New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method

Abstract

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

1. Introduction

Nonlinear partial differential equations (NLPDEs) play a prominent role in different branches of the applied sciences. In recent time, many researchers investigated exact traveling wave solutions of NLPDEs which play a crucial role to reveal the insight of complex physical phenomena. In the past several decades, a variety of effective and powerful methods, such as variational iteration method [1–3], tanh-coth method [4], homotopy perturbation method [5–7], Fan subequation method [8], projective Riccati equation method [9], differential transform method [10], direct algebraic method [11], first integral method [12], Hirota’s bilinear method [13], modified extended direct algebraic method [14], extended tanh method [15], Backlund transformation [16], bifurcation method [17], Cole-Hopf transformation method [18], sech-tanh method [19], (𝐺′/𝐺)-expansion method [20–22], modified (𝐺′/𝐺)-expansion method [23], multiwave method [24], extended (𝐺′/𝐺)-expansion method [25, 26], and others [27–33] were used to seek exact traveling wave solutions of the nonlinear evolution equations (NLEEs).

Recently, He and Wu [34] have presented a novel method called the Exp-function method for searching traveling wave solutions of the nonlinear evolution equations arising in mathematical physics. The Exp-function method is widely used to many kinds of NLPDEs, such as good Boussinesq equations [35], nonlinear differential equations [36], higher-order boundary value problems [37], nonlinear problems [38], Calogero-Degasperis-Fokas equation [39], nonlinear reaction-diffusion equations [40], 2D Bratu type equation [41], nonlinear lattice differential equations [42], generalized-Zakharov equations [43], (3 + 1)-dimensional Jimbo-Miwa equation [44], modified Zakharov-Kuznetsov equation [45], Brusselator reaction diffusion model [46], nonlinear heat equation [47], and the other important NLPDEs [48–51].

In this article, we apply the Exp-function method [34] to obtain the analytical solutions of the nonlinear partial differential equation, namely, (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation.

2. Description of the Exp-Function Method

Consider the general nonlinear partial differential equation𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,𝑢_𝑧,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑦,𝑢_𝑦𝑦,𝑢_𝑦𝑡,𝑢_𝑧𝑧,𝑢_𝑧𝑡,𝑢_𝑧𝑥,𝑢_𝑧𝑦,…=0.(2.1) The main steps of the Exp-function method [34] are as follows.

Step 1. Consider a complex variable as 𝑢(𝑥,𝑦,𝑧,𝑡)=𝑢(𝜂),𝜂=𝑥+𝑦+𝑧−𝑉𝑡.(2.2) Now using (2.2), (2.1) converts to a nonlinear ordinary differential equation for 𝑢(𝜂)𝑄𝑢,𝑢^,𝑢^,𝑢^,…=0,(2.3) where primes denote the ordinary derivative with respect to 𝜂.

Step 2. We assume that the traveling wave solution of (2.3) can be expressed in the form [34] 𝑢(𝜂)=∑^{𝑑𝑛=−𝑐}𝑎_𝑛exp(𝑛𝜂)∑^{𝑞𝑚=−𝑝}𝑏_𝑚exp(𝑚𝜂)=𝑎^−𝑐exp(−𝑐𝜂)+⋯+𝑎_𝑑exp(𝑑𝜂)𝑏_−𝑝exp(−𝑝𝜂)+⋯+𝑏_𝑞exp(𝑞𝜂),(2.4) where 𝑐,𝑑,𝑝, and 𝑞 are positive integers to be determined later, and 𝑎_𝑛 and 𝑏_𝑚 are unknown constants. Equation (2.4) can be rewritten in the following equivalent form: 𝑢(𝜂)=𝑎^𝑐exp(𝑐𝜂)+⋯+𝑎_−𝑑exp(−𝑑𝜂)𝑏_𝑝exp(𝑝𝜂)+⋯+𝑏_−𝑞exp(−𝑞𝜂).(2.5)

Step 3. In order to determine the values of 𝑐 and 𝑝, we balance the highest order linear term with the highest order nonlinear term, and, determining the values of 𝑑 and 𝑞, we balance the lowest order linear term with the lowest order nonlinear term in (2.3). Thus, we obtain the values of 𝑐,𝑑,𝑝, and 𝑞.

