23077, and C=D=1.
(a)
3D graph (b)
2D graph whent=0Figure 4
The singular wave solutions ofv24(x,t) in (78) when k=1, β=2, γ=2, μ=2, λ=-7/8, c=2, α=-1, C=-3, and D=5.
(a)
3D graph (b)
2D graph whent=0Again implicitly solving (56) and (61) via using k=1,β=2,γ=2,μ=2,λ=-7/8, we obtain c=2,α=-1. Then the graphical representations of the chosen exact solution(78)v24x,t=-k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D+2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2αusing the parameters as mentioned above and C=-3,D=5 are simulated in Figure 4 describing the singular wave solutions.


## 3.1. The Benney-Luke Equation

In this section, we will provide a use of theG′/G2-expansion method for seeking exact solitary wave solutions of the Benney-Luke equation, which is used to approximate the full water wave equations and appropriately described two-way water wave propagation with surface tension. The equation can be written in the following form [44, 45]:(11)utt-uxx+αuxxxx-βuxxtt+utuxx+2uxuxt=0,where α,β are the positive integers such that their difference is in terms of the inverse bond number capturing the effects of gravity forces and surface tension.Using the traveling wave transformationξ=kx-ct, (11) is converted into the following ODE in the variable U=U(ξ):(12)c2-k2U′′+αk4-βk2c2U4-3ck2U′U′′=0.Integrating (12) with respect to ξ once and then choosing the constant of integration to be zero, we obtain the following ODE:(13)2c2-k2U′+2αk4-βk2c2U′′′-3ck2U′2=0,for which the homogeneous balance principle is applied. Following Step 3 of the mentioned method, the highest order derivative U′′′ and the nonlinear term of the highest order (U′)2 are balanced via using formula (6) as follows:(14)degU′′′=N+3=degU′2=2N+1,which leads to N=1. Hence, the form of exact solutions of the ODE in (13) using the G′/G2-expansion method can be expressed as(15)Uξ=a0+a1G′G2+b1G′G2-1,where a0,a1,b1 are unknown constants. Substituting (15) into (13) along with (5), then collecting all the coefficients with the same power of G′/G2i,(i=0,±1,±2,…), and finally setting these resulting coefficients to be zero, we consequently obtain the following system of algebraic equations in a0,a1,b1,k,c,μ,λ,α,β:(16)G′G2-4:-12αk4μ3b1+12βc2k2μ3b1-3ck2μ2b12=0,G′G2-2:-16αk4λμ2b1+16βc2k2λμ2b1-6ck2λμb12+6ck2μ2a1b1-2c2μb1+2k2μb1=0,G′G20:-4αk4λ2μb1+4αk4λμ2a1+4βc2k2λ2μb1-4βc2k2λμ2a1-3ck2λ2b12+12ck2λμa1b1-3ck2μ2a12-2c2λb1+2c2μa1+2k2λb1-2k2μa1=0,G′G22:16αk4λ2μa1-16βc2k2λ2μa1+6ck2λ2a1b1-6ck2λμa12+2c2λa1-2k2λa1=0,G′G24:12αk4λ3a1-12βc2k2λ3a1-3ck2λ2a12=0.Solving the obtained algebraic system (16) by use of Maple, we get the following three cases.Case 1.
(17) a 0 = a 0 , a 1 = 0 , b 1 = ∓ 4 k α - β μ 4 α k 2 λ μ + 1 4 β k 2 λ μ + 1 , k = k , c = ± 4 α k 2 λ μ + 1 k 4 β k 2 λ μ + 1 ,where a0,k,α,β,μ,λ are arbitrary constants.Case 2.
(18) a 0 = a 0 , a 1 = ± 4 k α - β λ 4 α k 2 λ μ + 1 4 β k 2 λ μ + 1 , b 1 = 0 , k = k , c = ± 4 α k 2 λ μ + 1 k 4 β k 2 λ μ + 1 ,where a0,k,α,β,μ,λ are arbitrary constants.Case 3.
