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Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 153974, 8 pages

http://dx.doi.org/10.1155/2014/153974

## Exp-Function Method for a Generalized MKdV Equation

School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Received 15 December 2013; Accepted 24 April 2014; Published 15 May 2014

Academic Editor: Cengiz Çinar

Copyright © 2014 Yuzhen Chai et al. 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.

#### Abstract

Under investigation in this paper is a generalized MKdV equation, which describes the propagation of shallow water in fluid mechanics. In this paper, we have derived the exact solutions for the generalized MKdV equation including the bright soliton, dark soliton, two-peak bright soliton, two-peak dark soliton, shock soliton and periodic wave solution via Exp-function method. By figures and symbolic computations, we have discussed the propagation characteristics of those solitons under different values of those coefficients in the generalized MKdV equation. The method constructing soliton solutions in this paper may be useful for the investigations on the other nonlinear mathematical physics model and the conclusions of this paper can give theory support for the study of dynamic features of models in the shallow water.

#### 1. Introduction

Nonlinear science studies nonlinear phenomena of the world, which is cross-disciplinary [1–4]. In fact, any science, whether natural science or social science, has its own nonlinear phenomena and problems. To study these phenomena and problems, many branches of nonlinear science have been promoted to be built and developed [5–7]. Obviously, most of the phenomena are not linear in the natural sciences and engineering practice; thus many problems cannot be researched and solved by the linear methods, which makes the study of nonlinear science very significant. Actually, nonlinear science, which mainly consists of soliton, chaos, and fractal [3, 4, 7], is not the simple superposition and comprehensive of these nonlinear branches but a comprehensive subject to study the various communist rules in nonlinear phenomena. With the rapid development of nonlinear science, the study of exact solutions of nonlinear evolution equations has attracted much attention of many mathematicians and physicists.

In 1834, the solitary wave phenomenon was observed by Huang et al. [8] and Russell [9]. Later, people named the isolation water peak, which kept moving with constant shape and speed on the surface of the water, as a solitary wave [8]. The discovery of a solitary wave turned people into a new field in the study of the waves of the convection. In 1895, Holland’s Professor Kortewrg and his disciples Vries derived the famous KdV equation from the research of shallow water wave motion [10]. The analysis to the KdV equation has improved to recover inverse scattering method, on which people expanded the new research directions of algebra and geometry [8]: MKdV equation plays an important role in describing the plasmas and phonon in anharmonic lattice. In the nineteen fifties, physicist Fermi, Pasta, and Ulam made a famous FPU experiment, connecting the 64 particles by nonlinear spring, thereby forming a nonlinear vibrating string. Although the FPU experiment did not gain the solutions of solitary wave, it will expand the study of the solitary wave to the field outside the mechanics [11]. Later, Fermi et al. studied the nonlinear vibration problem of FPU model and obtained the solitary wave solutions [11], which is the right answer for the question of FPU. It is the first time to find solitary wave solutions in the field outside mechanics after the solution was found in the KdV equation, which give rise to the scientists’ interest in researching the solitary wave phenomenon. In 1965, Zabusky and Kruskal studied solitary waves in plasma [12]; they found that the waveform does not change nature before and after collision of nonlinear solitary waves, which is similar to particle collisions, so Zabusky and Kruskal named the solitary wave with the impact properties of collisions as soliton [12]. Soliton concept is an important milepost on the history of the development of the soliton theory. The next few decades, the soliton theory had a rapid development and penetrated into many areas, such as fluid mechanics, nonlinear optical fiber communication, plasma physics, fluid physics, chemistry, life science, and marine science [13–15].

Recently many new approaches to nonlinear wave equations have been proposed, such as Tanh-function method [16–18], F-expansion method [19], Jacobian elliptic function method [20], Darboux transformation method [21–26], adomian method [27–29], variational approach [30], and homotopy perturbation method [31]. All methods mentioned above have their limitations in their applications. We will apply Exp-function method to a generalized MKdV equation to gain exact solutions: where and are real parameters. When ( is a constant) as , solitons can be obtained.

