New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the (2+1)-Dimensional Evolution Equation

Abstract

The generalized Riccati equation mapping is extended with the basic (𝐺^′/𝐺)-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation 𝐺^′(𝜂)=𝑤+𝑢𝐺(𝜂)+𝑣𝐺²(𝜂) is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.

1. Introduction

The study of analytical solutions for nonlinear partial differential equations (PDEs) has become more imperative and stimulating research fields in mathematical physics, engineering sciences, and other technical arena [1–47]. In the recent past, a wide range of methods have been developed to construct traveling wave solutions of nonlinear PDEs such as, the inverse scattering method [1], the Backlund transformation method [2], the Hirota bilinear transformation method [3], the bifurcation method [4, 5], the Jacobi elliptic function expansion method [6–8], the Weierstrass elliptic function method [9], the direct algebraic method [10], the homotopy perturbation method [11, 12], the Exp-function method [13–17], and others [18–28].

Recently, Wang et al. [29] presented a widely used method, called the (𝐺^′/𝐺)-expansion method to obtain traveling wave solutions for some nonlinear evolution equations (NLEEs). Further, in this method, the second-order linear ordinary differential equation 𝐺^′′(𝜂)+𝜆𝐺^′(𝜂)+𝜇𝐺(𝜂)=0 is implemented, as an auxiliary equation, where 𝜆 and 𝜇 are constant coefficients. Afterwards, many researchers investigated many nonlinear PDEs to construct traveling wave solutions via this powerful (𝐺^′/𝐺)-expansion method. For example, Feng et al. [30] applied the same method for obtaining exact solutions of the Kolmogorov-Petrovskii-Piskunov equation. In [31], Naher et al. concerned about this method to construct traveling wave solutions for the higher-order Caudrey-Dodd-Gibbon equation. Zayed and Al-Joudi [32] studied some nonlinear partial differential equations to obtain analytical solutions by using the same method whereas Gepreel [33] executed this method and found exact solutions of nonlinear PDEs with variable coefficients in mathematical physics. Abazari and Abazari [34] constructed exact solutions for the Hirota-Ramani equation by using this method. Ozis and Aslan [35] established some traveling wave solutions for the Kawahara type equations via the same method. Naher et al. [36] investigated higher dimensional nonlinear evolution equation for obtaining traveling wave solutions by applying the improved (𝐺^′/𝐺)-expansion method. Naher and Abdullah [37] implemented this method to construct some new traveling wave solutions of the nonlinear reaction diffusion equation whilst they [38] studied the combined KdV-MKdV equation for obtaining abundant solutions via the same method. And [39] they executed this method to establish some new traveling wave solutions of the (2+1)-dimensional modified Zakharov-Kuznetsov equation and so on.

Zhu [40] investigated the (2+1)-dimensional Boiti-Leon-Pempinelle equation by applying the generalized Riccati equation mapping with the extended tanh-function method. In addition, 𝐺^′(𝜂)=𝑤+𝑢𝐺(𝜂)+𝑣𝐺²(𝜂) is used, as an auxiliary equation and called generalized Riccati equation, where 𝑢, 𝑣, and 𝑤 are arbitrary constants. Bekir and Cevikel [41] concerned about the tanh-coth method combined with the Riccati equation to study nonlinear coupled equation in mathematical physics. Guo et al. [42] implemented the extended Riccati equation mapping method for solving the diffusion-reaction and the mKdV equation with variable coefficient whilst Li et al. [43] studied higher-dimensional Jimbo-Miwa equation via the generalized Riccati equation expansion method. Salas [44] obtained some exact solutions for the Caudrey-Dodd-Gibbon equation by applying the projective Riccati equation method whereas Naher and Abdullah [45] studied the modified Benjamin-Bona-Mahony equation via the generalized Riccati equation mapping with the basic (𝐺^′/𝐺)-expansion method for constructing traveling wave solutions and so on.

