A Generalized and Improved (𝐺^′/𝐺)-Expansion Method for Nonlinear Evolution Equations

Abstract

A generalized and improved (𝐺^′/𝐺)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering.

1. Introduction

The world around us is inherently nonlinear, and nonlinear evolution equations (NLEEs) are widely used as models to describe the complex physical phenomena. The exact solutions of NLEEs play a vital role in nonlinear science and engineering. One of the fundamental problems for these models is to obtain their travelling wave solutions. The interest of finding travelling wave solution of NLEEs is increasing and has now become a hot topic to researchers. In recent years, many researchers who are interested in the nonlinear physical phenomena investigated exact solutions of NLEEs. They established many powerful and direct methods. For instance, the inverse scattering method [1], the Backlund transform method [2, 3], the Hirota’s bilinear transformation method [4], the truncated Painleve expansion method [5, 6], the Exp-function method [7–11], the tanh-function method [12–15], the Weierstrass elliptic function method [16], the Jacobi elliptic function expansion method [17–23], and so on.

Recently, Wang et al. [24] introduced a widely used straightforward method called the (𝐺^′/𝐺)-expansion method for obtaining the travelling wave solutions of various NLEEs, where 𝐺(𝜉) satisfies the second-order linear ordinary differential equation (ODE) 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, and 𝜆and 𝜇 are arbitrary constants. Applications of the (𝐺^′/𝐺)-expansion method, to NLEEs can be found in the articles [25–29] for better understanding.

To show the effectiveness and reliability of the (𝐺′/𝐺)-expansion method and to expand the range of its applicability, further research has been carried out by several researchers. Such as, Guo and Zhou [30] proposed the extended (𝐺′/𝐺)-expansion method in which the solutions are presented in the form 𝑢=𝑎₀+∑^𝑛_𝑖=1{𝑎_𝑖(𝐺^′/𝐺)^𝑖+𝑏_𝑖(𝐺^′/𝐺)^𝑖−1𝜎(1+(1/𝜇)(𝐺^′/𝐺)²} and obtained new travelling wave solutions of the Whitham-Broer-Kaup-like equation and couple Hirota-Satsuma KdV equations. Applying this extended method Zayed and Al-Joudi [31] constructed the traveling wave solutions of some nonlinear evolution equations. Zayed and El-Malky [32] also applied the extended (𝐺^′/𝐺)-expansion method to higher-dimensional evolution equations. Zhang et al. [33] presented an improved (𝐺^′/𝐺)-expansion method to seek general travelling wave solutions. In the original method the solution is presented as nonnegative power of (𝐺′/𝐺), but in [33] Zhang et al. proposed that the power may be any integral number. Zayed and Gepreel [34] employed the improved (𝐺^′/𝐺)-expansion method to Konopelchenko-Dubrovsky equation, Karsten-Krasil’ Shchik equation, Whitham-Broer-Kaup equation, and the fifth-order KdV equations to construct traveling wave solutions. Zayed [35] presented a new approach of the (𝐺^′/𝐺)-expansion method where 𝐺(𝜉) satisfies the Jacobi elliptical equation [𝐺^′(𝜉)]²=𝑒₂𝐺⁴(𝜉)+𝑒₁𝐺²(𝜉)+𝑒₀,𝑒₂,𝑒₁,𝑒₀ that are arbitrary constants and obtained new exact solutions of some NLEEs. Zayed [36] again presented a further alternative approach of this method in which 𝐺(𝜉) satisfies the Riccati equation 𝐺^′(𝜉)=𝐴+𝐵𝐺²(𝜉), where 𝐴 and 𝐵 are arbitrary constants.

Still substantial work has to be done in order for the (𝐺^′/𝐺)-expansion method to be well established, since every nonlinear equation has its own physically significant rich structure. In this paper, we propose a generalized and improved (𝐺^′/𝐺)-expansion method for solving NLEEs in mathematical physics and engineering. To show the reliability and advantages of the proposed method, the KdV equation, the ZKBBM equation, and the strain wave equation in microstructured solids are solved, and further new families of exact solutions are found.

2. Description of the Generalized and Improved (𝐺^′/𝐺)-Expansion Method

Let us consider the nonlinear partial differential equation of the form𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑡𝑥,𝑢_𝑥𝑥,…=0, (2.1)

where 𝑢=𝑢(𝑥,𝑡) is an unknown function, 𝑃 is a polynomial in 𝑢(𝑥,𝑡) and its partial derivatives in which the highest order partial derivatives and the nonlinear terms are involved. The main steps of the method are as follows.

Step 1. Combining the real variables 𝑥 and 𝑡 by a complex variable 𝜉, we suppose that 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥±𝑉𝑡, (2.2)where 𝑉 is the speed of the travelling wave. The transformation (2.2) transforms (2.1) into an ODE for 𝑢=𝑢(𝜉)𝑄𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0, (2.3)where 𝑄 is a function of 𝑢(𝜉) and its total derivatives.

Step 2. Suppose the solution of the ODE (2.3) can be expressed by a polynomial in (𝑑+(𝐺′/𝐺)) as follows: 𝑢(𝜉)=^𝑚_𝑛=−𝑚𝑒_−𝑛𝑑+𝐺^′/𝐺^𝑛, (2.4)where either 𝑎_−𝑚 or 𝑎_𝑚 may be zero, but both 𝑎_−𝑚 and 𝑎_𝑚 can not be zero simultaneously, 𝑎_𝑛(𝑛=0,±1,±2,…,±𝑚) and 𝑑 are constants to be determined later, and 𝐺=𝐺(𝜉) satisfies the following second order linear ODE: 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0. (2.5)

Step 3. The value of the positive integer 𝑚 can be determined by considering the homogeneous balance between the highest-order derivatives and highest-order nonlinear terms appearing in (2.3). If the degree of 𝑢(𝜉) is 𝐷[𝑢(𝜉)]=𝑚, then the degree of the other expressions will be as follows: 𝐷𝑑^𝑝𝑢(𝜉)𝑑𝜉^𝑝=𝑚+𝑝,𝐷𝑢^𝑝𝑑^𝑞𝑢(𝜉)𝑑𝜉^𝑞𝑠=𝑚𝑝+𝑠𝑚+𝑞. (2.6)

Step 4. Substituting (2.4) along with (2.5) into (2.3), we obtain polynomials in (𝑑+𝐺′/𝐺)^𝑚 and (𝑑+𝐺′/𝐺)^−𝑚,(𝑚=0,1,2,3,…). Collecting each coefficient of the resulted polynomials to zero yields a set of algebraic equations for 𝑎_𝑛(𝑛=0,±1,±2,±3,…,±𝑚), 𝑑 and 𝑉.