This paper will be organized as follows. In Section 2, the basic idea of Exp-function method is introduced. In Section 3, carrying on calculating and illustrating by the mathematical software MATHEMATIC, we will solve the solitary wave solutions of (2) based on the Exp-function method. In Section 4, we will obtain shock soliton, bright soliton, two-peak bright soliton, dark soliton, two-peak dark soliton, and periodic wave solutions and analyze the dynamic features of soliton solutions by using some figures. Finally, our conclusions will be addressed in Section 5.

#### 2. Basic Idea of Exp-Function Method

In order to illustrate the basic idea of the suggested method, we consider firstly the following general partial differential equation: We aim at its exact solutions, so we introduce a complex variable, , defined as where and are constants unknown to be further determined. Therefore we can convert (3) into an ordinary differential equation with respect to : Very simple and straightforward, the Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form: where , , , and are positive integers which are unknown to be further determined and and are unknown constants.

We suppose that which is the solution of (5) can be expressed as

To determine the values of and , we balance the linear term of highest order in (5) with the highest order nonlinear term. Similarly we balance the lowest orders to confirm and . For simplicity, we set some particular values for , , , and and then change the left side of (5) into the polynomial of . Equating the coefficients of to be zero results in a set of algebraic equations. Then solving the algebraic system with symbolic computation system, we can gain the solution. Substituting it into (3), we have the general form of the exact solution expressed as the form of . Taking the form of the solutions into consideration, we can have more extensive solutions via Exp-function method.

#### 3. Solving Generalized MKdV Equation via Exp-Function Method

Now we consider the follwing equation [32, 33]: This equation is called modified KdV equation, which arises in the process of understanding the role of nonlinear dispersion and in the formation of structures like liquid drops. The KdV equation is one of the most familiar models for solitons and the foundation which studies other equations. Developing upon this foundation, the isolated theories are treated as the milestone of mathematics physical method. During researching the isolated theories, the remarkable application should be laser shooting practice and fiber-optic communication.

For researching the variety of the solutions for the modified KdV equation, we carry on the MKdV equation to expand. Thus, we have where and are real parameters. In the following, we consider (9).

Introduce a complex variable, , defined as By (10), (9) becomes which denotes the differential with respect to .

The Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form: where , , , and are positive integers unknown to be further determined and and are unknown constants.

We suppose that the solution of (11) can be expressed as To determine values of and , we balance the linear term of highest order in (11) with the highest order nonlinear term. By simple calculation, we have where are coefficients only for simplicity.

Balancing highest order of exponential function in (14), we have which leads to the result Similarly to make sure of the values of and , we balance the linear term of lowest order in (11): where are determined to be coefficients only for simplicity.

Balancing lowest order of exponential function in (17), we have which leads to the result For simplicity, we set and , so (13) reduces to Substituting (20) into (11), and by the help of Mathematica, we have where with .

Equating the coefficients of to be zero, we have Solving (23), simultaneously, we obtain Substituting (24) into (20) results in a compact-like solution, which reads where , , , , and are real parameters and . Besides, the obtained solution equation (25) is a generalized soliton solution of (9).

In case, is an imaginary parameter, the obtained soliton solution can be converted into the periodic solution or compact-like solution. We write where is a real parameter.

Use the transformations Equation (25) becomes where , , , , and are real parameters and . Simplifying the above formula, we have where , , , , and are real parameters and .

If we search for a periodic solution or compact-like solution, the imaginary part in the denominator of (29) must be zero, which requires that Solving from (30), we obtain Substituting (31) into (29) results in a compact-like solution, which reads where and are free parameters, and , , and it requires that .

#### 4. Discussing the Forms of the Solutions

In the above, we gain two cases of solution equations (25) and (32). In order to get the richness of solutions of (9), we have these two cases discussed in the following.

##### 4.1. The Solutions from (25)

Through discussions, we obtain the following results.