(19) a 0 = a 0 , a 1 = ± 4 k α - β λ 16 α k 2 λ μ + 1 16 β k 2 λ μ + 1 , b 1 = ∓ 4 k α - β μ 16 α k 2 λ μ + 1 16 β k 2 λ μ + 1 , k = k , c = ± 16 α k 2 λ μ + 1 k 16 β k 2 λ μ + 1 ,where a0,k,α,β,μ,λ are arbitrary constants.
When we substitute the above three cases of the obtained parameters along with the functionsG′/G2 specified in (7)-(10) into the solution form (15), we can write three results of solutions of (11) as follows.Result 1.
From Case 1 in (17), we have ξ=kx∓4αk2λμ+1k/4βk2λμ+1t and the following exact solutions.
Whenμλ>0, the trigonometric function solution corresponding to the parameter values in Case 1 can be written as(20)u11x,t=a0∓4kα-βμλ4αk2λμ+14βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ-1.
Whenμλ<0, the exponential function solution associated with the parameter values in Case 1 can be expressed as(21)u21x,t=a0∓8kα-βμλ4αk2λμ+14βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D-1.
Whenμ=0,λ≠0, the exact solution corresponding to the parameter values in Case 1 is u31(x,t)=a0, which is the constant solution.Result 2.
From Case 2 in (18), we have ξ=kx∓4αk2λμ+1k/4βk2λμ+1t and the following exact solutions.
Whenμλ>0, the trigonometric function solution corresponding to the parameter values in Case 2 can be written as(22)u12x,t=a0±4kα-βμλ4αk2λμ+14βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ.
Whenμλ<0, the exponential function solution associated with the parameter values in Case 2 can be expressed as(23)u22x,t=a0±2kα-β4αk2λμ+14βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D.
Whenμ=0,λ≠0, the rational function solution corresponding to the parameter values in Case 2 can be expressed as(24)u32x,t=a0∓4kα-βCCξ+D,for which the traveling wave transformation for this case is ξ=k(x∓t).Result 3.
From Case 3 in (19), we have ξ=kx∓16αk2λμ+1k/16βk2λμ+1t and the following exact solutions.
Whenμλ>0, the trigonometric function solution corresponding to the parameter values in Case 3 can be written as(25)u13x,t=a0±4kα-βμλ16αk2λμ+116βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ∓4kα-βμλ16αk2λμ+116βk2λμ+1Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ-1.
Whenμλ<0, the exponential function solution associated with the parameter values in Case 3 can be expressed as(26)u23x,t=a0±2kα-β16αk2λμ+116βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D∓8kα-βμλ16αk2λμ+116βk2λμ+12μλ-4Cμλe2ξμλCe2ξμλ-D-1.
Whenμ=0,λ≠0, the rational function solution corresponding to the parameter values in Case 3 can be expressed as(27)u33x,t=a0∓4kα-βCCξ+D,for which the traveling wave transformation for this case is ξ=k(x∓t).Akter and Akbar [44] utilized the modified simple equation method to obtain exact solutions of (11) which were written as fractions of exponential functions. They can be transformed into the tanh and coth functions for which the arbitrary constants are selected appropriately as shown in their paper. Islam et al. [45] demonstrated the application of the improved F-expansion method with Riccati equation to obtain exact traveling wave solutions of (11). Their results were expressed in terms of the following functions:(i)
tanh, coth, their reciprocals, and their summations(ii)
tan, cot, their reciprocals, and their summations(iii)
rational functions.However, our obtained results in (20)-(27) are more generalized than the ones described above; i.e., selecting the appropriate constants C and D in our solutions can lead to the solutions obtained by other existing methods mentioned previously. Here we present the plots of the exact solution u21(x,t) in (21) with the positive formula using the following parameter values: α=2,β=3,k=1,a0=1,μ=0.5,λ=-1,C=D=1. The three-dimensional and two-dimensional plots of this solution are portrayed in Figure 1 demonstrating the solitary wave solution of kink type.Figure 1
The solitary wave solutions of kink type foru21(x,t) in (21) when α=2, β=3, k=1, a0=1, μ=0.5, λ=-1, and C=D=1.