Step 5. Suppose that the value of the constants 𝑎_𝑛(𝑛=0,±1,±2,±3,…,±𝑚), 𝑑, and 𝑉 can be obtained by solving the algebraic equations obtained in Step 4. Since the general solution of (2.5) is well known for us, substituting the values of 𝑎_𝑛(𝑛=0,±1,±2,±3,…,±𝑚), 𝑑 and 𝑉 into (2.4), we obtain more general type and new exact traveling wave solutions of the nonlinear evolution equation (2.1).

3. Applications of the Proposed Method

In this section, we employ the proposed method to obtain some new and more general exact travelling wave solutions of the celebrated KdV equation, the ZKBBM equation, and the strain wave equation in microstructured solids.

3.1. The KdV Equation

Let us consider the KdV equation,𝑢_𝑡+𝑢𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0. (3.1)

Making use of the travelling wave transformation 𝜉=𝑥−𝑉𝑡, (3.1) is converted into the following ODE:−𝑉𝑢^′+𝑢𝑢^′+𝛿𝑢^′′′=0. (3.2)

Equation (3.2) is integrable, therefore, integrating we obtain that1𝐶−𝑉𝑢+2𝑢²+𝛿𝑢^′′=0, (3.3)

where 𝐶 is an integral constant. Substituting (2.4) into (3.3) and considering the homogeneous balance between the highest-order derivative 𝑢^′′ and the nonlinear term 𝑢², we obtain 𝑚=2.

Therefore, the solution of (3.3) is of the form𝑢(𝜉)=𝑒₂𝑑+𝐺′𝐺²+𝑒₁𝑑+𝐺′𝐺+𝑒₀+𝑒₋₁𝑑+𝐺^′+𝑒/𝐺𝑑+𝐺^−2′/𝐺². (3.4)

Substituting (3.4) into (3.3), the left hand side is transformed into polynomials in (𝑑+(𝐺^′/𝐺))^𝑚 and (𝑑+(𝐺^′/𝐺))^−𝑚,(𝑚=0,1,2,3,…). Equating each coefficient of these polynomials to zero, we obtain a set of simultaneous algebraic equations (we will omit to display them for simplicity) for 𝑒₀, 𝑒₁, 𝑒₂, 𝑒₋₁, 𝑒₋₂, 𝑑, 𝐶, and 𝑉. Solving the set of simultaneous algebraic equations by using the symbolic computation systems, such as Maple 13, we obtain the following.

Case 1. 𝑒₂=−12𝛿, 𝑒₁=12𝛿(2𝑑−𝜆), 𝑒₀=𝑒₀, 𝑒₋₁=0, 𝑒₋₂=0, 𝑑=𝑑, 𝑉=𝛿𝜆²+8𝛿𝜇+𝑒₀+12𝛿𝑑(𝑑−𝜆),𝐶=𝑒²₀/2+8𝛿𝜇𝑒₀+𝛿𝜆²𝑒₀+24𝛿²𝜇²+12𝛿²𝜆²𝜇−144𝛿²𝜆𝑑³+72𝛿²𝑑⁴−96𝛿²𝜆𝜇𝑑−12𝛿𝜆𝑒₀𝑑+12𝛿𝑒₀𝑑²+84𝛿²𝜆²𝑑²+96𝛿²𝜇𝑑²−12𝛿²𝜆³𝑑, (3.5)where 𝑒₀, 𝑑, 𝜆, and 𝜇 are free parameters.

Case 2. 𝑒₂=0, 𝑒₁=0, 𝑒₀=𝑒₀, 𝑒₋₁=−12𝛿𝜆𝜇+𝑑(24𝛿𝑑²−36𝛿𝜆𝑑+24𝛿𝜇+12𝛿𝜆²), 𝑒₋₂=−12𝛿𝜇²+𝑑(24𝛿𝜆𝜇−12𝛿𝑑³−12𝛿𝜆²𝑑−24𝛿𝜇𝑑+24𝛿𝜆𝑑²), 𝑑=𝑑, 𝑉=8𝛿𝜇+𝜆²𝛿+𝑒₀(),𝑒+12𝛿𝑑𝑑−𝜆𝐶=²⁰2+8𝛿𝑒₀𝜇+𝛿𝜆²𝑒₀+24𝛿²𝜇²+12𝛿²𝜆²𝜇−144𝛿²𝜆𝑑³+72𝛿²𝑑⁴−96𝛿²𝜆𝜇𝑑−12𝛿𝜆𝑒₀𝑑+12𝛿𝑒₀𝑑²+84𝛿²𝜆²𝑑²+96𝛿²𝜇𝑑²−12𝛿²𝜆³𝑑, (3.6)where 𝑒₀, 𝑑, 𝜆, and 𝜇 are free parameters.

Case 3. 𝑒₂=−12𝛿, 𝑒₁=0, 𝑒₀=𝑒₀, 𝑒₋₁=0, 𝑒₋₂=−(3/4)𝛿𝜆⁴+6𝛿𝜆²𝜇−12𝛿𝜇², 𝑑=𝜆/2, 𝑉=−2𝛿𝜆²+8𝛿𝜇+𝑒₀,𝐶=−6𝛿²𝜆⁴+48𝛿²𝜆²𝜇−96𝛿²𝜇²−2𝛿𝜆²𝑒₀+8𝛿𝜇𝑒₀+12𝑒²₀, (3.7)where 𝑒₀, 𝜆, and 𝜇 are free parameters.

For Case 1, substituting (3.5) into (3.4) and simplifying, we obtain the following.

When 𝜆²−4𝜇>0, 𝑢₁₁(𝑥,𝑡)=−3𝛿𝜆²−4𝜇𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉²+3𝛿𝜆²+12𝛿𝑑(𝑑−𝜆)+𝑒₀, (3.8)

where 𝜉=𝑥−{(𝛿𝜆²+8𝛿𝜇+𝑒₀)+12𝛿𝑑(𝑑−𝜆)}𝑡, and 𝐴, 𝐵 are arbitrary constants. If 𝐴, 𝐵, 𝑒₀, 𝑑, 𝜆, and 𝜇 take special values, various known results in the literature can be rediscovered.

Suppose that 𝐴>0 and 𝐴²>𝐵², then solution (3.8) reduces to𝑢₁₁(𝑥,𝑡)=3𝛿𝜆²−4𝜇sech²𝜆²−4𝜇2𝜉+𝜉₀+12𝛿𝜇+𝛿𝑑(𝑑−𝜆)+𝑒₀, (3.9)

where 𝜉₀=tanh⁻¹(𝐵/𝐴).

