Research Article | Open Access

Naila Sajid, Ghazala Akram, "The Application of the -Expansion Method for Finding the Exact Solutions of Two Integrable Equations", *Mathematical Problems in Engineering*, vol. 2018, Article ID 5191736, 10 pages, 2018. https://doi.org/10.1155/2018/5191736

# The Application of the -Expansion Method for Finding the Exact Solutions of Two Integrable Equations

**Academic Editor:**Maria L. Gandarias

#### Abstract

In this article, the method is connected to search for new hyperbolic, periodic, and rational solutions of -dimensional fifth-order nonlinear integrable equation and -dimensional Date-Jimbo-Kashiwara-Miwa equation. The obtained solutions consist of trigonometric, hyperbolic, rational functions and W-shaped soliton. Furthermore, 3D and 2D graphs are plotted by choosing the suitable values of the parameters involved.

#### 1. Introduction

Firstly, consider the -dimensional fifth-order nonlinear integrable equation. In [1], Wazwaz proposed a new -dimensional fifth-order nonlinear integrable equation of the formwhere stands for wave propagation of physical quantity and subscripts represent partial differentiation with respect to the given variable and obtained multiple soliton solutions using the simplified Hirota’s method established by Hereman and Nuseir [2].

Furthermore, Yuan* et al.* [3] consider the -dimensional Date-Jimbo-Kashiwara-Miwa equation. To study the -dimensional Date-Jimbo-Kashiwara-Miwa equation, many researchers considered the following integrable equation:where is the real function of the variables , , and . With the help of the Hirota method and auxiliary variables, the bilinear Bäcklund transformation and N-soliton solutions are obtained.

Nonlinear evolution equations (NLEEs) are one of the fastest developing zones of research in the field of science and engineering, especially in mathematical biology, nonlinear optics, optical fiber, fluid mechanics, solid state physics, biophysics, chemical physics, chemical kinetics,* etc.* Many effective methods have been proposed to solve the NLEEs, such as the Hirota method [1], Hereman-Nuseir method [2], inverse scattering transformation [4], Painlevé technique [5], Bäcklund transformation [6], Darboux transformation [7, 8], Binary-Bell-polynomial scheme [9], first integral method [10, 11], expansion method [12], the expansion method [13], Exp-function method [14], ansatz method [15], sine-Gordon expansion method [16, 17], the trial equation method [18, 19], homotopy asymptotic [20], and so on.

The present paper organized as follows: In Section 2, description of the method for finding the exact traveling wave solutions of NLEEs is presented. Section 3 illustrates the method to solve the -dimensional fifth-order nonlinear integrable equation and -dimensional Date-Jimbo-Kashiwara-Miwa equation. Results and discussion are presented in Section 4. Finally, in Section 5, some conclusions are given.

#### 2. Traveling Wave Hypothesis

Consider a NLEE in two independent variables and of the form aswhere is an unknown function and is the polynomial in which have various partial derivatives. Under the wave transformationwhere and are the wave number and wave speed, respectively. Eq. (3) can be transformed into the following nonlinear ordinary differential equation aswhere prime denotes the derivatives with respect to .

##### 2.1. The Expansion Method: Quick Recapitulation

The key steps of method are given as

*Step 1. *According to method, the wave solution can be expressed aswhere are constants to be determined and satisfies the auxiliary ODE given asThe auxiliary Eq. (7) has the general solutions given as follows. *Case 1 *(hyperbolic function solutions). When and ,*Case 2 *(trigonometric function solutions). When and ,*Case 3 *(hyperbolic function solutions). When , and ,*Case 4 *(rational function solutions). When , and ,*Case 5*. When , and ,where is integration constant.

*Step 2. *The positive integer can be determined by balancing the highest derivative term with the highest order nonlinear term in Eq. (5). Substituting Eq. (6) into Eq. (5) yields an algebraic equation involving powers of . Equating the coefficients of each power of to zero gives a system of algebraic equations for , , , , and

*Step 3. *Substituting , , , , and , a variety of exact solutions of Eq.(3) can be constructed.