Step 4. Substituting the values of 𝑐,𝑑,𝑝, and 𝑞 into (2.5), and then substituted (2.5) into (2.3) and simplifying, we obtain _𝑖𝐶_𝑖exp(±𝑖𝜂)=0,𝑖=0,1,2,3,….(2.6) Then each coefficient 𝐶_𝑖=0 is to set, yields a system of algebraic equations for 𝑎_𝑐′𝑠 and 𝑏_𝑝′𝑠.

Step 5. We assume that the unknown 𝑎_𝑐′𝑠 and 𝑏_𝑝′𝑠 can be determined by solving the system of algebraic equations obtained in Step 4. Putting these values into (2.5), we obtain exact traveling wave solutions of the (2.1).

3. Application of the Method

In this section, we apply the method to construct the traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation. The obtained solutions will be displayed in Figures 1, 2, 3, 4, 5, and 6 by using the software Maple 13.

Figure 1 

Periodic solution for 𝛼=−1, 𝑦=0 and 𝑧=0.

Figure 2 

Periodic solution for 𝛽=2, 𝑦=0 and 𝑧=0.

Figure 3 

Solitons solution for 𝑦=0 and 𝑧=0.

Figure 4 

Solitons solution for 𝑦=0 and 𝑧=0.

Figure 5 

Solitons solution for 𝑦=0 and 𝑧=0.

Figure 6 

Solitons solution for 𝑦=0 and 𝑧=0.

We consider the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation𝑢_𝑡+𝛼𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥+𝑢_𝑥𝑦𝑦+𝑢_𝑥𝑧𝑧=0,(3.1) where 𝛼 is a nonzero constant.

Zayed [52] solved (3.1) using the (𝐺′/𝐺)-expansion method. Later, in article [53], he solved same equation by the generalized (𝐺′/𝐺)-expansion method.

Here, we will solve this equation by the Exp-function method.

Now, we use the transformation (2.2) into (3.1), which yields−𝑉𝑢^+𝛼𝑢²𝑢^+3𝑢^=0,(3.2) where primes denote the derivatives with respect to 𝜂.

According to Step 2, the solution of (3.2) can be written in the form of (2.5). To determine the values of 𝑐 and 𝑝, according to Step 3, we balance the highest order linear term of 𝑢′′′ with the highest order nonlinear term of 𝑢²𝑢′ in (3.2), that is, 𝑢′′′ and 𝑢²𝑢′. Therefore, we have the following:𝑢′′′=𝑐¹exp[(3𝑝+𝑐)𝜂]+⋯𝑢𝑐₂exp[4𝑝𝜂]+⋯₂,𝑢′=𝑐³exp[(𝑝+3𝑐)𝜂]+⋯𝑐₄exp[4𝑝𝜂]+⋯,(3.3) where 𝑐_𝑗 are coefficients only for simplicity; from (3.3), we obtain that 3𝑝+𝑐=𝑝+3𝑐,whichleads𝑝=𝑐.(3.4) To determine the values of 𝑑 and 𝑞, we balance the lowest order linear term of 𝑢^′′ with the lowest order nonlinear term of 𝑢²𝑢′ in (3.2). We have𝑢′′′=⋯+𝑑¹exp[−(𝑑−𝑞)𝜂]𝑢⋯+𝑑₂exp[−4𝑞𝜂₂],𝑢′=⋯+𝑑³exp[−3(𝑑−𝑞)𝜂]⋯+𝑑₄exp[−4𝑞𝜂],(3.5) where 𝑑_𝑗 are determined coefficients only for simplicity; from (3.5), we obtain −(𝑑−𝑞)=−3(𝑑−𝑞),whichleads𝑞=𝑑.(3.6) Any real values can be considered for 𝑐 and 𝑑, since they are free parameters. But the final solutions of (3.1) do not depend upon the choice of 𝑐 and 𝑑.