(a)
3D graph (b)
2D graph whent=0


## 3.2. Equation of Nanoionic Currents along Microtubules

In this section, we will show an application of theG′/G2-expansion method in nanobiosciences. One of the important models in such fields is the nonlinear transmission line model for nanoionic currents along microtubules (MTs) segmented into identical elementary rings (ERs). The model is playing an important role in cellular signalling and the elaborated details of derivation of the equation can be found in [46, 47]. The equation of nanoionic currents along MTs is described as follows [47]:(28)l23uxxx+Z3/2lwG0-2δC0uut+2ux+ZC0lut+1lRZ-1-G0Zu=0,where R=0.34×109  Ω is the resistance of the ER with length l=8×10-9 m,C0=1.8×10-15 F is the total maximal capacitance of the ER, G0=1.1×10-13 Si is conductance of pertaining nanopores (NPs), and Z=5.56×1010  Ω is the characteristic impedance of the system. Parameters ω and δ represent conductance of NPs in ER and nonlinearity of ER capacitor, respectively.Using the dimensionless wave variableξ=1/lx-c/τt, where c is the dimensionless velocity of wave and τ=RC0=0.6×10-6 s is the characteristic time of charging ER capacitor, we obtain the traveling wave transformation u(x,t)=U(ξ) and then (28) is converted to the following new ODE:(29)U′′′+αcUU′+6-βcU′+γU=0,where α=3Z3/2/τ(2δC0-wG0), β=3ZC0/τ, and γ=3(RZ-1-G0Z).Balancing between the highest order derivativeU′′′ and the nonlinear term of the highest order UU′ by using formula (6), we obtain(30)deg⁡U′′′=N+3=deg⁡UU′=2N+1⟹N=2.Thus, the form of exact solutions of (29) using the G′/G2-expansion method can be expressed as(31)Uξ=a0+a1G′G2+a2G′G22+b1G′G2-1+b2G′G2-2,where a0,a1,a2,b1,b2 are unknown constants. Substituting (31) into (29) along with (5), then collecting all the coefficients with the same power of G′/G2i,(i=0,±1,±2,…), and finally setting these resulting coefficients to be zero, we consequently obtain the following system of algebraic equations in a0,a1,a2,b1,b2,c,μ,λ,α,β,γ:(32)G′G2-5:-2αcμb22-24μ3b2=0,G′G2-4:-3αcμb1b2-6μ3b1=0,G′G2-3:-2αcλb22-2αcμa0b2-αcμb12+2βcμb2-40λμ2b2-12μb2=0,G′G2-2:-3αcλb1b2-αcμa0b1-αcμa1b2+βcμb1-8λμ2b1+γb2-6μb1=0,G′G2-1:-2αcλa0b2-αcλb12+2βcλb2-16λ2μb2+γb1-12λb2=0,G′G20:-αcλa0b1-αcλa1b2+αcμa0a1+αcμa2b1+βcλb1-βcμa1-2λ2μb1+2λμ2a1+γa0-6λb1+6μa1=0,G′G21:2αcμa0a2+αcμa12-2βcμa2+16λμ2a2+γa1+12μa2=0,G′G22:αcλa0a1+αcλa2b1+3αcμa1a2-βcλa1+8λ2μa1+γa2+6λa1=0,G′G23:2αcλa0a2+αcλa12+2αcμa22-2βcλa2+40λ2μa2+12λa2=0,G′G24:3αcλa1a2+6λ3a1=0,G′G25:2αcλa22+24λ3a2=0.Using Maple to solve algebraic system (32), we obtain the following three cases.Case 1.
(33) a 0 = β c - 8 λ μ - 6 α c , a 1 = 0 , a 2 = 0 , b 1 = 0 , b 2 = - 12 μ 2 α c , c = c ,with c≠0,μ,λ which are arbitrary constants.Case 2.
