Journal of Applied Mathematics

Volume 2014, Article ID 839485, 11 pages

http://dx.doi.org/10.1155/2014/839485

## Stability Analysis for Travelling Wave Solutions of the Olver and Fifth-Order KdV Equations

^{1}Mathematics Department, Faculty of Science, Taibah University, Al-Ula 41921-259, Saudi Arabia^{2}Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Received 8 November 2013; Revised 25 January 2014; Accepted 2 February 2014; Published 24 March 2014

Academic Editor: Michael Meylan

Copyright © 2014 A. R. Seadawy 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

The Olver equation is governing a unidirectional model for describing long and small amplitude waves in shallow water waves. The solitary wave solutions of the Olver and fifth-order KdV equations can be obtained by using extended tanh and sech-tanh methods. The present results are describing the generation and evolution of such waves, their interactions, and their stability. Moreover, the methods can be applied to a wide class of nonlinear evolution equations. All solutions are exact and stable and have applications in physics.

#### 1. Introduction

The research on the Korteweg-de Vries (KdV) equations attracted the interest of many scientists. The KdV equations describe nonlinear dispersive long waves; many other partial differential equations have been derived to model wave phenomena in diverse nonlinear systems. The KdV equation plays an important role in describing motions of long waves in shallow water under gravity, one-dimensional nonlinear lattice [1, 2], fluid mechanics [3, 4], quantum mechanics, plasma physics, nonlinear optics, and other areas. The KdV equation is a well-known model for the description of nonlinear long internal waves in a fluid stratified by both density and current. The steady-state version of this equation was produced by Long [5], while Benney [6] gave the integral expressions for calculation of the coefficients of the KdV equation for waves in a fluid with arbitrary stratification in the density and current.

There are many classical methods proposed to solve the KdV equations, including direct integration, Lyapunov approach, Hirota’s dependent variable transformation, the inverse scattering transform, and the Bäcklund transformation [7–9]. A direct algebraic approach has also been developed by Parkes and Duffy [10] in which the solutions to the particular equation are represented by an automated tanh-function method [10]. Recently, Wazwaz considered the abundant solitons solutions, compactons and solitary patterns solutions, some new solitons, and periodic solutions of the fifth-order KdV equation [11, 12]. The adiabatic parameter dynamics of 1-soliton solution of the generalized fifth-order nonlinear KdV equation is obtained by virtue of the soliton perturbation theory [13, 14]. The authors present a Mathematica package that deals with complicated algebraic system and outputs directly the required solutions for particular nonlinear equations [15–21].

Exact solutions to nonlinear evolution equations (NEEs) play an important role in nonlinear physical science, since the characteristics of these solutions may well simulate real-life physical phenomena [22–24]. The wave phenomena can be observed in fluid dynamics, plasma physics, elastic media, and so forth. The main task of this work is to show that our proposed methods, improved and methods, are very efficient in solving the Olver equation and the fifth-order KdV equation by using extended method and extended method [25–28].

This paper is organized as follows. An introduction in is presented in Section 1. In Section 2, an analysis of the extended method and extended method is formulated. In Section 3, the travelling wave solutions of the Olver and the fifth-order KdV equations are obtained. Finally, the paper ends with a conclusion in Section 4.

#### 2. An Analysis of the Methods

##### 2.1. The sech-tanh Method

We suppose that , where , and has the following formal travelling wave solution: where and are constants to be determined.

*Step 1. *Equating the highest-order nonlinear term and the highest-order linear partial derivative in the ordinary differential equations yields the value of .

*Step 2. *By setting the coefficients of for and to zero, we have the following set of over determined equations in the unknowns , and and for .

*Step 3. *By using Mathematica and Wu’s elimination methods, the algebraic equations in Step 2 can be solved.

##### 2.2. The Extended Method

The method developed and introduced an independent variable: that is introduced and leads to the change of the following derivatives:

The extended method admits the use of the finite expansion: where is a positive integer, in most cases, that will be determined. Expansion equation (4) reduces to the standard method for . The parameter is usually obtained by balancing the linear terms of highest-order in the resulting equation with the highest-order nonlinear terms. Substituting of (4) into the ODE results in an algebraic system of equations in powers of that will lead to the determination of the parameters and .

*Stability of Solution*. Hamiltonian system for the momentum is given by

The sufficient condition for discussing the stability of solution is , where is coefficient of time.

#### 3. Application of the Methods

##### 3.1. The Olver Equation

In this section, we will employ the proposed methods to solve Olver equation [23]: where the coefficients are real constants depending on surface tension. These coefficients are Here, represents a dimensionless surface tension coefficient, is the ratio of wave amplitude to undisturbed fluid depth, and is the square of the ratio of fluid depth to wave length.