Case 1. We set 𝑝=𝑐=1 and 𝑞=𝑑=1.
For this case, the trial solution (2.5) reduces to 𝑢(𝜂)=𝑎¹𝑒^𝜂+𝑎₀+𝑎₋₁𝑒^−𝜂𝑏₁𝑒^𝜂+𝑏₀+𝑏₋₁𝑒^−𝜂.(3.7) Since, 𝑏₁≠0, (3.7) can be simplified 𝑢(𝜂)=𝑎¹𝑒^𝜂+𝑎₀+𝑎₋₁𝑒^−𝜂𝑒^𝜂+𝑏₀+𝑏₋₁𝑒^−𝜂.(3.8) By substituting (3.8) into (3.2) and equating the coefficients of exp(±𝑛𝜂),𝑛=0,1,2,3,…, with the aid of Maple 13, we obtain a set of algebraic equations in terms of 𝑎₋₁,𝑎₀,𝑎₁,𝑏₋₁,𝑏₀, and 𝑉1𝐴𝐶³𝑒^3𝜂+𝐶₂𝑒^2𝜂+𝐶₁𝑒^𝜂+𝐶₀+𝐶₋₁𝑒^−𝜂+𝐶₋₂𝑒^−2𝜂+𝐶₋₃𝑒^−3𝜂=0.(3.9) And, setting each coefficient of exp(±𝑛𝜂),𝑛=0,1,2,3,…, to zero, we obtain 𝐶₃=0,𝐶₂=0,𝐶₁=0,𝐶₀=0,𝐶₋₁=0,𝐶₋₂=0,𝐶₋₃=0.(3.10) For determining unknowns, we solve the obtained system of algebraic (3.10) with the aid of Maple 13, and we obtain four different sets of solutions.Set 1. We obtain that 𝑏₋₁=𝑏₋₁,𝑎₋₁=∓6𝑏⁻¹√−2𝛼,𝑎⁰=0,𝑎₁=±6√−2𝛼,𝑏⁰=0,𝑉=−6,(3.11) where 𝑏₋₁ is free parameter.Set 2. We obtain that 𝑎₀=𝑎₀,𝑏₀=𝑏₀,𝑎₋₁=∓112√−2𝛼2𝛼𝑎²⁰𝑎₁+9𝑏²⁰,=±3√−2𝛼,𝑏⁻¹=118𝛼𝑎²⁰+14𝑏²⁰,𝑉=−32,(3.12) where 𝑎₀ and 𝑏₀ are free parameters.Set 3. We obtain that 𝑎₁=𝑎₁,𝑏₀=𝑏₀,𝑎₋₁=𝑏²⁰2𝛼𝑎²¹+98𝛼𝑎₁𝑎₀,=𝑏⁰𝛼𝑎²¹+9𝛼𝑎₁,𝑏₋₁=𝑏²⁰2𝛼𝑎²¹+98𝛼𝑎²¹,𝑉=3+𝛼𝑎²¹,(3.13) where 𝑎₁ and 𝑏₀ are free parameters.Set 4. We obtain that 𝑎₀=𝑎₀,𝑎₋₁=0,𝑎₁=0,𝑏₋₁=172𝛼𝑎²⁰,𝑏₀=0,𝑉=3,(3.14) where 𝑎₀ is free parameter.
Now, substituting (3.11) into (3.8), we obtain traveling wave solution 𝑢(𝜂)=±6𝑒^𝜂∓6𝑏₋₁𝑒^−𝜂√−2𝛼𝑒^𝜂+𝑏₋₁𝑒^−𝜂.(3.15) Equation (3.15) can be simplified as 𝑢(𝜂)=±6√−2𝛼1−2𝑏⁻¹(cosh𝜂−sinh𝜂)1+𝑏₋₁cosh𝜂+1−𝑏₋₁sinh𝜂,(3.16) where 𝜂=𝑥+𝑦+𝑧+6𝑡.
If 𝑏₋₁=1 from (3.16), we obtain 𝑢(𝜂)=±6𝑖√2𝛼tanh𝜂.(3.17) Substituting (3.12) into (3.8) and simplifying, we get traveling wave solution 𝑢(𝜂)=±3√−2𝛼⎡⎢⎢⎣1+12±𝑎⁰√−2𝛼+3𝑏₀−22𝛼𝑎²⁰+9𝑏²⁰(cosh𝜂−sinh𝜂)36+2𝛼𝑎²⁰+9𝑏²⁰cosh𝜂+36−2𝛼𝑎²⁰−9𝑏²⁰sinh𝜂+36𝑏₀⎤⎥⎥⎦,(3.18) where 𝜂=𝑥+𝑦+𝑧+(3/2)𝑡.
If 𝛼 is negative, that is, 𝛼=−𝛽, 𝛽>0, 𝑏₀=2 and 𝑎₀=0, then from (3.18), we obtain 𝑢(𝜂)=±3√2𝛽tanh𝜂2.(3.19) Substituting (3.13) into (3.8) and simplifying, we obtain 𝑢(𝜂)=𝑎₁1+8𝛼𝑎²¹+𝑏²⁰2𝛼𝑎²¹+9cosh𝜂+782𝛼𝑏𝑎⁰²¹−𝑏²⁰2𝛼𝑎²¹+9sinh𝜂+8𝛼𝑎²¹𝑏₀,(3.20) where 𝜂=𝑥+𝑦+𝑧−(3+𝛼𝑎²¹)𝑡.
If 𝑏₀=1,𝛼=6, and 𝑎₁=1/2, (3.20) becomes 𝑢(𝜂)=12+1+23cosh𝜂.(3.21) Substituting (3.14) into (3.8) and simplifying, we obtain 𝑢(𝜂)=72𝑎⁰72+𝛼𝑎²⁰cosh𝜂+72−𝛼𝑎²⁰sinh𝜂,(3.22) where 𝜂=𝑥+𝑦+𝑧−3𝑡.
If 𝑎₀=3 and 𝛼=8, (3.22) becomes 𝑢(𝜂)=32secℎ𝜂.(3.23)