(34) a 0 = β c - 8 λ μ - 6 α c , a 1 = 0 , a 2 = - 12 λ 2 α c , b 1 = 0 , b 2 = 0 , c = c ,with c≠0,μ,λ which are arbitrary constants.Case 3.
(35) a 0 = β c - 8 λ μ - 6 α c , a 1 = 0 , a 2 = - 12 λ 2 α c , b 1 = 0 , b 2 = - 12 μ 2 α c , c = c ,with c≠0,μ,λ which are arbitrary constants.Inserting the above three cases in (33)-(35) along with the functions G′/G2 described in (7)-(10) into the solution form (31), we attain three results of solutions of (28) as follows.Result 1.
Using parameter values of Case 1 in (33) and ξ=1/lx-c/τt, we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of (28) can be written as(36)u11x,t=βc-8μλ-6αc-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-2.
Whenμλ<0, the exponential function solution of (28) can be expressed as (37)u21x,t=βc-8μλ-6αc-48μ2λ2αc2μλ-4Cμλe2ξμλCe2ξμλ-D-2.
Whenμ=0,λ≠0, the exact solution of (28) is the constant solution u31(x,t)=βc-6/αc.Result 2.
Using parameter values of Case 2 in (34) and ξ=1/lx-c/τt, we have the following exact solutions.Whenμλ>0, the trigonometric function solution of (28) can be written as(38)u12x,t=βc-8μλ-6αc-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ2.Whenμλ<0, the exponential function solution of (28) can be expressed as(39)u22x,t=βc-8μλ-6αc-3αc2μλ-4Cμλe2ξμλCe2ξμλ-D2.Whenμ=0,λ≠0, the rational function solution of (28) can be expressed as(40)u32x,t=βc-6αc-12C2αcCξ+D2.Result 3.
Using parameter values of Case 3 in (35) and ξ=1/lx-c/τt, we have the following exact solutions.Whenμλ>0, the trigonometric function solution of (28) can be written as(41)u13x,t=βc-8μλ-6αc-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ2-12μλαcCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-2.Whenμλ<0, the exponential function solution of (28) can be expressed as(42)u23x,t=βc-8μλ-6αc-3αc2μλ-4Cμλe2ξμλCe2ξμλ-D2-48μ2λ2αc2μλ-4Cμλe2ξμλCe2ξμλ-D-2.Whenμ=0,λ≠0, the rational function solution of (28) can be expressed as(43)u33x,t=βc-6αc-12C2αcCξ+D2.Satarić et al. [46] firstly proposed model (28) and obtained its analytical solution by converting (28) with the change of variables and the use of the appropriate boundary conditions into the solvable ODE in a new variable. The analytical solution was expressed in terms of a square of the exponential secant function. Later, Sekulić et al. [47] applied the modified extended tanh-function method to solve (28) for exact traveling wave solutions. The solutions were written as the square of the following functions: tan, cot, tanh, and coth. It is not difficult to verify that the mathematical structures of their exact solutions are the particular cases of those of our solutions as shown in (36), (37), (38), (39), (41), and (42). Meanwhile our rational function solutions do not appear in their work. In addition, Zayed and Alurrfi [48] utilized the (G′/G,1/G)-expansion method to analytically obtain the exact traveling wave solutions of problem (28). Their solutions included the solitary wave solutions and the periodic wave solutions when the appropriate values of the parameters were particularly selected. Comparing the mathematical structures of their specific solutions with our obtained solutions, almost all of their specific solutions can be obtained from our solutions. For instance, solutions (20), (23), and (24) in their paper are structurally equivalent to the hyperbolic form of the solution in (39) via using C=D=1 and C=1,D=-1. The trigonometric solution (28) in their paper has the same mathematical structure as our solution in (38) with C=D=1 and C=-1,D=1. Solutions (29), (32), and (33) in their article can be equivalently transformed to our solution in (38) by choosing C=0,D=1 and C=1,D=0. Finally, rational solutions (36) and (37) in their paper with ξ of degree two in the denominators are structurally equivalent to our rational solution in (40). Furthermore, the exact solutions in (41) and (42) can generate other types of solutions different from the compared ones.In [46, 47], they used the estimated dimensionless parameter σ=1.67×102 and took σc=2.5,2δ-wG0/C0=0.1 for plotting their solutions. Using the parameter values mentioned above and choosing μ=0.5,λ=-1,C=1,D=-1, we obtain the graphical representations of the exact traveling wave solution u22(x,t) in (39) as demonstrated in Figure 2 describing the soliton solution of bell-type. The obtained three-dimensional graph shown in Figure 2(a) is similar to their results.Figure 2
The soliton solution of bell-type foru22(x,t) in (39) when α=1.8×107, β=500.4, c=0.01497, μ=0.5, λ=-1, C=1, and D=-1.