When 𝜆²−4𝜇<0,𝑢₁₁(𝑥,𝑡)=−3𝛿4𝜇−𝜆²−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉²+3𝛿𝜆²+12𝛿𝑑(𝑑−𝜆)+𝑒₀. (3.10)

When 𝜆²−4𝜇=0,𝑢₁₃()𝐵𝑥,𝑡=−12𝛿²(𝐴+𝐵𝜉)²+3𝛿𝜆²+12𝛿𝑑(𝑑−𝜆)+𝑒₀. (3.11)

Again for Case 2, substituting (3.6) into (3.4) and simplifying, we obtain the following.

When 𝜆²−4𝜇>0,𝑢₂₁()=𝑥,𝑡−12𝛿𝜇²+𝑑24𝛿𝜆𝜇−12𝛿𝑑³−12𝛿𝜆²𝑑−24𝛿𝜇𝑑+24𝛿𝜆𝑑²𝜆𝑑−2+𝜆²−4𝜇2𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh((𝜆²−4𝜇)/2)𝜉+𝐵sinh((𝜆²−4𝜇)/2)𝜉²+−12𝛿𝜆𝜇+𝑑24𝛿𝑑²−36𝛿𝜆𝑑+24𝛿𝜇+12𝛿𝜆²𝜆𝑑−2+𝜆²−4𝜇2𝐴sinh𝜆²−4𝜇/2𝜉+𝐵cosh𝜆²−4𝜇/2𝜉𝐴cosh𝜆²−4𝜇/2𝜉+𝐵sinh𝜆²−4𝜇/2𝜉+𝑒₀, (3.12)

where 𝜉=𝑥−{(8𝛿𝜇+𝜆²𝛿+𝑒₀)+12𝑑𝛿(𝑑−𝜆)}𝑡 and 𝐴, 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0,𝑢₂₂()=𝑥,𝑡−12𝛿𝜇²+12𝑑2𝛿𝜆𝜇−𝛿𝑑³−𝛿𝜆²𝑑−2𝛿𝜇𝑑+2𝛿𝜆𝑑²𝜆𝑑−2+4𝜇−𝜆²2−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉²+−12𝛿𝜆𝜇+12𝑑2𝛿𝑑²−3𝛿𝜆𝑑+2𝛿𝜇+𝛿𝜆²𝜆𝑑−2+4𝜇−𝜆²2−𝐴sin4𝜇−𝜆²/2𝜉+𝐵cos4𝜇−𝜆²/2𝜉𝐴cos4𝜇−𝜆²/2𝜉+𝐵sin4𝜇−𝜆²/2𝜉+𝑒₀, (3.13)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑢₂₃(𝑥,𝑡)=−12𝛿𝜇²+𝑑24𝛿𝜆𝜇−12𝛿𝑑³−12𝛿𝜆²𝑑−24𝛿𝜇𝑑+24𝛿𝜆𝑑²(𝑑−(𝜆/2)+(𝐵/(𝐴+𝐵𝜉)))²+−12𝛿𝜆𝜇+𝑑24𝛿𝑑²−36𝛿𝜆𝑑+24𝛿𝜇+12𝛿𝜆²(𝑑−(𝜆/2)+(𝐵/(𝐴+𝐵𝜉)))+𝑒₀, (3.14)

where 𝐴 and 𝐵 are arbitrary constants.

Finally for Case 3, substituting (3.7) into (3.4) and simplifying, we obtain the following.

When 𝜆²−4𝜇>0,𝑢₃₁(𝑥,𝑡)=−3𝛿𝜆²−4𝜇𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉²+−3𝛿𝜆⁴+24𝛿𝜆²𝜇−48𝛿𝜇²𝜆²−4𝜇𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉²+𝑒₀, (3.15)

where 𝜉=𝑥−(−2𝛿𝜆²+8𝛿𝜇+𝑒₀)𝑡, 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0,𝑢₃₂(𝑥,𝑡)=−3𝛿4𝜇−𝜆²−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉²+−3𝛿𝜆⁴+24𝛿𝜆²𝜇−48𝛿𝜇²4𝜇−𝜆²−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉²+𝑒₀. (3.16)

When 𝜆²−4𝜇=0, 𝑢₃₃()𝐵𝑥,𝑡=−12𝛿²(𝐴+𝐵𝜉)²+−3𝛿𝜆⁴+24𝛿𝜆²𝜇−48𝛿𝜇²(𝐴+𝐵𝜉)²4𝐵²+𝑒₀. (3.17)

Wang et al. [24] investigated solutions of the KdV equation by the basic (𝐺′/𝐺)-expansion method and obtained only four solutions. But by our proposed method, we obtain ten distinct solutions of the KdV equation with additional free parameter 𝑑. It is noted that if 𝑑=0 and/or 𝑑=𝜆, then our solutions 𝑢₁₁,𝑢₁₂, and 𝑢₁₃(solutions (3.8)–(3.11)) are identical to the solutions 𝑢₁, 𝑢₂ and 𝑢₃ obtained by Wang et al. [24] (see Section A). But if 𝑑≠0 and 𝑑≠𝜆, the solutions 𝑢₁₁,𝑢₁₂, and 𝑢₁₃ are unlike to Wang et al. [24] solutions. Besides, we obtain additional solutions (3.12)–(3.17) which were not obtained by Wang et al. Therefore, we may assert that the basic (𝐺′/𝐺)-expansion method is a particular case of the proposed generalized and improved (𝐺′/𝐺)-expansion method. It is noteworthy to mention that, if we set special values the parameters, then some of the solutions match to some known solutions obtained by other methods, and some new solutions of the KdV equation are constructed. Since every nonlinear equation has its own physically significant rich structure; therefore, these new solutions will help us to understand the internal mechanism of the complex physical phenomena. Thus, the proposed generalized and improved (𝐺′/𝐺)-expansion method is promising in the discipline of mathematical physics and engineering.

3.2. The ZKBBM Equation

Now we construct the traveling wave solutions of the ZKBBM equation by the proposed method. Let us consider the ZKBBM equation𝑢_𝑡+𝑢_𝑥−2𝑎𝑢𝑢_𝑥−𝑏𝑢_𝑥𝑥𝑡=0. (3.18)

The travelling wave variable 𝜉=𝑥+𝑉𝑡 permits us to reduce (3.18) into the following ODE: (1+𝑉)𝑢^′−2𝑎𝑢𝑢^′−𝑏𝑉𝑢^′′′=0, (3.19)

where prime denotes the derivatives with respect to 𝜉. Equation (3.19) is integrable, therefore, integrating we obtain𝐶+(1+𝑉)𝑢−𝑎𝑢²−𝑏𝑉𝑢^′′=0, (3.20)

where 𝐶 is an integral constant.

Considering the homogeneous balanced between the nonlinear term 𝑢² and the highest-order derivative 𝑢^′′ in (3.20), we get 𝑚=2. Therefore, the formation of solution of (3.20) is same of (3.4).