Many researchers implemented various methods to investigate the (2+1)-dimensional modified Zakharov-Kuznetsov equation. For instance, Khalfallah [46] used homogeneous balance method to establish traveling wave solutions of this equation. In [47], Bekir applied basic (𝐺^′/𝐺)-expansion method for obtaining exact traveling wave solutions for the same equation. In this basic (𝐺^′/𝐺)-expansion method, they employed second-order linear ordinary differential equation (LODE) with constant coefficients, as an auxiliary equation instead of generalized Riccati equation.

The importance of our present work is, in order to construct many new traveling wave solutions including solitons, periodic, and rational solutions, a (2+1)-dimensional Modified Zakharov-Kuznetsov equation considered by applying the extended generalized Riccati equation mapping method.

2. The Extended Generalized Riccati Equation Mapping Method

Suppose the general nonlinear partial differential equation 𝐻𝑣,𝑣_𝑡,𝑣_𝑥,𝑣_𝑦,𝑣_𝑥𝑡,𝑣_𝑦𝑡,𝑣_𝑥𝑦,𝑣_𝑡𝑡,𝑣_𝑥𝑥,𝑣_𝑦𝑦,…=0, (2.1) where 𝑣=𝑣(𝑥,𝑦,𝑡) is an unknown function, 𝐻 is a polynomial in 𝑣(𝑥,𝑦,𝑡), and the subscripts indicate the partial derivatives.

The most important steps of the generalized Riccati equation mapping together with the (𝐺^′/𝐺)-expansion method [29, 40] are as follows.

Step 1. Consider the traveling wave variable: 𝑣𝑥,𝑦,𝑡=𝑔𝜂,𝜂=𝑥+𝑦−𝐶𝑡, (2.2) where 𝐶 is the speed of the traveling wave. Now using (2.2), (2.1) is converted into an ordinary differential equation for 𝑔(𝜂): 𝐹𝑔,𝑔^′,𝑔^′′,𝑔^′′′,…=0, (2.3) where the superscripts stand for the ordinary derivatives with respect to 𝜂.

Step 2. Equation (2.3) integrates term by term one or more times according to possibility and yields constant(s) of integration. The integral constant(s) may be zero for simplicity.

Step 3. Suppose that the traveling wave solution of (2.3) can be expressed in the form [29, 40] 𝑔𝜂=^𝑛_𝑗=0𝑒_𝑗𝐺^′𝐺^𝑗, (2.4) where 𝑒_𝑗(𝑗=0,1,2,…,𝑛) and 𝑒_𝑛≠0, with 𝐺=𝐺(𝜂) is the solution of the generalized Riccati equation: 𝐺^′=𝑤+𝑢𝐺+𝑣𝐺², (2.5) where 𝑢, 𝑣, 𝑤 are arbitrary constants and 𝑣≠0.

Step 4. To decide the positive integer 𝑛, consider the homogeneous balance between the nonlinear terms and the highest order derivatives appearing in (2.3).

Step 5. Substitute (2.4) along with (2.5) into the (2.3), then collect all the coefficients with the same order, the left hand side of (2.3) converts into polynomials in 𝐺^𝑘(𝜂) and 𝐺^−𝑘(𝜂),(𝑘=0,1,2,…). Then equating each coefficient of the polynomials to zero and yield a set of algebraic equations for 𝑒_𝑗(𝑗=0,1,2,…,𝑛), 𝑢, 𝑣, 𝑤, and 𝐶.

Step 6. Solve the system of algebraic equations which are found in Step 5 with the aid of algebraic software Maple to obtain values for 𝑒_𝑗(𝑗=0,1,2,…,𝑛) and 𝐶 then, substitute obtained values in (2.4) along with (2.5) with the value of 𝑛, we obtain exact solutions of (2.1).
In the following, we have twenty seven solutions including four different families of (2.5).