(a)
3D graph (b)
2D graph whent=0


## 3.3. The Generalized Hirota-Satsuma Coupled KdV System

In 1982 Satsuma and Hirota proposed the new system of equations which is called the generalized Hirota-Satsuma coupled KdV system as follows [49]:(44)Ut=14Uxxx+3UUx+3-V2+Wx,Vt=-12Vxxx-3UVx,Wt=-12Wxxx-3UWx.The above system can be obtained from the four reductions of KP hierarchy. In particular, the well-known Hirota-Satsuma coupled KdV system [50], which was derived in 1981 by Hirota and Satsuma to describe interactions of two long waves with different dispersion relations, can be obtained by setting W=0 in (44). We want to obtain traveling wave solutions for system (44) which are in the following form:(45)Ux,t=uξ,Vx,t=vξ,Wx,t=wξ,ξ=kx-ct,where k and c are nonzero arbitrary constants to be determined later. Substituting (45) into (44), we yield a system of ODEs as follows:(46)-cku′=14k3u′′′+3kuu′+3k-v2+w′,(47)-ckv′=-12k3v′′′-3kuv′,(48)-ckw′=-12k3w′′′-3kuw′.Let [51](49)u=αv2+βv+γ,w=Av+Bwhere α,β,γ,A, and B are constants to be also determined later.Substituting (49) into (47) and (48) and then integrating once, we know that (47) and (48) give the same equation as follows:(50)k2v′′=-2αv3-3βv2+2c-3γv+c1,where c1 is a constant of integration. Multiplying (50) by v′ and then integrating the resulting equation with respect to ξ, we obtain(51)k2v′2=-αv4-2βv3+2c-3γv2+2c1v+c2,where c2 is also a constant of integration.From (49)-(51), we obtain(52)k2u′′=2αk2v′2+k22αv+βv′′,=2α-αv4-2βv3+2c-3γv2+2c1v+c2+2αv+β-2αv3-3βv2+2c-3γv+c1.Integrating (46) once, we get(53)14k2u′′+32u2+cu+3-v2+w+c3=0,where c3 is a constant of integration. Substituting (49) and (52) into (53), we yield the following system:(54)3αc-3αγ+34β2-3=0,12αc1+βc+γβ+A=0,142αc2+βc1+32γ2+cγ+3B+c3=0.Let(55)c1=12α2β3+2cαβ-6αβγ,vξ=aPξ-β2α.We find from (54) that(56)α=β2-44γ-c,A=4βc-γβ2-4,B=16-γ+cβ2-4216c3cβ2-2c3cβ4-16c3γβ2+2c3γβ4+56c2γβ2-48γ2cβ2-16c2+14c2β6-3c2β4+12c2β2-16γ2c-32c2γ-8c3β2+β4γ3-2β4c3+32c3γ-32c3c+48γ3+β4γ2c.From (50), we therefore obtain(57)ak2P′′-a2c-6γ+3β22αP+2αa3P3=0.Applying the homogeneous balance principle and (6) mentioned in Step 3 to the terms P′′ and P3, we then have that(58)deg⁡P′′=N+2=deg⁡P3=3N,which leads to N=1. Hence, the form of exact solutions of the ordinary differential equation in (57) using the G′/G2-expansion method is(59)Pξ=a0+a1G′G2+b1G′G2-1,where a0,a1,b1 are unknown constants. Substituting (59) into (57) along with (5), then collecting all the coefficients with the same power of G′/G2i,(i=0,±1,±2,…), and finally setting these resulting coefficients to be zero, we consequently attain the following system of algebraic equations in a0,a1,b1,k,c,a,α,β,γ:(60)G′G2-3:2αa3b13+2ab1k2μ2=0,G′G2-2:6αa0a3b12=0,G′G2-1:2ab1μλk2-3ab1β22α-2acb1+6aγb1+6αb1a3a02+6αa1a3b12=0,G′G20:-3aβ2a02α-2aca0+6aγa0+2αa3a03+12αa3a0a1b1=0,G′G21:2k2aa1λμ-3aβ2a12α-2aca1+6aγa1+6αa3a02a1+6αa3a12b1=0,G′G22:6αa3a0a12=0,G′G23:2a3αa13+2ak2λ2a1=0.By solving the nonlinear system in (60) with Maple, we obtain the following cases.Case 1.