Substituting (3.4) into (3.20), and collecting all the terms of the same power, the left hand side of (3.20) is converted into polynomials in (𝑑+𝐺′/𝐺)^𝑚 and (𝑑+𝐺′/𝐺)^−𝑚,(𝑚=0,1,2,3,…). Setting each coefficient of these polynomials to zero, we obtain an overdetermined set of algebraic equations (we will omit them to display for simplicity) for 𝑒₀, 𝑒₁, 𝑒₂, 𝑒₋₁, 𝑒₋₂, 𝑑, 𝐶, and 𝑉. Solving this overdetermined set of algebraic equations, we obtain the following.

Case 1. 𝑒₂=0, 𝑒₁=0, 𝑒₀=−(𝑏𝑉𝜆²−𝑉−1+8𝑏𝑉𝜇+12𝑏𝑉𝑑²−12𝑏𝑉𝜆𝑑)/2𝑎, 𝑒₋₁=6𝑏𝑉(−𝜆𝜇+𝜆²𝑑 + 2𝑑³−3𝜆𝑑²+2𝑑𝜇)/𝑎, 𝑒₋₂=−6𝑏𝑉(𝜇²+𝜆²𝑑²+𝑑⁴−2𝜆𝑑³+2𝑑²𝜇−2𝜆𝜇𝑑)/𝑎, 𝑑=𝑑, 𝑉=𝑉, 𝐶=−1−𝑉²−2𝑉+16𝑏²𝑉²𝜇²−8𝑏²𝑉²𝜇𝜆²+𝑏²𝑉²𝜆⁴,4𝑎 (3.21)where 𝑑, 𝑉, 𝜆 and 𝜇 are free parameters.

Case 2. 𝑑=𝑑, 𝑉=𝑉, 𝑒₋₂=0, 𝑒₋₁=0, 𝑒₂=−6𝑏𝑉/𝑎, 𝑒₁=6𝑏𝑉(−𝜆+2𝑑)/𝑎, 𝑒₀=−(8𝑏𝑉𝜇−𝑉−1 + 𝑏𝑉𝜆²+12𝑏𝑉𝑑²−12𝑏𝑉𝜆𝑑)/2𝑎, 𝐶=−8𝑏²𝑉²𝜇𝜆²−2𝑉−1+𝑏²𝑉²𝜆⁴+16𝑏²𝑉²𝜇²−𝑉²,4𝑎 (3.22)where 𝑑, 𝑉, 𝜆, and 𝜇 are free parameters.

Case 3. 𝑑=𝜆/2, 𝑉=𝑉, 𝑒₋₁=0, 𝑒₋₂=3𝑏𝑉(8𝜆²𝜇−𝜆⁴−16𝜇²)/8𝑎, 𝑒₂=−6𝑏𝑉/𝑎, 𝑒₁=0, 𝑒₀=(𝑉+1 + 2𝑏𝑉𝜆²−8𝑏𝑉𝜇𝑉)/2𝑎, 𝐶=−128𝑏²𝑉²𝜆²𝜇−2𝑉−1+16𝑏²𝑉²𝜆⁴+256𝑏²𝑉²𝜇²−𝑉²,4𝑎 (3.23)where 𝑉, 𝜆, and 𝜇 are free parameters.

Now substituting (3.21)–(3.23) into (3.4), we obtain the following solutions of (3.20):𝑢₁(𝜉)=−𝑏𝑉𝜆²−𝑉−1+8𝑏𝑉𝜇+12𝑑(𝑏𝑉𝑑−𝑏𝑉𝜆)−𝜆2𝑎6𝑏𝑉𝜆𝜇−𝑑²+2𝑑²−3𝜆𝑑+2𝜇𝑎𝑑+𝐺^′−𝜇/𝐺6𝑏𝑉²+𝑑𝜆²𝑑+𝑑³−2𝜆𝑑²+2𝑑𝜇−2𝜆𝜇𝑎𝑑+𝐺^′/𝐺²,𝑢₂(𝜉)=−6𝑏𝑉𝑎𝐺𝑑+^′𝐺²+6𝑏𝑉(−𝜆+2𝑑)𝑎𝐺𝑑+^′𝐺−8𝑏𝑉𝜇−𝑉−1+𝑏𝑉𝜆²+12𝑑(𝑏𝑉𝑑−𝑏𝑉𝜆),2𝑎𝑢₃(𝜉)=−6𝑏𝑉𝑎𝜆2+𝐺^′𝐺²+3𝑏𝑉8𝜆²𝜇−𝜆⁴−16𝜇²8𝑎(𝜆/2)+𝐺^′/𝐺²+𝑉+1+2𝑏𝑉𝜆²−8𝑏𝑉𝜇𝑉,2𝑎 (3.24)(3.25)(3.26)

where 𝜉=𝑥+𝑉𝑡.

Substituting the general solutions of (2.5) into (3.24), we obtain three types of travelling wave solutions of the ZKBBM equation as the following.

When 𝜆²−4𝜇>0,𝑢₁₁()=𝜇𝑥,𝑡−6𝑏𝑉²+𝑑𝜆²𝑑+𝑑³−2𝜆𝑑²+2𝑑𝜇−2𝜆𝜇𝑎𝜆𝑑−2+𝜆²−4𝜇2𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉²−6𝑏𝑉𝜆𝜇−𝑑𝜆²+2𝑑²−2𝜆𝑑+2𝜇𝑎𝜆𝑑−2+𝜆²−4𝜇2𝐴cosh𝜆²−4𝜇/2𝜉+𝐵sinh𝜆²−4𝜇/2𝜉𝐴sinh𝜆²−4𝜇/2𝜉+𝐵cosh𝜆²𝜉−−4𝜇/2𝑏𝑉𝜆²+8𝑏𝑉𝜇−𝑉−1+12𝑏𝑉𝑑(𝑑−𝜆),2𝑎 (3.27)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0,𝑢₁₂()=𝜇𝑥,𝑡−6𝑏𝑉²+𝑑𝜆²𝑑+𝑑³−2𝜆𝑑²+2𝑑𝜇−2𝜆𝜇𝑎𝜆𝑑−2+4𝜇−𝜆²2𝐴cos(4𝜇−𝜆²)/2𝜉−𝐵sin(4𝜇−𝜆²)/2𝜉𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉²−6𝑏𝑉𝜆𝜇−𝑑𝜆²+2𝑑²−2𝜆𝑑+2𝜇𝑎𝜆𝑑−2+4𝜇−𝜆²2𝐴cos4𝜇−𝜆²/2𝜉−𝐵sin4𝜇−𝜆²/2𝜉𝐴sin4𝜇−𝜆²/2𝜉+𝐵cos4𝜇−𝜆²𝜉−/2𝑏𝑉𝜆²+8𝑏𝑉𝜇−𝑉−1+12𝑏𝑉𝑑(𝑑−𝜆),2𝑎 (3.28)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑢₁₃()=𝜇𝑥,𝑡−6𝑏𝑉²+𝑑𝜆²𝑑+𝑑³−2𝜆𝑑²+2𝑑𝜇−2𝜆𝜇𝑎(𝑑−(𝜆/2)+(𝐵/(𝐴+𝐵𝜉)))²−6𝑏𝑉𝜆𝜇−𝑑𝜆²+2𝑑²−2𝜆𝑑+2𝜇𝑎(()+((−𝑑−𝜆/2𝐵/𝐴+𝐵𝜉)))𝑏𝑉𝜆²+8𝑏𝑉𝜇−𝑉−1+12𝑏𝑉𝑑(𝑑−𝜆),2𝑎 (3.29)

where 𝐴 and 𝐵 are arbitrary constants.