Family 1. When 𝑢²−4𝑣𝑤>0 and 𝑢𝑣≠0or 𝑣𝑤≠0, the solutions of (2.5) are: 𝐺₁=−1√2𝑣𝑢+𝑢²√−4𝑣𝑤tanh𝑢²−4𝑣𝑤2𝜂,𝐺₂=−1√2𝑣𝑢+𝑢²√−4𝑣𝑤coth𝑢²−4𝑣𝑤2𝜂,𝐺₃=−1√2𝑣𝑢+𝑢²−4𝑣𝑤tanh√𝑢²−4𝑣𝑤𝜂±isech√𝑢²,𝐺−4𝑣𝑤𝜂₄=−1√2𝑣𝑢+𝑢²−4𝑣𝑤coth√𝑢²−4𝑣𝑤𝜂±csch√𝑢²,𝐺−4𝑣𝑤𝜂₅=−1√4𝑣2𝑢+𝑢²√−4𝑣𝑤tanh𝑢²−4𝑣𝑤4𝜂√+coth𝑢²−4𝑣𝑤4𝜂,𝐺₆=1±2𝑣−𝑢+𝐷²+𝐸²𝑢²−4𝑣𝑤−𝐷√𝑢²−4𝑣𝑤cosh√𝑢²−4𝑣𝑤𝜂𝐷sinh√𝑢²,𝐺−4𝑣𝑤𝜂+𝐸₇=1±2𝑣−𝑢−𝐷²+𝐸²𝑢²−4𝑣𝑤+𝐷√𝑢²−4𝑣𝑤cosh√𝑢²−4𝑣𝑤𝜂𝐷sinh√𝑢²,−4𝑣𝑤𝜂+𝐸 (2.6) where 𝐷 and 𝐸 are two nonzero real constants. 𝐺₈=2𝑤cosh√𝑢²−4𝑣𝑤/2𝜂√𝑢²−4𝑣𝑤sinh√𝑢²−4𝑣𝑤/2𝜂−𝑢cosh√𝑢²𝜂,𝐺−4𝑣𝑤/2₉=−2𝑤sinh√𝑢²−4𝑣𝑤/2𝜂𝑢sinh√𝑢²−4𝑣𝑤/2𝜂−√𝑢²−4𝑣𝑤cosh√𝑢²𝜂,𝐺−4𝑣𝑤/2₁₀=2𝑤cosh√𝑢²−4𝑣𝑤𝜂√𝑢²−4𝑣𝑤sinh√𝑢²−4𝑣𝑤𝜂−𝑢cosh√𝑢²−4𝑣𝑤𝜂±𝑖√𝑢²,𝐺−4𝑣𝑤₁₁=2𝑤sinh√𝑢²−4𝑣𝑤𝜂−𝑢sinh√𝑢²−4𝑣𝑤𝜂+√𝑢²−4𝑣𝑤cosh√𝑢²−4𝑣𝑤𝜂±√𝑢²,𝐺−4𝑣𝑤₁₂=4𝑤sinh√𝑢²−4𝑣𝑤/4𝜂cosh√𝑢²−4𝑣𝑤/4𝜂−2𝑢sinh√𝑢²−4𝑣𝑤/4𝜂cosh√𝑢²−4𝑣𝑤/4𝜂+Δ₁, (2.7) where Δ₁=2√𝑢²−4𝑣𝑤cosh²((√𝑢²−4𝑣𝑤/4)𝜂)−√𝑢²−4𝑣𝑤.