#### 3. Application of the Method

##### 3.1. The -Dimensional Fifth-Order Nonlinear Integrable Equation

The -dimensional fifth-order nonlinear integrable equation is given asApplying the wave transformation , to Eq. (13) yields the following nonlinear ODE:Set to getwhere the integration constant is taken as zero. Balance the linear term of highest order with the highest order nonlinear term in Eq. (15), to get balancing number as . Thus the solution isSubstituting Eq. (16) into Eq. (15) and collecting the coefficient of each power of and then setting each of coefficients to zero give a system of algebraic equations as Solving the above system of equations yieldsConsequently, the following different cases are obtained for the exact solutions of -dimensional fifth-order nonlinear integrable equation.

*Case 1 *(hyperbolic function solutions). When and ,

*Case 2 *(trigonometric function solutions). When and ,

*Case 3 *(hyperbolic function solutions). When , and ,

*Case 4 *(rational function solutions). When , and ,

*Case 5*. When , and ,where and is the integration constant.

The solutions () of the -dimensional fifth-order nonlinear integrable equation are graphically presented by Figures 1, 2, 3, 4, and 5.

##### 3.2. -Dimensional Date-Jimbo-Kashiwara-Miwa Equation

The -dimensional Date-Jimbo-Kashiwara-Miwa equation is given asApplying the wave transformation , to Eq. (24) yields the following nonlinear ODE:Set to getwhere the integration constant is taken as zero. Balance the linear term of highest order with the highest order nonlinear term in Eq. (26), to get balancing number as . Thus, the solution can be written asSubstituting Eq. (27) into Eq. (26) and collecting the coefficient of each power of and then setting each of coefficient to zero give a system of algebraic equations asSolving the above system of equations yieldsConsequently, the following different cases are obtained for the exact solutions of -dimensional Date-Jimbo-Kashiwara-Miwa equation.

*Case 1 *(hyperbolic function solutions). When and ,

*Case 2 *(trigonometric function solutions). When and ,

*Case 3 *(hyperbolic function solutions). When , and ,

*Case 4 *(rational function solutions). When , and ,

*Case 5*. When , and ,where and is the integration constant.

The graphical representation of the solutions for the -dimensional Date-Jimbo-Kashiwara-Miwa equation is shown by Figures 6–10.

#### 4. Results and Discussion

In this manuscript, several traveling wave solutions are developed. A new kind of W-shaped soliton solution is demonstrated and the other obtained solutions consist of trigonometric, hyperbolic, rational functions which are also new. On comparing our results with the well-known results obtained in [1, 3], it may be concluded that the obtained results for Eq. (13) and Eq. (24) are newly constructed. Wazwaz [1] reported some multiple soliton solutions to Eq. (13) using simplified Hirota’s method and Yuan* et al.* [2] acquired N-soliton solutions to Eq. (24) using Hirota method and auxiliary variables. In both equations for case , the absolute behavior of the solution and is shown in Figures 1 and 6, respectively. The graphical representation of the other solutions () for the -dimensional fifth-order nonlinear integrable equation and for the -dimensional Date-Jimbo-Kashiwara-Miwa equation is shown by Figures 2–5 and 7–10, respectively.

#### 5. Conclusion

The method is used to investigate the exact solutions of the -dimensional fifth-order nonlinear integrable equation and -dimensional Date-Jimbo-Kashiwara-Miwa equation. With the implementation of the method, many exact traveling wave solutions are obtained including trigonometric, hyperbolic, rational, and W-shaped soliton. The reported solutions in this article may be useful in explaining the physical meaning of the studied models and other nonlinear models arising in the field of fiber optics and other related fields. These results indicate that the proposed method is very useful and effective in performing solution to the NLEEs.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### References