###### 3.1.1. Using a Method

Equation (6) was equivalently obtained upon using the wave variable . Balancing with in (8) gives . method equation (1) admits the use of the finite expansion: and by substituting from (9) into (8) and setting the coefficients of for and to zero, we have the following set of overdetermined equations in the unknowns , , , , , and .

By solving the set of result equations by using Mathematica, we obtain the following solutions.

*Case I*. Consider
In this case, the generalized soliton solution can be written as
where , and .

Figure 1(a) shows the stability dark solitary wave solutions with (, , and ) in the interval and time in the interval . Figure 1(b) shows the stability contour of solitary wave solution with (, , and ) in the interval and time in the interval .

*Case II*. Consider
In this case, the generalized soliton solution can be written as

*Case III*. Consider
In this case, the generalized soliton solution can be written as

###### 3.1.2. Using the Extended Method

We have By substituting (16) into (8) and collecting the coefficient of , we obtain a system of algebraic equations for , , , , , and . Solving this system gives the following solution.

*Case I*. Consider
In this case, the generalized soliton solution can be written as
where , and .

Figure 2(a) shows the stability dark solitary wave solutions with (, , and ) in the interval and time in the interval . Figure 2(b) shows the stability contour of solitary wave solution with (, , and ) in the interval and time in the interval .

*Case II*. Consider
In this case, the generalized soliton solution can be written as

*Case III*. Consider

In this case, the generalized soliton solution can be written as

##### 3.2. Solving the Fifth-Order Korteweg-de Vries Equation

In this section we will employ the proposed methods to solve the fifth-order Korteweg-de Vries equation: Or equivalently is obtained upon using the wave variable , when the higher-order coefficients are given by

###### 3.2.1. Using a Method

Balancing with in (24) gives . method equation (1) admits the use of the finite expansion: and by substituting from (27) into (24) and setting the coefficients of for and to zero, we have the following set of overdetermined equations in the unknowns , , , , , and .

Solve the set of result equations by using Mathematica; we obtain the following solutions.

*Case I*. Consider
In this case, the generalized soliton solution can be written as
where and .

Figure 3(a) shows the stability bright solitary wave solutions with in the interval and time in the interval .

Figure 3(b) shows the stability contour of solitary wave solution with in the interval and time in the interval .

*Case II*. Consider
In this case, the generalized soliton solution can be written as

*Case III*. Consider
In this case, the generalized soliton solution can be written as

###### 3.2.2. Using the Extended Method

We have By substituting (33) into (23) and collecting the coefficient of , we obtain a system of algebraic equations for , , , , , and . Solving this system gives the following solution.

*Case I*. Consider
In this case, the generalized soliton solution can be written as
where and .

Figure 4(a) shows the stability bright solitary wave solutions with in the interval and time in the interval .

Figure 4(b) shows the stability contour of solitary wave solution with in the interval and time in the interval .

*Case II*. Consider

In this case, the generalized soliton solution can be written as

Figure 5(a) shows the stability bright solitary wave solutions with in the interval and time in the interval .

Figure 5(b) shows the stability contour of solitary wave solution with in the interval and time in the interval .

*Case III*. Consider

In this case, the generalized soliton solution can be written as

Figure 6(a) shows the stability bright solitary wave solutions with in the interval and time in the interval . Figure 6(b) shows the stability contour of solitary wave solution with in the interval and time in the interval .

#### 4. Conclusion

Contour plots produced by Mathematica are drawn shaded, in such a way that regions with higher values of , , and , , are drawn lighter. As with all Mathematica graphics commands, options allow you to control the appearance of the graph. Contours plot allows you to determine the number of contours to be drawn. The default is ten equally spaced curves. An analytic study was conducted on the Olver and fifth-order KdV equations. We formally derived travelling wave solutions for the Olver and fifth-order KdV equations. However, by using another distinct approach, we derived one traveling wave solutions for each Olver and fifth-order KdV equations. The structures of the obtained solutions are distinct and stable.

#### Conflict of Interests

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

#### Acknowledgment

This project was supported by the Deanship of Scientific Research, Taibah University, KSA, Project no. 4086/1434, year 2013.