(61) a 0 = 0 , a 1 = 0 , b 1 = ± - 1 / α μ k a , k = k , c = 1 4 α 4 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Case 2.
(62) a 0 = 0 , a 1 = ± - 1 / α λ k a , b 1 = 0 , k = k , c = 1 4 α 4 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Case 3.
(63) a 0 = 0 , a 1 = ± - 1 / α λ k a , b 1 = ± - 1 / α μ k a , k = k , c = 1 4 α - 8 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Case 4.
(64) a 0 = 0 , a 1 = ± - 1 / α λ k a , b 1 = ∓ - 1 / α μ k a , k = k , c = 1 4 α 16 k 2 λ μ α + 12 α γ - 3 β 2 ,where k,a≠0,β,γ,μ,λ are arbitrary constants, and α≠0 is expressed in (56).Inserting the above four cases shown in (61)-(64) along with the functions G′/G2 described in (7)-(10) into the solution form (59), we obtain four results of solutions of system (44) as follows.Result 1.
Using parameter values specified in Case 1 as shown in (61) and ξ=k(x-1/4α(4k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(65)v11x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,u11x,t=αv11x,t+βv11x,t+γ,w11x,t=Av11x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(66)v21x,t=±2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2α,u21x,t=αv21x,t+βv21x,t+γ,w21x,t=Av21x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as (67)v31x,t=-β2α,u31x,t=αv31x,t+βv31x,t+γ,w31x,t=Av31x,t+B.Result 2.
Using parameter values specified in Case 2 as shown in (62) and ξ=k(x-1/4α(4k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(68)v12x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-β2α,u12x,t=αv12x,t+βv12x,t+γ,w12x,t=Av12x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(69)v22x,t=±k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-β2α,u22x,t=αv22x,t+βv22x,t+γ,w22x,t=Av22x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as(70)v32x,t=∓kC-1/αCξ+D-β2α,u32x,t=αv32x,t+βv32x,t+γ,w32x,t=Av32x,t+B,for which the traveling wave transformation for this case is ξ=k(x-6γ/β2+2t).Result 3.
Using parameter values specified in Case 3 as shown in (63) and ξ=k(x-1/4α(-8k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(71)v13x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,u13x,t=αv13x,t+βv13x,t+γ,w13x,t=Av13x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(72)v23x,t=±k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D±2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2α,u23x,t=αv23x,t+βv23x,t+γ,w23x,t=Av23x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as(73)v33x,t=∓kC-1/αCξ+D-β2α,u33x,t=αv33x,t+βv33x,t+γ,w33x,t=Av33x,t+B,for which the traveling wave transformation for this case is ξ=k(x-6γ/β2+2t).Result 4.