Again substituting the general solutions of (2.5) into (3.25), we obtain three types of travelling wave solutions of the ZKBBM equation of the following.

When 𝜆²−4𝜇>0,𝑢₂₁()𝜆𝑥,𝑡=−3𝑏𝑉²−4𝜇2𝑎𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉²+2𝑏𝑉𝜆²−8𝑏𝑉𝜇+𝑉+1,2𝑎 (3.30)

where 𝐴 and 𝐵 are arbitrary constants.

If 𝐵>0 and 𝐴²<𝐵², then we can obtain soliton solutions𝑢₂₁()=𝜆𝑥,𝑡3𝑏𝑉²−4𝜇2𝑎sech²𝜆²−4𝜇2𝜉+𝜉₀−𝑏𝑉𝜆²−4𝑏𝑉𝜇−𝑉−1,2𝑎 (3.31)

where 𝜉₀=tanh⁻¹(𝐴/𝐵).

When 𝜆²−4𝜇<0,𝑢₂₂(𝑥,𝑡)=−3𝑏𝑉2𝑎4𝜇−𝜆²𝐴cos(4𝜇−𝜆²)/2𝜉−𝐵sin(4𝜇−𝜆²)/2𝜉𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉²+2𝑏𝑉𝜆²−8𝑏𝑉𝜇+𝑉+1,2𝑎 (3.32)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑢₂₃(𝑥,𝑡)=−6𝑏𝑉𝑎𝐵𝐴+𝐵𝜉²+2𝑏𝑉𝜆²−8𝑏𝑉𝜇+𝑉+1,2𝑎 (3.33)

where 𝐴 and 𝐵 are arbitrary constants.

Finally substituting the general solutions of (2.5) into (3.26), we obtain the travelling wave solutions of the ZKBBM equation as the following.

When 𝜆²−4𝜇>0,𝑢₃₁()=𝜆𝑥,𝑡−3𝑏𝑉²−4𝜇²𝜆8𝑎𝑑+−2+𝜆²−4𝜇2𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉²−6𝑏𝑉𝑎𝜆𝑑+−2+𝜆²−4𝜇2𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉²+𝑉+1+2𝑏𝑉𝜆²−8𝑏𝑉𝜇𝑉,2𝑎 (3.34)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0,𝑢₃₂(𝑥,𝑡)=−6𝑏𝑉𝑎𝜆𝑑−2+4𝜇−𝜆²2𝐴cos(4𝜇−𝜆²)/2𝜉−𝐵sin(4𝜇−𝜆²)/2𝜉𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉²−3𝑏𝑉𝜆²−4𝜇²𝜆8𝑎𝑑−2+4𝜇−𝜆²2𝐴cos(4𝜇−𝜆²)/2𝜉−𝐵sin(4𝜇−𝜆²)/2𝜉𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉²+𝑉+1+2𝑏𝑉𝜆²−8𝑏𝑉𝜇𝑉,2𝑎 (3.35)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑢₃₃(𝑥,𝑡)=−6𝑏𝑉𝑎𝜆𝑑−2+𝐵𝐴+𝐵𝜉²−3𝑏𝑉𝜆²−4𝜇²8𝑎(𝑑−(𝜆/2)+(𝐵/(𝐴+𝐵𝜉)))²+𝑉+1+2𝑏𝑉𝜆²−8𝑏𝑉𝜇𝑉,2𝑎 (3.36)

where 𝐴 and 𝐵 are arbitrary constants.

It is observed that if 𝑑=0, then our solutions (3.27)–(3.33) are identical to Zhang et al. [33] solutions obtained by the improved (𝐺′/𝐺)-expansion method (see Section B). On the other hand, if 𝑑≠0, then the solutions (3.27)–(3.33) are dissimilar to Zhang et al. [33] solutions. Moreover, we obtain solutions (3.34)–(3.36) which were not obtained by Zhang et al. [33]. Therefore, we may stress that the improved (𝐺′/𝐺)-expansion method is also a particular case of our proposed method. It is noticed that the proposed generalized and improved (𝐺′/𝐺)-expansion method performs as a viable tool for finding generalized traveling wave solutions. The performance of the proposed method is trustworthy and efficient and gives more new solutions of nonlinear partial differential equations.

3.3. The Strain Waves Equation in Microstructured Solids

Let us consider an engineering application problem of nonlinear bell-shaped and kink-shaped strain waves in microstructured solids as discussed by Porubov and Pastrone [37]. The governing equation for the strain waves in microstructured solids is given by𝑣_𝑡𝑡−𝑣_𝑥𝑥−𝜀𝛼₁𝑣²_𝑥𝑥−𝛾𝛼₂𝑣_𝑥𝑥𝑡+𝛿𝛼₃𝑣_{𝑥𝑥𝑥𝑥}−𝛿𝛼₄−𝛾²𝛼₇𝑣_{𝑥𝑥𝑡𝑡}+𝛾𝛿𝛼₅𝑣_{𝑥𝑥𝑥𝑥𝑡}+𝛼₆𝑣_{𝑥𝑥𝑡𝑡𝑡}=0. (3.37)

If 𝛾=0, we have the nondissipative case, and governed by the double dispersive equation (see [38] for details),𝑣_𝑡𝑡−𝑣_𝑥𝑥−𝜀𝛼₁𝑣²_𝑥𝑥+𝛿𝛼₃𝑣_{𝑥𝑥𝑥𝑥}−𝛿𝛼₄𝑣_{𝑥𝑥𝑡𝑡}=0. (3.38)

The balance between nonlinearity and dispersion takes place when𝛿=𝑂(𝜀). Therefore, (3.38) becomes,𝑣_𝑡𝑡−𝑣_𝑥𝑥−𝜀𝛼₁𝑣²_𝑥𝑥−𝛼₃𝑣_{𝑥𝑥𝑥𝑥}+𝛼₄𝑣_{𝑥𝑥𝑡𝑡}=0. (3.39)

The travelling wave transformation 𝜉=𝑥−𝑉𝑡 allows us to convert (3.39) into the following ODE:𝑉²−1𝑣^′′−𝜀𝛼₁𝑣^2′′+𝜀𝛼₃−𝑉²𝛼₄𝑣⁽⁴⁾=0, (3.40)

where 𝑣^′′ denotes the second derivative, and 𝑣⁽⁴⁾ denote the fourth derivative with respect to 𝜉. Equation(3.40) is integrable, therefore, integrating we obtain𝑉²−1𝑣−𝜀𝛼₁𝑣²+𝜀𝛼₃−𝑉²𝛼₄𝑣^′′+𝐶=0, (3.41)

where 𝐶 is an integral constant.