###### 4.1.1. When and

when and , (9) becomes . Then we can obtain shock solitons. The solution formula is shown as follows: where , , and are real parameters.

*Case 1 (when , , ). *
Consider
Description is as shown in Figure 1(a).

*Case 2 (when , , and ). *
Consider
Description is as shown in Figure 1(b).

###### 4.1.2. When and

when and , (9) becomes . Then we can obtain one-peak bright solitons. The solution formula is shown as follows: where , , and are real parameters and .

*Case 3 (when , , and ). *
Consider
Description is as shown in Figure 2(a).

*Case 4 (when , , and ). *
Consider
Description is as shown in Figure 2(b).

###### 4.1.3. When and

when and , (9) becomes . Then we can obtain bright solitons and dark solitons. The solution formula is shown as follows: where , , and are real parameters and .

*Case 5 (when , , and ). *
Consider
Description is as shown in Figure 3(a).

*Case 6 (when , , and ). *
Consider
Description is as shown in Figure 3(b).

*Case 7 (when , , and ). *
Consider
Description is as shown in Figure 4(a).

*Case 8 (when , , and ). *
Consider
Description is as shown in Figure 4(b).

##### 4.2. The Solutions from (32)

Based on the form of (32), if we set and at the same time, the solution should be zero. Therefore, it is not significative to study and analyse. So, in the following, we consider and under the circumstance that and are not zero in the meantime.

When and , (9) becomes . Then we can obtain periodic wave solutions. The solution formula is shown as follows: where and .

To illustrate the analysis, we obtain the following results.

*Case 9 (when , ). *
Consider
Description is as shown in Figure 5.

#### 5. Conclusions

In this paper, our main attention has been focused on seeking the soliton solutions of (9) via Exp-function method. By applying Exp-function method, we have obtained shock soliton, bright soliton, two-peak bright soliton, dark soliton, two-peak dark soliton, and periodic wave solution. In addition, with figures and symbolic computations, we have described the propagation characteristics of those solitons under different values of those coefficients in the generalized MKdV equation. Furthermore, the problem solving process and the algorithm, by the help of Mathematica, can be easily extended to all kinds of nonlinear equations.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the National Natural Science Foundation of China (grant no. 11172194), by the Special Funds of the National Natural Science Foundation of China under Grant No. 11347165, by Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi under Grant No. 2013110.