Family 2. When 𝑢²−4𝑣𝑤<0 and 𝑢𝑣≠0or 𝑣𝑤≠0, the solutions of (2.5) are: 𝐺₁₃=1√2𝑣−𝑢+4𝑣𝑤−𝑢²√tan4𝑣𝑤−𝑢²2𝜂,𝐺₁₄=−1√2𝑣𝑢+4𝑣𝑤−𝑢²cot√4𝑣𝑤−𝑢²2𝜂,𝐺₁₅=1√2𝑣−𝑢+4𝑣𝑤−𝑢²tan√4𝑣𝑤−𝑢²𝜂±sec√4𝑣𝑤−𝑢²𝜂,𝐺₁₆=−1√2𝑣𝑢+4𝑣𝑤−𝑢²cot√4𝑣𝑤−𝑢²𝜂±csc√4𝑣𝑤−𝑢²𝜂,𝐺₁₇=1√4𝑣−2𝑢+4𝑣𝑤−𝑢²√tan4𝑣𝑤−𝑢²4𝜂−cot√4𝑣𝑤−𝑢²4𝜂,𝐺₁₈=1±2𝑣−𝑢+𝐷²−𝐸²4𝑣𝑤−𝑢²−𝐷√4𝑣𝑤−𝑢²cos√4𝑣𝑤−𝑢²𝜂𝐷sin√4𝑣𝑤−𝑢²𝜂,𝐺+𝐸₁₉=1±2𝑣−𝑢−𝐷²−𝐸²4𝑣𝑤−𝑢²+𝐷√4𝑣𝑤−𝑢²cos√4𝑣𝑤−𝑢²𝜂𝐷sin√4𝑣𝑤−𝑢²𝜂,+𝐸 (2.8) where 𝐷 and 𝐸 are two nonzero real constants and satisfy 𝐷²−𝐸²>0. 𝐺₂₀=−2𝑤cos√4𝑣𝑤−𝑢²/2𝜂√4𝑣𝑤−𝑢²sin√4𝑣𝑤−𝑢²/2𝜂+𝑢cos√4𝑣𝑤−𝑢²𝜂,𝐺/2₂₁=2𝑤sin√4𝑣𝑤−𝑢²/2𝜂−𝑢sin√4𝑣𝑤−𝑢²/2𝜂+√4𝑣𝑤−𝑢²cos√4𝑣𝑤−𝑢²𝜂,𝐺/2₂₂=−2𝑤cos√4𝑣𝑤−𝑢²𝜂√4𝑣𝑤−𝑢²sin√4𝑣𝑤−𝑢²𝜂+𝑢cos√4𝑣𝑤−𝑢²𝜂±√4𝑣𝑤−𝑢²,𝐺₂₃=2𝑤sin√4𝑣𝑤−𝑢²𝜂−𝑢sin√4𝑣𝑤−𝑢²𝜂+√4𝑣𝑤−𝑢²cos√4𝑣𝑤−𝑢²𝜂±√4𝑣𝑤−𝑢²,𝐺₂₄=4𝑤sin√4𝑣𝑤−𝑢²/4𝜂cos√4𝑣𝑤−𝑢²/4𝜂−2𝑢sin√4𝑣𝑤−𝑢²/4𝜂cos√4𝑣𝑤−𝑢²/4𝜂+Δ₂, (2.9) where Δ₂=2√4𝑣𝑤−𝑢²cos²((√4𝑣𝑤−𝑢²/4)𝜂)−√4𝑣𝑤−𝑢².

Family 3. When 𝑤=0 and 𝑢𝑣≠0, the solution (2.5) becomes: 𝐺₂₅=−𝑢𝑓₁𝑣𝑓₁,𝐺+cosh𝑢𝜂−sinh𝑢𝜂₂₆=−𝑢cosh𝑢𝜂+sinh𝑢𝜂𝑣𝑓₁,+cosh𝑢𝜂+sinh𝑢𝜂 (2.10) where 𝑓₁ is an arbitrary constant.

Family 4. when 𝑣≠0 and 𝑤=𝑢=0, the solution of (2.5) becomes: 𝐺₂₇=−1𝑣𝜂+𝑙₁, (2.11) where 𝑙₁ is an arbitrary constant.

3. Applications of the Method

In this section, we have constructed new traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by using the method.

3.1. The (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation

We consider the (2+1)-dimensional Modified Zakharov-Kuznetsov equation followed by Bekir [47] 𝑢_𝑡+𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥+𝑢_𝑥𝑦𝑦=0. (3.1) Now, we use the wave transformation (2.2) into the (3.1), which yields: −𝐶𝑔^′+𝑢²𝑔^′+2𝑔^′′′=0. (3.2) Equation (3.2) is integrable, therefore, integrating with respect 𝜂 once yields: 1𝑄−𝐶𝑔+3𝑔³+2𝑔^′′=0, (3.3) where 𝑄 is an integral constant which is to be determined later.