Using parameter values specified in Case 4 as shown in (64) and ξ=k(x-1/4α(16k2λμα+12αγ-3β2)t), we have the following exact solutions.
Whenμλ>0, the trigonometric function solution of system (44) can be written as(74)v14x,t=±k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ∓k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,u14x,t=αv14x,t+βv14x,t+γ,w14x,t=Av14x,t+B.
Whenμλ<0, the exponential function solution of system (44) can be expressed as(75)v24x,t=±k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D∓2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2α,u24x,t=αv24x,t+βv24x,t+γ,w24x,t=Av24x,t+B.
Whenμ=0,λ≠0, the rational function solution of system (44) can be written as(76)v34x,t=∓kC-1/αCξ+D-β2α,u34x,t=αv34x,t+βv34x,t+γ,w34x,t=Av34x,t+B,for which the traveling wave transformation for this case is ξ=k(x-6γ/β2+2t).Lu et al. [52] applied the new improved Riccati equation method to system (44) and then found that its exact solutions were expressed in terms of fractions of trigonometric functions and fractions of hyperbolic functions. El-Wakil and Abdou [53] found some exact solutions for system (44) by the modified extended tanh-function method. All of the obtained solutions had the forms of the following functions: tanh, coth, tan, and cot. It is not difficult to check that our exact solutions of system (44), which are shown in (65)-(76), are the functions whose mathematical forms are equivalent to those of most of the solutions found previously by choosing suitable parameters and some of them are obviously novel.In Figures3 and 4, we present the three-dimensional and two-dimensional plots of some exact solutions, which are v14(x,t) and v24(x,t), expressed in (74) and (75), respectively. They are depicted using the sets of the appropriate parameter values as described below. Employing c=-7.25,α=-0.23077, which are solved implicitly using (56), (61) and the parameter values k=0.5,β=1,γ=-4,μ=1.5,λ=1, and choosing C=D=1, we obtain the plots of the selected exact traveling wave solution(77)v14x,t=-k-μλαCcosμλξ+DsinμλξDcosμλξ-Csinμλξ-Ccosμλξ+DsinμλξDcosμλξ-Csinμλξ-1-β2α,as shown in Figure 3 which represents the periodic wave solutions.Figure 3
The periodic wave solutions ofv14(x,t) in (77) when k=0.5, β=1, γ=-4, μ=1.5, λ=1, c=-7.25, α=-0.23077, and C=D=1.
(a)
3D graph (b)
2D graph whent=0Figure 4
The singular wave solutions ofv24(x,t) in (78) when k=1, β=2, γ=2, μ=2, λ=-7/8, c=2, α=-1, C=-3, and D=5.
(a)
3D graph (b)
2D graph whent=0Again implicitly solving (56) and (61) via using k=1,β=2,γ=2,μ=2,λ=-7/8, we obtain c=2,α=-1. Then the graphical representations of the chosen exact solution(78)v24x,t=-k2-1α2μλ-4Cμλe2ξμλCe2ξμλ-D+2μλk-1α2μλ-4Cμλe2ξμλCe2ξμλ-D-1-β2αusing the parameters as mentioned above and C=-3,D=5 are simulated in Figure 4 describing the singular wave solutions.


## 4. Conclusions

In this paper, theG′/G2-expansion method has been applied to find some new forms of the explicit exact solutions of the three problems, i.e., the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The explicit exact solutions of the problems, which are obtained by the method, can be considered as a part of the gigantic variety of possible solution forms; however, they have provided physical representations in each problem. As shown in Section 3, we have found that the obtained exact solutions of the problems are expressed in terms of trigonometric, exponential (or equivalently hyperbolic), and rational functions. The investigation demonstrates that the method is considerably efficient and practically appropriate for analytically solving such problems with the aid of Maple. All of the obtained exact solutions of each problem have ensured the correctness by substituting them back into the original equations. Moreover, the method could also be employed efficiently for a broad range of NPDEs of integer orders.



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*Source: 7628651-2018-06-03.xml*