#### References

- L. Jin, “Application of variational iteration method to the fifth-order KdV equation,”
*International Journal of Contemporary Mathematical Sciences*, vol. 3, no. 5, pp. 213–221, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Khater, M. M. Hassan, and R. S. Temsah, “Cnoidal wave solutions for a class of fifth-order KdV equations,”
*Mathematics and Computers in Simulation*, vol. 70, no. 4, pp. 221–226, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kawahara, “Oscillatory solitary waves in dispersive media,”
*Journal of the Physical Society of Japan*, vol. 33, pp. 260–264, 1972. View at Publisher · View at Google Scholar - M. A. Helal, “Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics,”
*Chaos, Solitons and Fractals*, vol. 13, no. 9, pp. 1917–1929, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. R. Long, “Solitary waves in one- and two fluids systems,”
*Tellus*, vol. 8, Article ID 460471, 1956. View at Google Scholar - D. J. Benney, “Long nonlinear waves in fluid flows,”
*Journal of Mathematical Physics*, vol. 45, article 5263, 1966. View at Google Scholar - R. S. Drazin and P. G. Johnson,
*Solutions: An Introduction*, Cambridge University Press, Cambridge, UK, 1989. - V. I. Karpman, “Lyapunov approach to the soliton stability in highly dispersive systems. II. KdV-type equations,”
*Physics Letters A*, vol. 215, no. 5-6, pp. 257–259, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Liu and C. Bai, “Exact solutions of some fifth-order nonlinear equations,”
*Applied Mathematics B*, vol. 15, no. 1, pp. 28–32, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. J. Parkes and B. R. Duffy, “An automated Tanh-function method for finding solitary wave solutions to non-linear evolution equations,”
*Computer Physics Communications*, vol. 98, pp. 288–300, 1996. View at Publisher · View at Google Scholar - A. M. Wazwaz, “Compactons and solitary patterns solutions to fifth-order KdV like equations,”
*Physica A*, vol. 371, pp. 273–279, 2006. View at Publisher · View at Google Scholar - A.-M. Wazwaz, “Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method,”
*Applied Mathematics and Computation*, vol. 182, no. 1, pp. 283–300, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Biswas and S. Konar, “Soliton perturbation theory for the compound KdV equation,”
*International Journal of Theoretical Physics*, vol. 46, no. 2, pp. 237–243, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Biswas and E. Zerrad, “Soliton perturbation theory for the generalized fifth-order nonlinear equation,”
*Contemporary Engineering Science*, vol. 1, no. 2, pp. 63–69, 2008. View at Google Scholar - M. A. Helal and A. R. Seadawy, “Variational method for the derivative nonlinear Schrödinger equation with computational applications,”
*Physica Scripta*, vol. 80, pp. 350–360, 2009. View at Google Scholar - M. A. Helal and A. R. Seadawy, “Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 62, no. 5, pp. 839–847, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. R. Seadawy, “New exact solutions for the KdV equation with higher order nonlinearity by using the variational method,”
*Computers & Mathematics with Applications*, vol. 62, no. 10, pp. 3741–3755, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. R. Seadawy, “Exact solutions of a two-dimensional nonlinear Schrödinger equation,”
*Applied Mathematics Letters*, vol. 25, no. 4, pp. 687–691, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. R. Seadawy, “Traveling wave solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method,”
*Applied Mathematical Sciences*, vol. 6, pp. 4081–4090, 2012. View at Google Scholar - M. A. Helal and A. R. Seadawy, “Benjamin-Feir instability in nonlinear dispersive waves,”
*Computers & Mathematics with Applications*, vol. 64, no. 11, pp. 3557–3568, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Kaya, “An explicit and numerical solutions of some fifth-order KdV equation by decomposition method,”
*Applied Mathematics and Computation*, vol. 144, no. 2-3, pp. 353–363, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. J. Olver, “Hamiltonian and non-Hamiltonian models for water waves,” in
*Trends and Applications of Pure Mathematics to Mechanics*, vol. 195 of*Lecture Notes in Physics*, pp. 273–290, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. A. Kudryashov, M. B. Soukharev, and M. V. Demina, “Elliptic traveling waves of the Olver equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 11, pp. 4104–4114, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. R. Seadawy and K. El-Rashidy, “Traveling wave solutions for some coupled nonlinear evolution equations,”
*Mathematical and Computer Modelling*, vol. 57, no. 5-6, pp. 1371–1379, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. A. Helal, A. R. Seadawy, and R. S. Ibrahim, “Variational principle for Zakharov-Shabat equations in two-dimensions,”
*Applied Mathematics and Computation*, vol. 219, no. 10, pp. 5635–5648, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. R. Seadawy and A. Sayed, “Travelling wave solution of two-dimensional nonlinear KdV-Burgers equation,”
*Applied Mathematical Sciences*, vol. 7, no. 68, pp. 3367–3377, 2013. View at Google Scholar · View at MathSciNet - M. A. Helal, A. R. Seadawy, and M. H. Zekry, “Stability analysis solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave equation,”
*Applied Mathematical Sciences*, vol. 7, no. 65-68, pp. 3355–3365, 2013. View at Google Scholar · View at MathSciNet - A. R. Seadawy, “Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma,”
*Computers & Mathematics with Applications*, vol. 67, no. 1, pp. 172–180, 2014. View at Publisher · View at Google Scholar · View at MathSciNet