Balancing the nonlinear term 𝑣² with the highest-order derivative term 𝑣^′′, from (3.41), we obtain that 𝑚=2. Therefore, the shape of solution of (3.41) is also same as (3.4).

Substituting (3.4) into (3.41), and collecting all the terms of the same power, the left hand side of (3.41) is changed into polynomials in (𝑑+𝐺′/𝐺)^𝑚 and (𝑑+𝐺′/𝐺)^−𝑚,(𝑚=0,1,2,3,…). Setting each coefficient of these polynomials to zero, we obtain a set of simultaneous algebraic equations (we will omit them for simplicity) for 𝑒₀, 𝑒₁, 𝑒₂, 𝑒₋₁, 𝑒₋₂, 𝑑, 𝐶, and 𝑉. Solving this overdetermined set of algebraic equations, we obtain the following.

Case 1. 𝑉=𝑉, 𝑒₋₁=0, 𝑒₋₂=0, 𝑒₂=−6(𝑉²𝛼₄−𝛼₃)/𝛼₁, 𝑒₁=6(2𝑑−𝜆)(𝑉²𝛼₄−𝛼₃)/𝛼₁, 𝑒₀=((−1)/(2𝜀𝛼₁)){𝜀(𝜆²+8𝜇)(𝑉²𝛼₄−𝛼₃)+12𝜀𝑑(𝑑−𝜆)(𝑉²𝛼₄−𝛼₃)+1−𝑉²}, 1𝐶=4𝜀𝛼₁𝜀²𝑉²𝛼₄−𝛼₃𝜆²−4𝜇²−𝑉²−1², (3.42)where 𝑑, 𝑉, 𝜆, and 𝜇 are free parameters.

Case 2. 𝑉=𝑉, 𝑒₂=0, 𝑒₁=0, 𝑒₋₂=−6(𝑉²𝛼₄−𝛼₃)(𝑑²−𝑑𝜆+𝜇)²/𝛼₁, 𝑒₋₁=6(2𝑑−𝜆)(𝑑²−𝑑𝜆+𝜇)(𝑉²𝛼₄−𝛼₃)/𝛼₁, 𝑒₀=−{𝜀(𝜆²+8𝜇)(𝑉²𝛼₄−𝛼₃)+12𝜀𝑑(𝑑−𝜆)(𝑉²𝛼₄−𝛼₃)+1−𝑉²}/2𝜀𝛼₁, 𝜀𝐶=²𝑉²𝛼₄−𝛼₃𝜆²−4𝜇²−𝑉²−1²4𝜀𝛼₁, (3.43)where 𝑑, 𝑉, 𝜆, and 𝜇 are free parameters.

Now substituting (3.42)–(3.43) into (3.4), we obtain the following solutions of (3.41):𝑣₁(𝜉)=−6𝛼₁𝑉²𝛼₄−𝛼₃𝑑+𝐺′𝐺²+6𝛼₁(2𝑑−𝜆)𝑉²𝛼₄−𝛼₃𝑑+𝐺′𝐺−12𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.44)𝑣₂(𝜉)6=−𝛼₁𝑉²𝛼₄−𝛼₃𝑑²−𝑑𝜆+𝜇²𝑑+𝐺′𝐺⁻²+6𝛼₁(2𝑑−𝜆)𝑑²−𝑑𝜆+𝜇𝑉²𝛼₄−𝛼₃𝑑+𝐺′𝐺⁻¹−12𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.45)

where 𝜉=𝑥−𝑉𝑡.

Substituting the general solutions of (2.5) into (3.44), we obtain three types of travelling wave solutions of the strain wave equation in microstructured solids as follows.

When 𝜆²−4𝜇>0,𝑣₁₁()6𝑥,𝑡=−𝛼₁𝑉²𝛼₄−𝛼₃×𝜆𝑑−2+𝜆²−4𝜇2𝐴sinh((𝜆²−4𝜇/2)𝜉)+𝐵cosh((𝜆²−4𝜇/2)𝜉)𝐴cosh((𝜆²−4𝜇/2)𝜉)+𝐵sinh((𝜆²−4𝜇/2)𝜉)²+6𝛼₁(2𝑑−𝜆)𝑉²𝛼₄−𝛼₃×𝜆𝑑−2+𝜆²−4𝜇2𝐴sinh𝜆²−4𝜇/2𝜉+𝐵cosh𝜆²−4𝜇/2𝜉𝐴cosh𝜆²−4𝜇/2𝜉+𝐵sinh𝜆²𝜉−1−4𝜇/22𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.46)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0,𝑣₁₂()6𝑥,𝑡=−𝛼₁𝑉²𝛼₄−𝛼₃×𝜆𝑑−2+4𝜇−𝜆²2−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉²+6𝛼₁(2𝑑−𝜆)𝑉²𝛼₄−𝛼₃×𝜆𝑑−2+4𝜇−𝜆²2−𝐴sin4𝜇−𝜆²/2𝜉+𝐵cos4𝜇−𝜆²/2𝜉𝐴cos4𝜇−𝜆²/2𝜉+𝐵sin4𝜇−𝜆²𝜉−1/22𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.47)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑣₁₃()6𝑥,𝑡=−𝛼₁𝑉²𝛼₄−𝛼₃𝜆𝑑−2+𝐵𝐴+𝐵𝜉²+6𝛼₁(2𝑑−𝜆)𝑉²𝛼₄−𝛼₃𝜆𝑑−2+𝐵−1𝐴+𝐵𝜉2𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.48)

where 𝐴 and 𝐵 are arbitrary constants.

Again substituting the general solutions of (2.5) into (3.45), we obtain the solutions of the stain wave in microstructured solids as follows.