Taking the homogeneous balance between 𝑔³ and 𝑔^′′ in (3.3), we obtain 𝑛=1.

Therefore, the solution of (3.3) is of the form: 𝑔𝜂=𝑒₁𝐺^′𝐺+𝑒₀,𝑒₁≠0. (3.4)

Using (2.5), (3.4) can be rewritten as 𝑔𝜂=𝑒₁𝑢+𝑤𝐺⁻¹+𝑣𝐺+𝑒₀, (3.5) where 𝑢, 𝑣, and 𝑤 are free parameters.

By substituting (3.5) into (3.3), the left hand side is converted into polynomials in 𝐺^𝑘 and 𝐺^−𝑘(𝑘=0,1,2,…). Setting each coefficient of these resulted polynomials to zero, we obtain a set of algebraic equations for 𝑒₀, 𝑒₁, 𝑢, 𝑣, 𝑤, 𝑄, and 𝐶 (algebraic equations are not shown, for simplicity). Solving the system of algebraic equations with the help of algebraic software Maple, we obtain 𝑒₀=∓𝑢𝑖√3,𝑒₁=±2𝑖√3,𝐶=−𝑢²−8𝑣𝑤,𝑄=8𝑢𝑣𝑤𝑖√3. (3.6)

Family 5. The soliton and soliton-like solutions of (3.1) (when 𝑢²−4𝑣𝑤>0 and 𝑢𝑣≠0or 𝑣𝑤≠0) are: 𝑔₁=±2𝑖√32Ψ²sech²Ψ𝜂√𝑢+2ΨtanhΨ𝜂∓𝑢𝑖3, (3.7) where Ψ=(1/2)√𝑢²−4𝑣𝑤, 𝜂=𝑥+𝑦+(𝑢²+8𝑣𝑤)𝑡 and 𝑢, 𝑣, 𝑤 are arbitrary constants. 𝑔₂=∓2𝑖√32Ψ²csch²Ψ𝜂√𝑢+2ΨcothΨ𝜂∓𝑢𝑖𝑔3,₃=±2𝑖√34Ψ²sech2Ψ𝜂1∓𝑖sinh2Ψ𝜂√𝑢cosh2Ψ𝜂+2Ψsinh2Ψ𝜂±𝑖2Ψ∓𝑢𝑖𝑔3,₄=∓2𝑖√32Ψ²cscℎΨ𝜂√𝑢sinhΨ𝜂+2ΨcoshΨ𝜂∓𝑢𝑖𝑔3,₅=∓2𝑖√34Ψ²cscℎ2Ψ𝜂√𝑢tanhΨ𝜂+2Ψ∓𝑢𝑖𝑔3,₆=∓2𝑖√34𝐷Ψ²𝐷−𝐸sinh2Ψ𝜂−𝐷²+𝐸²cosh2Ψ𝜂𝐷sinh2Ψ𝜂+𝐸Ω₁∓𝑢𝑖√𝑔3,₇=∓2𝑖√34𝐷Ψ²𝐷−𝐸sinh2Ψ𝜂+𝐷²+𝐸²cosh2Ψ𝜂𝐷sinh2Ψ𝜂+𝐸Ω₂∓𝑢𝑖√3, (3.8) where Ω₁=𝑢𝐷sinh(2Ψ𝜂)+𝑢𝐸−2Ψ(𝐷²+𝐸²)+2𝐷Ψcosh(2Ψ𝜂), Ω₂=𝑢𝐷sinh(2Ψ𝜂)+𝑢𝐸+2Ψ(𝐷²+𝐸²)+2𝐷Ψcosh(2Ψ𝜂), 𝐷 and 𝐸 are two nonzero real constants. 𝑔₈=∓2𝑖√32Ψ²sechΨ𝜂√2ΨsinhΨ𝜂−𝑢coshΨ𝜂∓𝑢𝑖𝑔3,₉=±2𝑖√32Ψ²cschΨ𝜂√2ΨcoshΨ𝜂−𝑢sinhΨ𝜂∓𝑢𝑖𝑔3,₁₀=±2𝑖√34Ψ²sech2Ψ𝜂1∓𝑖sinh2Ψ𝜂√𝑢cosh2Ψ𝜂−2Ψsinh2Ψ𝜂∓𝑖2Ψ∓𝑢𝑖𝑔3,₁₁=±2𝑖√34Ψ²csch2Ψ𝜂1±cosh2Ψ𝜂√2Ψcosh2Ψ𝜂−𝑢sinh2Ψ𝜂±2Ψ∓𝑢𝑖𝑔3,₁₂=±2𝑖√32Ψ²cschΨ𝜂√2ΨcoshΨ𝜂−𝑢sinhΨ𝜂∓𝑢𝑖3. (3.9)