#### References

- E. Infeld and G. Rowlands,
*Nonlinear Waves, Solitons and Chaos*, Cambridge University Press, Cambridge, UK, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - A. Scott,
*Encyclopedia of Nonlinear Science*, Routledge, New York, NY, USA, 2005. View at MathSciNet - S. J. Li,
*Nonlinear Science and Its Application*, Harbin Institute of Technology Press, Harbin, China, 2011. - L. S. Shun,
*Forefront Issues of Nonlinear Science*, China Science and Technology Press, Beijing, China, 2009. - S. K. Liu and S. D. Liu,
*Nonlinear Equations in Physics*, Peking University Press, Beijing, China, 2000. - S. K. Liu, S. D. Liu, and B. D. Tan,
*Nonlinear Atmospherics Dynamics*, China National Defense Industry Press, Bejing, China, 1996. - G. Chen,
*Some Issues on the Theroy of Nonlinear Science and Its Application. [Academic Dissertation]*, Zhejiang University Press, Zhejiang, China, 1996. - J. T. Huang, J. Z. Xu, and Y. T. Xiong,
*Solitons Concepts, Principles and Applications*, Higher Education Press, Beijing, China, 2004. - J. S. Russell, “Report on waves,” in
*Proceedings of the 14th Meeting of the Bntish Association for the Advancement of Science*, pp. 311–390, John Murray, London, UK, 1844. - D. J. Kortewrg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave,”
*Philosophical Magazine*, vol. 39, no. 240, pp. 422–443, 1895. View at Publisher · View at Google Scholar - E. Fermi, J. Pasta, and S. Ulam, “Studies of nonlinear problems,” Los Alamos Document, Los Alamos, NM, USA, 1955.
- N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the recurrence of initial states,”
*Physical Review Letters*, vol. 15, no. 6, pp. 240–243, 1965. View at Publisher · View at Google Scholar · View at Scopus - T. X. Chen,
*Introduction to Nonlinear Physics*, University of Science and Technology of China Press, Hefei, China, 2002. - Z. L. Pan,
*The Mathematic Methods of the Nonlinear Problem and Its Application*, Zhejiang University Press, Zhejiang, China, 2002. - D. Y. Wang, D. J. Wu, and G. L. Huang,
*Solitary Wave in Space Plasma*, Shanghai Scientific and Technological Education Publishing House, Hangzhou, China, 2000. - H. A. Abdusalam, “On an improved complex tanh-function method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 2, pp. 99–106, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. A. Abdou and A. A. Soliman, “Modified extended tanh-function method and its application on nonlinear physical equations,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 353, no. 6, pp. 487–492, 2006. View at Publisher · View at Google Scholar · View at Scopus - D. L. Sekulić, M. V. Satarić, and M. B. Živanov, “Symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh-function method,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3499–3506, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W.-W. Li, Y. Tian, and Z. Zhang, “
*F*method and its application for finding new exact solutions to the sine-Gordon and sinh-Gordon equations,”*Applied Mathematics and Computation*, vol. 219, no. 3, pp. 1135–1143, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - A. H. Bhrawy, M. A. Abdelkawy, and A. Biswas, “Cnoidal and snoidal wave solutions to coupled nonlinear wave equations by the extended Jacobi's elliptic function method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, no. 4, pp. 915–925, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Guo and H.-Q. Hao, “Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, no. 9, pp. 2426–2435, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - F. Qi, B. Tian, X. Lü, R. Guo, and Y. Xue, “Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 6, pp. 2372–2381, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. Guo and B. Tian, “Integrability aspects and soliton solutions for an inhomogeneous nonlinear system with symbolic computation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 8, pp. 3189–3203, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. Guo, B. Tian, L. Wang, F.-H. Qi, and Y. Zhan, “Darboux transformation and soliton solutions for a system describing ultrashort pulse propagation in a multicomponent nonlinear medium,”
*Physica Scripta*, vol. 81, no. 2, Article ID 025002, 2010. View at Publisher · View at Google Scholar - R. Guo, B. Tian, L. Xing, H.-Q. Zhang, and T. Xu, “Integrability aspects and soliton solutions for a system describing ultrashort pulse propagation in an inhomogeneous multi-component medium,”
*Communications in Theoretical Physics*, vol. 54, no. 3, pp. 536–544, 2010. View at Google Scholar - R. Guo, B. Tian, X. Lü, H.-Q. Zhang, and W.-J. Liu, “Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrodinger-Maxwell-Bloch system with symbolic computation,”
*Computational Mathematics and Mathematical Physics*, vol. 52, no. 4, pp. 565–577, 2012. View at Google Scholar - L. Bougoffa and R. C. Rach, “Solving nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations by the Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 225, pp. 50–61, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - T. S. El-Danaf, M. A. Ramadan, and F. E. I. Abd Alaal, “The use of adomian decomposition method for solving the regularized long-wave equation,”
*Chaos, Solitons & Fractals*, vol. 26, no. 3, pp. 747–757, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. M. El-Sayed, D. Kaya, and S. Zarea, “The decomposition method applied to solve high-order linear Volterra-Fredholm integro-differential equations,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 5, no. 2, pp. 105–112, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Liu, “Variational Approach to Nonlinear Electrochemical System,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 5, no. 1, pp. 95–96, 2004. View at Google Scholar · View at Scopus - J. He, “Homotopy perturbation method for bifurcation of nonlinear problems,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 2, pp. 207–208, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. He and X. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,”
*Chaos, Solitons & Fractals*, vol. 29, no. 1, pp. 108–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,”
*Physical Review Letters*, vol. 70, no. 5, pp. 564–567, 1993. View at Publisher · View at Google Scholar · View at Scopus