When 𝜆²−4𝜇>0,𝑣₂₁()6𝑥,𝑡=−𝛼₁𝑉²𝛼₄−𝛼₃𝑑²−𝑑𝜆+𝜇²×𝜆𝑑−2+𝜆²−4𝜇2𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉⁻²+6𝛼₁(2𝑑−𝜆)𝑑²−𝑑𝜆+𝜇𝑉²𝛼₄−𝛼₃×𝜆𝑑−2+𝜆²−4𝜇2𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉⁻¹−12𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.49)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0,𝑣₂₂()6𝑥,𝑡=−𝛼₁𝑉²𝛼₄−𝛼₃𝑑²−𝑑𝜆+𝜇²×𝜆𝑑−2+4𝜇−𝜆²2−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉⁻²+6𝛼₁(2𝑑−𝜆)𝑑²−𝑑𝜆+𝜇𝑉²𝛼₄−𝛼₃×𝜆𝑑−2+4𝜇−𝜆²2−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉⁻¹−12𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.50)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑣₂₃()6𝑥,𝑡=−𝛼₁𝑉²𝛼₄−𝛼₃𝑑²−𝑑𝜆+𝜇²𝜆𝑑−2+𝐵𝐴+𝐵𝜉⁻²+6𝛼₁(2𝑑−𝜆)𝑑²−𝑑𝜆+𝜇𝑉²𝛼₄−𝛼₃𝜆𝑑−2+𝐵𝐴+𝐵𝜉⁻¹−12𝜀𝛼₁𝜀𝜆²+8𝜇𝑉²𝛼₄−𝛼₃+12𝜀𝑑(𝑑−𝜆)𝑉²𝛼₄−𝛼₃+1−𝑉², (3.51)

where 𝐴 and 𝐵 are arbitrary constants.

Porubov and Pastrone [37] investigated solutions of strain waves equation in microstructured solids, and they found a solution for (3.39) as follows:𝑣=6𝑘²𝑉²𝛼₄−𝛼₃cosh⁻²(𝑘(𝑥−𝑉𝑡)), (3.52)

where 𝑘²=(𝑉²−1)/(4𝜀(𝑉²𝛼₄−𝛼₃)).

In this paper we obtain six solutions with more free parameters by our proposed expansion method. As a result, the proposed method might be an advance and efficient tool in solving nonlinear equations that arise in the field of engineering problems. If 𝛾≠0, (3.37) can also be solved by the proposed expansion method.

4. Discussions

The advantages and limitations of the proposed expansion method over the basic (𝐺′/𝐺)-expansion method and the improved (𝐺′/𝐺)-expansion method are discussed below.

Advantages: The main advantage of the proposed expansion method over the basic (𝐺′/𝐺)-expansion method is that it provides more new exact traveling wave solutions along with additional free parameters. All the solutions obtained by basic (𝐺′/𝐺)-expansion method are through the proposed expansion method as a particular case, and in addition we obtain some new solutions. To clarify the fact, we apply the method in three equations: two are important in mathematical physics and another one is related to engineering problem, and, in all cases, we obtain some additional new exact solutions. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. Apart from the physical relevance, the close-form solutions of nonlinear evolution equations facilitate the numerical solvers to compare the accuracy of their results and help them in the stability analysis.

In the basic (𝐺′/𝐺)-expansion method, if the order of the reduced ODE (the ODE obtained from the PDE by using traveling wave variable) is equal or less than three, with the help of symbolic computation software, such as Maple 13, it is mostly possible to find out a useful solution of the algebraic equations resulted in Step 4 of Section 2. Otherwise, it is generally unable to guarantee the existence of a solution of the resulted algebraic equations; this is because the number of the equations included in the set of algebraic equations is generally greater than the number of unknowns. But the proposed generalized and improved (𝐺′/𝐺)-expansion method might be used less than or equal to fourth-order-reduced ODE, since it contains more arbitrary constants compared to the basic (𝐺′/𝐺)-expansion method.

Algebraic manipulation of the proposed expansion method with the help of Maple is much easier than the other methods.

Limitations: Sometimes the method gives solutions in disguised versions of known solutions that may be found by other methods, and if the order of the reduced ordinary differential equation is large enough, then the method unable to guarantee the existence of solutions of the resulted algebraic equations.

In [24], Wang et al. studied exact solutions of the KdV equation by using the basic (𝐺′/𝐺)-expansion method and obtained only four solutions (solutions (A.1)–(A.4), see Section A for lucidity). On the other hand by using the proposed expansion, we obtain ten solutions (solutions (3.8)–(3.17)). It is important to point out that, if we set 𝑑=0 and/or 𝑑=𝜆, then our solutions 𝑢₁₁,𝑢₁₂, and 𝑢₁₃(solutions (3.8)–(3.11)) are identical to the solutions 𝑢₁,𝑢₂, and 𝑢₃ attained by Wang et al. [24]. But if 𝑑≠0 and 𝑑≠𝜆, our solutions 𝑢₁₁,𝑢₁₂, and 𝑢₁₃ are different from Wang et al. [24] solutions. Moreover, we obtain additional solutions (3.12)–(3.17). These solutions are new and were not obtained by Wang et al. [24].

By the improved (𝐺′/𝐺)-expansion method, Zhang et al. [33] obtained seven solutions (solutions (B.3)–(B.9), see Section B for details) of the ZKBBM equations, but by means of the proposed expansion method we obtained ten solutions (solutions (3.27)–(3.36)). It is noteworthy to observe that if 𝑑=0, then our solutions (3.27)–(3.33) are identical to Zhang et al. [33] solutions (B.3)–(B.9). On the other hand if 𝑑≠0, then the solutions (3.27)–(3.33) are dissimilar to Zhang et al. [33] solutions. Furthermore, we obtain solutions (3.34)–(3.36). These solutions are new and were not obtained by Zhang et al. [33].

Not only for these equations, the proposed expansion provides more new exact solutions, and it also provides more new solutions for the Broer-Kaup equation, the Sharma-Tasso-Olver equation, the Gardner equation, the Burgers equation, the KdV Burgers equation, the approximate long-wave equation, the Boussinesq equation, and so on. We have prepared another articles in which the above equations are considered and found much more new exact solutions than the basic (𝐺′/𝐺)-expansion method and the improved (𝐺′/𝐺)-expansion method.

Therefore, the proposed expansion method is promising for solving nonlinear partial differential equations in mathematical physical and engineering problems.

5. Conclusion

A generalized and improved (𝐺′/𝐺)-expansion method has been proposed and applied in three equations, such as the KdV equation, the ZKBBM equation, and the strain wave equation in microstructured solids. The obtained solutions are more general, and many known solutions are only a special case of them. Further, this study shows that the proposed method is quite efficient and practically well suited to be used in finding exact solutions of NLEEs. Although the method is applied to only a small number (three) of nonlinear equations, it can be applied to many other equations, and this is our task in the future.

Appendices

A. Wang et al.’s Solutions [24]

Wang et al. [24] investigated solutions of the KdV equation by the (𝐺′/𝐺)-expansion method, and they obtained the following solutions.