Family 6. The periodic form solutions of (3.1) (when 𝑢²−4𝑣𝑤<0 and 𝑢𝑣≠0or 𝑣𝑤≠0) are: 𝑔₁₃=±2𝑖√32Θ²sec²Θ𝜂√−𝑢+2ΘtanΘ𝜂∓𝑢𝑖3, (3.10) where Θ=(1/2)√4𝑣𝑤−𝑢²,  𝜂=𝑥+𝑦+(𝑢²+8𝑣𝑤)𝑡 and 𝑢, 𝑣, 𝑤 are arbitrary constants. 𝑔₁₄=∓2𝑖√32Θ²csc²Θ𝜂𝑢+2Θcot√Θ𝜂∓𝑢𝑖𝑔3,₁₅=±2𝑖√34Θ²sec2Θ𝜂1±sin2Θ𝜂√−𝑢cos2Θ𝜂+2Θsin2Θ𝜂±2Θ∓𝑢𝑖𝑔3,₁₆=∓2𝑖√32Θ²secΘ𝜂√𝑢cosΘ𝜂+2ΘsinΘ𝜂∓𝑢𝑖𝑔3,₁₇=∓2𝑖√32Θ²cscΘ𝜂√𝑢sinΘ𝜂+2ΘcosΘ𝜂∓𝑢𝑖𝑔3,₁₈=∓2𝑖√34𝐷Θ²𝐷²−𝐸²cos2Θ𝜂−𝐸sin2Θ𝜂−𝐷𝐷sin2Θ𝜂+𝐸Ω₃∓𝑢𝑖√𝑔3.₁₉=∓2𝑖√34𝐷Θ²𝐷²−𝐸²cos2Θ𝜂+𝐸sin2Θ𝜂+𝐷𝐷sin2Θ𝜂+𝐸Ω₄∓𝑢𝑖√3, (3.11) where Ω₃=𝑢𝐷sin(2Θ𝜂)+2𝐷Θcos(2Θ𝜂)+𝑢𝐸−2Θ(𝐷²−𝐸²), Ω₄=𝑢𝐷sin(2Θ𝜂)+2𝐷Θcos(2Θ𝜂)+𝑢𝐸+2Θ(𝐷²−𝐸²), 𝐷 and 𝐸 are two nonzero real constants and satisfies 𝐷²−𝐸²>0. 𝑔₂₀=∓2𝑖√34Θ²csc2Θ𝜂𝑢cot√Θ𝜂+2Θ∓𝑢𝑖𝑔3,₂₁=∓2𝑖√34Θ²csc2Θ𝜂√𝑢tanΘ𝜂+2Θ∓𝑢𝑖𝑔3,₂₂=∓2𝑖√32Θ²sec2Θ𝜂1±sin2Θ𝜂𝑢cos2Θ𝜂+2Θsin2Θ𝜂±2Θ𝑢²−2𝑣𝑤cos²√2Θ𝜂+2Θ1±sin2Θ𝜂2Θ±𝑢cos2Θ𝜂∓𝑢𝑖𝑔3,₂₃=±2𝑖√32Θ²csc2Θ𝜂−𝑢sin2Θ𝜂+2Θcos2Θ𝜂±2Θ2𝑣𝑤−𝑢²√cos2Θ𝜂−2𝑢Θsin2Θ𝜂±2𝑣𝑤∓𝑢𝑖𝑔3,₂₄=∓2𝑖√32Θ²cscΘ𝜂√𝑢sinΘ𝜂+2ΘcosΘ𝜂∓𝑢𝑖3. (3.12)

