On the Complex and Hyperbolic Structures for the (2 + 1)-Dimensional Boussinesq Water Equation

1 Department of Office Management and Manager Assistant, Bolvadin Vocational School, AfyonKocatepe University, Afyonkarahisar 03300, Turkey 2 Department of Computer Engineering, Faculty of Engineering, Tunceli University, Tunceli 62100, Turkey; hmbaskonus@gmail.com 3 Department of Mathematics, Faculty of Science, Firat University, Elazig 23119, Turkey; hbulut@firat.edu.tr * Correspondence: fozpinar@aku.edu.tr; Tel.: +90-272-612-6353


Introduction
The (2 + 1) Boussinesq equation was founded to describe some physical facts such as the propagation of small-amplitude long waves in shallow water in 1987 [1]. Some authors have investigated the physical and analytical structures of the (2 + 1)-dimensional Boussinesq water equation by using various methods [1][2][3]. Moleleki and Khalique have considered the simplest equation method for solving the (2 + 1)-dimensional Boussinesq equation [4]. Zhang, Meng, Li, and Tian have studied the soliton resonance condition of the (2 + 1)-dimensional Boussinesq equation which is used to describe the propagation of gravity waves on the surface of water [5]. The homogeneous balance method has been successfully applied to the (2 + 1)-dimensional Boussinesq equation [6,7]. Allen and Rowlands have discussed the stability of solitary waves of the (2 + 1)-dimensional Boussinesq water equation and found that pulse-like solutions to the (2 + 1)-dimensional Boussinesq water equation are stable against linear perturbations [3]. The most general methods along this direction such as the exp(´Φ(η))-expansion method [8][9][10], the transformed rational function method [11], Bäcklund transformations [12], Frobenius integrable decompositions [13], and the multiple exp-functions method [14,15] have been applied to the various differential equations by Ma, Zhu, Huang and Zhang et al. Moreover, Wronskian solutions to the (1 + 1)-dimensional Boussinesq equation have been systematically presented in [16].
The main aim of this paper is to determine whether or not the new analytical method will be a powerful tool for obtaining new exponential, hyperbolic and complex analytical solutions to the (2 + 1)-dimensional Boussinesq water equation defined by [1][2][3]: u px, y, tq " U pξq , ξ " k px`y´ctq .
Using Equation (3), we can convert Equation (2) into a nonlinear ordinary differential equation (NODE) defined by: NODE`U, U 1 , U 2 , U 3 ,¨¨¨˘" 0, where NODE is a polynomial of U and its derivatives and the superscripts indicate the ordinary derivatives with respect to ξ.
Step 3: Substituting Equations (6) and (7)(8)(9)(10)(11) into Equation (5), we get a polynomial of exp p´Ω pξqq . We equate all the coefficients of same power of exp p´Ω pξqq to zero. This procedure yields a system of equations which can be solved to find A 0 , (5), the general solutions of Equation (5) complete the determination of the solution of Equation (1).

Applications
In this sub-section of the study, we apply the above-mentioned method to the (2 + 1)-dimensional Boussinesq water equation [1][2][3] for obtaining new analytical solutions such as a new hyperbolic function solution and a complex function solution.
Example 1. When we consider the (2 + 1)-dimensional Boussinesq water equation along with Equations (3) and (5), we obtain the following nonlinear ordinary differential equation: where c, k are constants and U " U pξq. Using the balance principle for determining the relationship between U 2 and U 2 , we derive the following equation: By using this relationship, we can attain some new analytical solutions for Equation (1) as follows: Case 1: Let M " 1 and N " 3, and we can write; where A 3 ‰ 0 and B 1 ‰ 0. Substituting Equations (14) and (15) in Equation (12), we get an equation including exp p´Ω pξqq and its various powers. Therefore, we have a system of equations from the coefficients of polynomial of exp p´Ω pξqq . Solving this system of equations yields the following coefficients:

Consider using Equation (18) along with Equations
Substituting Equation (19) along with Equations (3) and (7) into Equation (14), we find a new exponential function solution for Equation (1) as follows:

Case 2:
Letting M " 2 and N " 4, we can write the following: where A 4 ‰ 0 and B 2 ‰ 0. Substituting Equations (24) and (25) in Equation (12), we get an equation including exp p´Ω pξqq and its various powers. Therefore, we have a system of algebraic equations from the coefficients of the polynomial of exp p´Ω pξqq. Solving this system of equations yields the following coefficients; Case 2.1:

Physical Expressions and Discussions and Remarks
In this subsection of the manuscript, we introduce some basic properties of the MEFM and the physical meaning of the complex, dark solitonand hyperbolic function solutions found for Equation (1) obtained in this paper.
MEFM is more comprehensive according to the exp p´Ω pξqq-expansion method because MEFM includes one more parameter such as M. This gives many coefficients, which leads to many more traveling wave solutions as evidenced by the fact that we have obtained so many analytical solutions to the (2 + 1)-dimensional Boussinesq water equation for only M " 1 and N " 3. If we take M " 3 and N " 5, we can write the following equations: and where A 5 ‰ 0, B 3 ‰ 0. When we use Equations (30) and (32) in Equation (12), we obtain a system of algebraic equations. By solving this system via Wolfram Mathematica 9, we can obtain other analytical solutions which cannot be obtained by using only the exp p´Ω pξqq-expansion method. Therefore, this procedure of Equation (6) will contribute to more analytical solutions and to a better understanding of engineering and physical problems along with new physical predictions.
To the best of our knowledge, when we conduct a comparison with analytical solutions obtained by Ma [11][12][13][14][15], we have obtained similar hyperbolic solutions under the terms of M " 1 and N " 3; moreover, we have found new complex hyperbolic function solutions by using MEFM.When we compare these analytical solutions with solutions obtained by Lai, Wu, Zhou [1], Alam, Hafez, Akbar, Roshid [2], and Allen, Rowlands [3], and Chen, Yan, Zhang [7], they are new and have not been submitted to literature previously.
Secondly, hyperbolic functions are circular functions as well [20]. They arise in many problems of mathematics and mathematical physics. For instance, the hyperbolic sinearises in the gravitational potential of a cylinder. The hyperbolic cosine function is the shape of a hanging cable. The hyperbolic tangent arises in the calculation of and rapidity of special relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates in general relativity [20]. The hyperbolic secant arises in the profile of a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization [20]. It is estimated that all these analytical solutions are related to such physical problems.
In consideration of the surfaces depicted here, shown in Figures 1-9 they have been constructed using suitable parameters. These values of parameters are consistent with the physical meaning of the problem.

7
To the best of our knowledge, when we conduct a comparison with analytical solutions obtained by Ma [11][12][13][14][15], we have obtained similar hyperbolic solutions under the terms of 1 M  and 3 N  ; moreover, we have found new complex hyperbolic function solutions by using MEFM.When we compare these analytical solutions with solutions obtained by Lai, Wu, Zhou [1], Alam, Hafez, Akbar, Roshid [2], and Allen, Rowlands [3], and Chen, Yan, Zhang [7], they are new and have not been submitted to literature previously.
Secondly, hyperbolic functions are circular functions as well [20]. They arise in many problems of mathematics and mathematical physics. For instance, the hyperbolic sinearises in the gravitational potential of a cylinder. The hyperbolic cosine function is the shape of a hanging cable. The hyperbolic tangent arises in the calculation of and rapidity of special relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates in general relativity [20]. The hyperbolic secant arises in the profile of a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization [20]. It is estimated that all these analytical solutions are related to such physical problems.
In consideration of the surfaces depicted here, shown in Figures 1-9, they have been constructed using suitable parameters. These values of parameters are consistent with the physical meaning of the problem.       To the best of our knowledge, when we conduct a comparison with analytical solutions obtained by Ma [11][12][13][14][15], we have obtained similar hyperbolic solutions under the terms of 1 M  and 3 N  ; moreover, we have found new complex hyperbolic function solutions by using MEFM.When we compare these analytical solutions with solutions obtained by Lai, Wu, Zhou [1], Alam, Hafez, Akbar, Roshid [2], and Allen, Rowlands [3], and Chen, Yan, Zhang [7], they are new and have not been submitted to literature previously.
Secondly, hyperbolic functions are circular functions as well [20]. They arise in many problems of mathematics and mathematical physics. For instance, the hyperbolic sinearises in the gravitational potential of a cylinder. The hyperbolic cosine function is the shape of a hanging cable. The hyperbolic tangent arises in the calculation of and rapidity of special relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates in general relativity [20]. The hyperbolic secant arises in the profile of a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization [20]. It is estimated that all these analytical solutions are related to such physical problems.
In consideration of the surfaces depicted here, shown in Figures 1-9, they have been constructed using suitable parameters. These values of parameters are consistent with the physical meaning of the problem.

Conclusions
In this paper we have applied the application of MEFM to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential, complex and rational function solutions. We have observed that all analytical solutions obtained in this paper have verified to the Equation (1) by using Wolfram Mathematica 9. This method has provided many coefficients for Equations (14) and (24). Some of them have been considered in this paper to obtain new analytical solutions. If other coefficients are considered, of course, one can obtain different prototype solutions for Equation (1). Therefore, it can be said that this method is a powerful tool for obtaining solutions of the same type as Equation (1).

Conclusions
In this paper we have applied the application of MEFM to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential, complex and rational function solutions. We have observed that all analytical solutions obtained in this paper have verified to the Equation (1) by using Wolfram Mathematica 9. This method has provided many coefficients for Equations (14) and (24). Some of them have been considered in this paper to obtain new analytical solutions. If other coefficients are considered, of course, one can obtain different prototype solutions for Equation (1). Therefore, it can be said that this method is a powerful tool for obtaining solutions of the same type as Equation (1).