When 𝜆²−4𝜇>0,𝑢₁=−3𝛿𝜆²−4𝜇𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉²+3𝛿𝜆²+𝑒₀, (A.1)

where 𝜉=𝑥−(𝛿𝜆²+8𝛿𝜇+𝑒₀)𝑡, and 𝐴, 𝐵 are arbitrary constants.

For 𝐴>0 and 𝐴²>𝐵², solution (A.1) reduces to𝑢₁=3𝛿𝜆²−4𝜇sech²𝜆²−4𝜇2𝜉+𝜉₀+12𝛿𝜇+𝑒₀, (A.2)

where 𝜉₀=tanh⁻¹(𝐵/𝐴).

When 𝜆²−4𝜇<0,𝑢₂=−3𝛿4𝜇−𝜆²−𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉𝐴cos(4𝜇−𝜆²)/2𝜉+𝐵sin(4𝜇−𝜆²)/2𝜉²+3𝛿𝜆²+𝑒₀. (A.3)

When 𝜆²−4𝜇=0,𝑢₃𝐵=−12𝛿²(𝐴+𝐵𝜉)²+3𝛿𝜆²+𝑒₀. (A.4)

B. Zhang et al.’s Solutions [33]

Zhang et al. [33] proposed an improved (𝐺′/𝐺)-expansion method and solved the ZKBBM equation by the proposed method. They obtained the solutions of the overdetermined set of algebraic equation as follows.

Case 1. 𝑒₂=𝑒₁=0, 𝑒₀=−(𝑏𝑉𝜆²−𝑉−1+8𝑏𝑉𝜇)/2𝑎, 𝑒₋₁=−6𝑏𝑉𝜆𝜇/𝑎, 𝑒₋₂=−6𝑏𝑉𝜇²/𝑎, 𝑉=𝑉, 𝐶=−1−𝑉²−2𝑉+16𝑏²𝑉²𝜇²−8𝑏²𝑉²𝜇𝜆²+𝑏²𝑉²𝜆⁴,4𝑎 (B.1)where 𝑉, 𝜆, and 𝜇 are arbitrary constants.

Case 2. 𝑉=𝑉, 𝑒₂=−6𝑏𝑉/𝑎, 𝑒₁=−6𝑏𝑉𝜆/𝑎, 𝑒₀=−(8𝑏𝑉𝜇−𝑉−1+𝑏𝑉𝜆²)/2𝑎, 𝑒₋₂=𝑒₋₁=0, 𝐶=−8𝑏²𝑉²𝜇𝜆²−2𝑉−𝑉²+𝑏²𝑉²𝜆⁴+16𝑏²𝑉²𝜇²−1,4𝑎 (B.2)where 𝑉, 𝜆, and 𝜇 are arbitrary constants.

For Case 1, Zhang et al. [33] obtained the following solutions.

When 𝜆²−4𝜇>0,𝑢₁₁=−6𝑏𝑉𝜇²𝑎−𝜆2+𝜆²−4𝜇2𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉²−6𝑏𝑉𝜆𝜇𝑎−𝜆2+𝜆²−4𝜇2𝐴cosh𝜆²−4𝜇/2𝜉+𝐵sinh𝜆²−4𝜇/2𝜉𝐴sinh𝜆²−4𝜇/2𝜉+𝐵cosh𝜆²𝜉−−4𝜇/2𝑏𝑉𝜆²+8𝑏𝑉𝜇−𝑉−1,2𝑎 (B.3)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇<0, 𝑢₁₂=−6𝑏𝑉𝜇²𝑎−𝜆2+4𝜇−𝜆²2𝐴cos(4𝜇−𝜆²)/2𝜉−𝐵sin(4𝜇−𝜆²)/2𝜉𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉²−6𝑏𝑉𝜆𝜇𝑎−𝜆2+4𝜇−𝜆²2𝐴cos4𝜇−𝜆²/2𝜉−𝐵sin4𝜇−𝜆²/2𝜉𝐴sin4𝜇−𝜆²/2𝜉+𝐵cos4𝜇−𝜆²𝜉−/2𝑏𝑉𝜆²+8𝑏𝑉𝜇−𝑉−1,2𝑎 (B.4)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑢₁₃=−6𝑏𝑉𝜇²𝑎(−(𝜆/2)+(𝐵/(𝐴+𝐵𝜉)))²−6𝑏𝑉𝜆𝜇𝑎(−()+((−𝜆/2𝐵/𝐴+𝐵𝜉)))𝑏𝑉𝜆²+8𝑏𝑉𝜇−𝑉−1,2𝑎 (B.5)

where 𝐴 and 𝐵 are arbitrary constants.

For Case 2, Zhang et al. [33] obtained the following solutions.

When 𝜆²−4𝜇>0, 𝑢₂₁𝜆=−3𝑏𝑉²−4𝜇2𝑎𝐴cosh(𝜆²−4𝜇)/2𝜉+𝐵sinh(𝜆²−4𝜇)/2𝜉𝐴sinh(𝜆²−4𝜇)/2𝜉+𝐵cosh(𝜆²−4𝜇)/2𝜉²+2𝑏𝑉𝜆²−8𝑏𝑉𝜇+𝑉+1,2𝑎 (B.6)

where 𝐴 and 𝐵 are arbitrary constants.

If 𝐵>0 and 𝐴²<𝐵², then they obtained the soliton solutions𝑢₂₁=3𝑏𝑉𝜆²−4𝜇2𝑎sech²𝜆²−4𝜇2𝜉+𝜉₀−𝑏𝑉𝜆²−4𝑏𝑉𝜇−𝑉−1,2𝑎 (B.7)

where 𝜉₀=tanh⁻¹(𝐴/𝐵).

When 𝜆²−4𝜇<0,𝑢₂₂=−3𝑏𝑉2𝑎4𝜇−𝜆²𝐴cos(4𝜇−𝜆²)/2𝜉−𝐵sin(4𝜇−𝜆²)/2𝜉𝐴sin(4𝜇−𝜆²)/2𝜉+𝐵cos(4𝜇−𝜆²)/2𝜉²+2𝑏𝑉𝜆²−8𝑏𝑉𝜇+𝑉+1,2𝑎 (B.8)

where 𝐴 and 𝐵 are arbitrary constants.

When 𝜆²−4𝜇=0,𝑢₂₃=−6𝑏𝑉𝑎𝐵𝐴+𝐵𝜉²+2𝑏𝑉𝜆²−8𝑏𝑉𝜇+𝑉+1,2𝑎 (B.9)

where 𝐴 and 𝐵 are arbitrary constants.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referee(s) for their useful and valuable comments and suggestions. The authors also acknowledge the research grant under the Government of Malaysia to support this research work.