Case 2. We set 𝑝=𝑐=2 and 𝑞=𝑑=1.
For this case, the trial solution (2.5) reduces to 𝑢(𝜂)=𝑎²𝑒^2𝜂+𝑎₁𝑒^𝜂+𝑎₀+𝑎₋₁𝑒^−𝜂𝑏₂𝑒^2𝜂+𝑏₁𝑒^𝜂+𝑏₀+𝑏₋₁𝑒^−𝜂.(3.24) Since, there are some free parameters in (3.24), for simplicity, we may consider that 𝑏₂=1 and 𝑏₋₁=0. Then the solution (3.24) is simplified as 𝑢(𝜂)=𝑎²𝑒^2𝜂+𝑎₁𝑒^𝜂+𝑎₀+𝑎₋₁𝑒^−𝜂𝑒^2𝜂+𝑏₁𝑒^𝜂+𝑏₀.(3.25) Performing the same procedure as described in Case 1, we obtain four sets of solutions.

Set 1. We obtain that𝑏₀=𝑏₀,𝑎₋₁=0,𝑎₀=∓6𝑏⁰√−2𝛼,𝑎¹=0,𝑎₂=±6√−2𝛼,𝑏¹=0,𝑉=−6,(3.26) where 𝑏₀ is free parameter.

Set 2. We obtain that 𝑎₁=𝑎₁,𝑏₁=𝑏₁,𝑎₋₁=0,𝑎₀=∓112√−2𝛼2𝛼𝑎²¹𝑎₂+9𝑏²¹,=±3√−2𝛼,𝑏⁰=118𝛼𝑎²¹+14𝑏²¹,𝑉=−32,(3.27) where 𝑎₁ and 𝑏₁ are free parameters.

Set 3. We obtain that 𝑎₂=𝑎₂,𝑏₁=𝑏₁,𝑎₋₁=0,𝑎₀=𝑏²¹2𝛼𝑎²²+98𝛼𝑎₂𝑎₁,=𝑏¹𝛼𝑎²²+9𝛼𝑎₂,𝑏₀=𝑏²¹2𝛼𝑎²²+98𝛼𝑎²²,𝑉=𝛼𝑎²²+3,(3.28) where 𝑎₂ and 𝑏₁ are free parameters.