Family 7. The soliton and soliton-like solutions of (3.1) (when 𝑤=0 and 𝑢𝑣≠0) are: 𝑔₂₅=±2𝑖√3𝑢cosh𝑢𝜂−sinh𝑢𝜂𝑓₁√+cosh𝑢𝜂−sinh𝑢𝜂∓𝑢𝑖𝑔3,₂₆=±2𝑖√3𝑢𝑓₁𝑓₁√+cosh𝑢𝜂+sinh𝑢𝜂∓𝑢𝑖3, (3.13) where 𝑓₁ is an arbitrary constant, 𝜂=𝑥+𝑦+(𝑢²+8𝑣𝑤)𝑡.

Family 8. The rational function solution (when 𝑣≠0 and 𝑤=𝑢=0) is: 𝑔₂₇=∓2𝑖𝑣√3𝑣𝜂+𝑙₁, (3.14) where 𝑙₁ is an arbitrary constant and 𝜂=𝑥+𝑦+(𝑢²+8𝑣𝑤)𝑡.

4. Results and Discussion

It is significant to mention that one of our solutions is coincided for some special case with already published results which are presented in Table 1. Furthermore, some of newly constructed solutions are illustrated in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

Table 1 

Comparison between Bekir [47] solutions and newly obtained solutions.

Figure 1 

Solitons solution for 𝑢=3, 𝑣=0.5, 𝑤=0.25.

Figure 2 

Periodic solution for 𝑢=1, 𝑣=1, 𝑤=0.125.

Figure 3 

Solitons solution for 𝑢=4, 𝑣=0.5, 𝑤=1.

Figure 4 

Solitons solution for 𝑢=0.5, 𝑣=25.10−6, 𝑤=25.10−5.

Figure 5 

Periodic solution for 𝑢=1, 𝑣=25.10−5, 𝑤=25.10−3.

Figure 6 

Solitons solution for 𝑢=1, 𝑣=0.45, 𝑤=3.

Figure 7 

Solitons solution for 𝑢=2, 𝑣=25.10−4, 𝑤=25.10−3.

Figure 8 

Solitons solution for 𝑢=1, 𝑣=1, 𝑤=1.

Figure 9 

Periodic solution for 𝑢=0.5, 𝑣=7, 𝑤=0, 𝑓1=0.125.

Figure 10 

Solitons solution for 𝑢=1, 𝑣=0.5, 𝑤=1.

Figure 11 

Solitons solution for 𝑢=3, 𝑣=25.10−4, 𝑤=25.10−3.

Figure 12 

Periodic solution for 𝑢=0.25, 𝑣=−5, 𝑤=0, 𝑓1=0.25.

As in Table 1, we have newly constructed traveling wave solutions 𝑔₁ to 𝑔₂₆ which are not being stated in the earlier literature.

4.1. Graphical Depictions of Newly Obtained Traveling Wave Solutions

The graphical descriptions of some solutions are represented in Figures 1–12 with the aid of commercial software Maple.

5. Conclusions

In this paper, we have investigated the (2+1)-dimensional modified Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method. Twenty seven exact traveling wave solutions are constructed including solitons and periodic wave solutions by applying this powerful method. In addition, newly obtained solutions are depicted in terms of the hyperbolic, the trigonometric, and the rational functional form. The obtained solutions reveal that this method is a promising mathematical tool because it can establish a variety of new solutions of dissimilar physical structures if compared with existing methods. The correctness of newly constructed solutions is verified to be compared with already published results. Consequently, nonlinear evolution equations which regularly arise in many scientific real-time application fields can be studied by applying the extended generalized Riccati equation mapping method.

Acknowledgments

This paper is supported by the USM short-term Grant (Ref. no. 304/PMATHS/6310072), and the authors would like to express their thanks to the School of Mathematical Sciences, USM for providing related research facilities.