- A. M. Wazwaz, “A new fifth-order nonlinear integrable equation: multiple soliton solutions,”
*Physica Scripta*, vol. 83, p. 015012, 2011. View at: Google Scholar - W. Hereman and A. Nuseir, “Symbolic methods to construct exact solutions of nonlinear partial differential equations,”
*Mathematics and Computers in Simulation*, vol. 43, no. 1, pp. 13–27, 1997. View at: Publisher Site | Google Scholar | MathSciNet - Y.-Q. Yuan, B. Tian, W.-R. Sun, J. Chai, and L. Liu, “Wronskian and Grammian solutions for a -dimensional Date-Jimbo-Kashiwara-Miwa equation,”
*Computers & Mathematics with Applications. An International Journal*, vol. 74, no. 4, pp. 873–879, 2017. View at: Publisher Site | Google Scholar | MathSciNet - M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,”
*Studies in Applied Mathematics*, vol. 53, no. 4, pp. 249–315, 1974. View at: Publisher Site | Google Scholar | MathSciNet - J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,”
*Journal of Mathematical Physics*, vol. 24, no. 3, pp. 522–526, 1983. View at: Publisher Site | Google Scholar | MathSciNet - X.-Y. Gao, “Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation,”
*Applied Mathematics Letters*, vol. 73, pp. 143–149, 2017. View at: Publisher Site | Google Scholar | MathSciNet - M. Li, T. Xu, and D. Meng, “Rational solitons in the parity-time-symmetric nonlocal nonlinear Schrödinger model,”
*Journal of the Physical Society of Japan*, vol. 85, no. 12, p. 124001, 2016. View at: Google Scholar - T. Xu, H. Li, H. Zhang, M. Li, and S. Lan, “Darboux transformation and analytic solutions of the discrete PT-symmetric nonlocal nonlinear Schrödinger equation,”
*Applied Mathematics Letters*, vol. 63, pp. 88–94, 2017. View at: Publisher Site | Google Scholar | MathSciNet - C. Gilson, F. Lambert, J. Nimmo, and R. Willox, “On the combinatorics of the Hirota -operators,”
*Proceedings of the Royal Society A Mathematical, Physical and Engineering Sciences*, vol. 452, no. 1945, pp. 223–234, 1996. View at: Publisher Site | Google Scholar | MathSciNet - F. Batool and G. Akram, “On the solitary wave dynamics of complex Ginzburg–Landau equation with cubic nonlinearity,”
*Optical and Quantum Electronics*, vol. 49, no. 4, 2017. View at: Publisher Site | Google Scholar - G. Akram and N. Mahak, “Analytical solution of the Korteweg–de Vries equation and microtubule equation using the first integral method,”
*Optical and Quantum Electronics*, vol. 50, no. 3, 2018. View at: Google Scholar - G. Akram and F. Batool, “A class of traveling wave solutions for space–time fractional biological population model in mathematical physics,”
*Indian Journal of Physics*, vol. 91, no. 10, pp. 1145–1148, 2017. View at: Publisher Site | Google Scholar - A. Akbulut, M. Kaplan, and F. Tascan, “The investigation of exact solutions of nonlinear partial differential equations by using exp(−Φ(
*ξ*)) method,”*Optik - International Journal for Light and Electron Optics*, vol. 132, pp. 382–387, 2017. View at: Publisher Site | Google Scholar - M. Dehghan, J. Manafian Heris, and A. Saadatmandi, “Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics,”
*International Journal of Numerical Methods for Heat & Fluid Flow*, vol. 21, no. 6-7, pp. 736–753, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Ö. Güner and A. Bekir, “Solving nonlinear space-time fractional differential equations via ansatz method,”
*Computational Methods for Differential Equations*, vol. 6, no. 1, pp. 1–11, 2018. View at: Google Scholar | MathSciNet - D. Kumar, K. Hosseini, and F. Samadani, “The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics,”
*Optik - International Journal for Light and Electron Optics*, vol. 149, pp. 439–446, 2017. View at: Publisher Site | Google Scholar - F. Batool and G. Akram, “Solitary wave solutions of (2 + 1)-dimensional soliton equation arising in mathematical physics,”
*Optik - International Journal for Light and Electron Optics*, vol. 144, pp. 156–162, 2017. View at: Publisher Site | Google Scholar - A. Biswas, Y. Yildirim, E. Yasar et al., “Optical soliton perturbation with full nonlinearity by trial equation method,”
*Optik - International Journal for Light and Electron Optics*, vol. 157, pp. 1366–1375, 2018. View at: Publisher Site | Google Scholar - M. Ekici, M. Mirzazadeh, A. Sonmezoglu et al., “Dispersive optical solitons with Schrödinger–Hirota equation by extended trial equation method,”
*Optik - International Journal for Light and Electron Optics*, vol. 136, pp. 451–461, 2017. View at: Publisher Site | Google Scholar - A. K. Gupta and S. Saha Ray, “Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq-BURger equations,”
*Computers & Fluids. An International Journal*, vol. 103, pp. 34–41, 2014. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2018 Naila Sajid and Ghazala Akram. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.