Set 4. We obtain that𝑎₁=𝑎₁,𝑎₋₁=0,𝑎₀=0,𝑎₂=0,𝑏₀=𝛼𝑎²¹72,𝑏¹=0,𝑉=3,(3.29) where 𝑎₁ is a free parameter.
Using (3.26) into (3.25) and simplifying, we obtain that 𝑢(𝜂)=±6√−2𝛼1−2𝑏⁰(cosh𝜂−sinh𝜂)1+𝑏₀cosh𝜂+1−𝑏₀sinh𝜂.(3.30) If 𝑏₀=1, from (3.30), we obtain that 𝑢(𝜂)=±6𝑖√2𝛼tanh𝜂,(3.31) where 𝜂=𝑥+𝑦+𝑧+6𝑡.
Substituting (3.27) into (3.25) and simplifying, we obtain that 𝑢(𝜂)=±3√−2𝛼⎡⎢⎢⎣1+12±𝑎¹√−2𝛼+3𝑏₁−22𝛼𝑎²¹+9𝑏²¹(cosh𝜂−sinh𝜂)36+2𝛼𝑎²¹+9𝑏²¹cosh𝜂+36−2𝛼𝑎²¹−9𝑏²¹sinh𝜂+36𝑏₁⎤⎥⎥⎦.(3.32) If 𝛼 is negative, that is, 𝛼=−𝛽, 𝛽>0, 𝑏₁=2 and 𝑎₁=0, (3.32) can be simplified as 𝑢(𝜂)=±3√2𝛽tanh𝜂2,(3.33) where 𝜂=𝑥+𝑦+𝑧+(3/2)𝑡.
Substituting (3.28) into (3.25) and simplifying, we obtain that 𝑢(𝜂)=𝑎₂1+8𝛼𝑎²²+𝑏²¹2𝛼𝑎²²+9cosh𝜂+782𝛼𝑏𝑎¹²²−𝑏²¹2𝛼𝑎²²+9sinh𝜂+8𝛼𝑎²²𝑏₁.(3.34) If 𝑏₁=1,𝛼=6, and 𝑎₂=1/2, (3.34) becomes 𝑢(𝜂)=12+1+23cosh𝜂,(3.35) where 𝜂=𝑥+𝑦+𝑧−(3+𝛼𝑎²²)𝑡.
Using (3.29) into (3.25) and simplifying, we obtain that 𝑢(𝜂)=72𝑎¹72+𝛼𝑎²¹cosh𝜂+72−𝛼𝑎²¹sinh𝜂.(3.36) If 𝑎₁=3, and 𝛼=8, (3.36) becomes 𝑢(𝜂)=32secℎ𝜂,(3.37) where 𝜂=𝑥+𝑦+𝑧−3𝑡.

Case 3. We set 𝑝=𝑐=2 and 𝑞=𝑑=2.
For this case, the trial solution (2.5) reduces to 𝑢(𝜂)=𝑎²𝑒^2𝜂+𝑎₁𝑒^𝜂+𝑎₀+𝑎₋₁𝑒^−𝜂+𝑎₋₂𝑒^−2𝜂𝑏₂𝑒^2𝜂+𝑏₁𝑒^𝜂+𝑏₀+𝑏₋₁𝑒^−𝜂+𝑏₋₂𝑒^−2𝜂.(3.38) Since, there are some free parameters in (3.38), we may consider 𝑏₂=1,𝑎₋₂=0,𝑏₋₂=0, and 𝑏₋₁=0 so that the (3.38) reduces to the (3.25). This indicates that the Case 3 is equivalent to the Case 2. Equation (3.38) can be rewritten as 𝑢(𝜂)=𝑎²𝑒^𝜂+𝑎₁+𝑎₀𝑒^−𝜂+𝑎₋₁𝑒^−2𝜂+𝑎₋₂𝑒^−3𝜂𝑏₂𝑒^𝜂+𝑏₁+𝑏₀𝑒^−𝜂+𝑏₋₁𝑒^−2𝜂+𝑏₋₂𝑒^−3𝜂.(3.39)
If we put 𝑎₋₂=0,𝑎₋₁=0,𝑏₂=1,𝑏₋₂=0, and 𝑏₋₁=0 into (3.39), we obtain the solution form as (3.8). This implies that the Case 3 is equivalent to the Case 1.
Also, if we consider 𝑝=𝑐=3 and 𝑞=𝑑=3, it can be shown that this Case is also equivalent to the Cases 1 and 2.
Therefore, we think that no need to find the solutions again.

It is noted that the solution (3.17) and (3.31) are identical, solution (3.19) and (3.33) are identical, solution (3.21) and (3.35) are identical, and solution (3.23) and (3.37) are identical.

Beyond Table 1, Zayed [52] obtained another solution (3.39). But, we obtain two more new solutions (3.21) and (3.23).

Table 1 

Comparison between Zayed [52] solutions and our solutions.

Graphical Representations of the Solutions
The above solutions are shown with the aid of Maple 13 in the graphs.

4. Conclusions

Using the Exp-function method, with the aid of symbolic computation software Maple 13, new exact traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation are constructed. It is important that some of the obtained solutions are identical to the solutions available in the literature and some are new. These solutions can be used to describe the insight of the complex physical phenomena.

Acknowledgment

This paper is supported by USM short-term Grant 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. The authors are also grateful to the referee(s) for their valuable comments and suggestions.