Mathematical Problems in Engineering

Volume 2010 (2010), Article ID 797084, 10 pages

http://dx.doi.org/10.1155/2010/797084

## Exact Solutions to KdV6 Equation by Using a New Approach of the Projective Riccati Equation Method

^{1}Departamento de Matemáticas, Universidad Nacional de Colombia, Calle 45, Carrera 30, P.O. Box: Apartado Aéreo: 52465, Bogotá, Colombia^{2}Departamento de Matemáticas, Universidad Nacional de Colombia, Carrera 27 no. 64–60, P.O. Box: Apartado Aéreo 127, Manizales, Colombia^{3}Departamento de Matemáticas, Universidad de Caldas, Calle 65 no. 26–10, Caldas, P.O. Box: Apartado Aéreo 275, Manizales, Colombia

Received 21 January 2010; Revised 23 May 2010; Accepted 8 July 2010

Academic Editor: David Chelidze

Copyright © 2010 Cesar A. Gómez S 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

We study a new integrable KdV6 equation from the point of view of its exact solutions by using an improved computational method. A new approach to the projective Riccati equations method is implemented and used to construct traveling wave solutions for a new integrable system, which is equivalent to KdV6 equation. Periodic and soliton solutions are formally derived. Finally, some conclusions are given.

#### 1. Introduction

The sixth-order nonlinear wave equation has been recently derived by Karasu-Kalkanl1 et al. [1] as a new integrable particular case of the general sixth-order wave equation where, , , , , , , are arbitrary parameters, and , is a differentiable function. By means of the change of variable equation (1.1) converts to the Korteweg-de Vries equation with a source satisfying a third-order ordinary differential equation (KdV6) which is regarded as a nonholonomic deformation of the KdV equation [2]. Setting the system (1.4) reduces to [2, 3] A first study on the integrability of (1.6) has been done by Kupershmidt [2]. However, only at the end of the last year, Yao and Zeng [4] have derived the integrability of (1.6). More exactly, they showed that (1.6) is equivalent to the Rosochatius deformations of the KdV equation with self-consistent sources (RD-KdVESCS). This is a remarkable fact because the soliton equations with self-consistent sources (SESCS) have important physical applications. For instance, the KdV equation with self-consistent sources (KdVESCS) describes the interaction of long and short capillary-gravity waves [5]. On the other hand, when the system (1.6) reduces to potential KdV equation, so that solutions of the potential KdV equation are solutions to (1.1). Furthermore, solving (1.6) we can obtain new solutions to (1.1). In the soliton theory, several computational methods have been implemented to handle nonlinear evolution equations. Among them are the tanh method [6], generalized tanh method [7, 8], the extended tanh method [9–11], the improved tanh-coth method [12, 13], the Exp-function method [14–16], the projective Riccati equations method [17], the generalized projective Riccati equations method [18–23], the extended hyperbolic function method [24], variational iteration method [25–27], He's polynomials [28], homotopy perturbation method [29–31], and many other methods [32–35], which have been used in a satisfactory way to obtain exact solutions to NLPDEs. Exact solutions to system (1.6) and (1.1) have been obtained using several methods [3, 4, 36–38]. In this paper, we obtain exact solutions to system (1.6). However, our idea is based on a new version of the projective Riccati method which can be considered as a generalized method, from which all other methods can be derived. This paper is organized as follows. In Section 2 we briefly review the new improved projective Riccati equations method. In Section 3 we give the mathematical framework to search exact for solutions to the system (1.6). In Section 4, we mention a new sixth-order KdV system from which novel solutions to (1.6) can be derived. Finally, some conclusions are given.

#### 2. The Method

In the search of the traveling wave solutions to nonlinear partial differential equation of the form
the first step consists in use the wave transformation
where is a constant. With (2.2), equation (2.1) converts to an ordinary differential equation (ODE) for the function
To find solutions to (2.3), we suppose that can be expressed as
where is a *rational function* in the new variables , which are solutions to the system
being an arbitrary constant to be determinate and a rational function in the variable . Taking
where , and , then (2.5) reduces to
From (2.7) we obtain

Let and , with . In this case, (2.8) reduces to and (2.5) are transformed into The following are solutions to (2.9): Therefore, solutions to (2.10) are given by In all cases .

#### 3. Exact Solutions to the Integrable KdV6 System

Using the traveling wave transformation the system (1.6) reduces to Integrating (3.2) with respect to and setting the constant of integration to zero we obtain Using the idea of the projective Riccati equations method [19–22], we seek solutions to (3.4) as follows: where and satisfy the system given by (2.10) (with ). Substituting (3.5) into (3.4), after balancing we have that and is an arbitrary positive constant. By simplicity we take . Therefore, (3.5) reduce to Substituting this last two equations into (3.4), using (2.10) we obtain an algebraic system in the unknowns , , , , , , , , , , , , , and . Solving it and using (3.7), (2.12), and (3.1) we have the following set of new nontrivial solutions to KdV6 system (1.6). In all cases, A combined formal soliton solution is: where are arbitrary constants, and

Furthermore, A soliton solution is given by where are arbitrary constants and

##### 3.1. A New System

A direct calculation shows that (1.1) reduces to On the other hand, it is easy to see that (3.12) can be written as Using the analogy between KdV equation and MKdV equation and motivated by the structure of (3.13), the authors in [38] have introduced the so-called MKdV6 equation and they showed that where is the Miura transformation between KdV6 equation (1.1) and MKdV6 equation (3.14). Therefore, solving (3.14), according to (3.15), solutions to (1.1) are obtained. Setting , then the new MKdV6 equation is equivalent to new system In equivalent form, with , , from (3.14) the following system is derived: We believe that traveling wave solutions to these systems can be obtained using the method used here. By reasons of space, we omit them.

#### 4. Conclusions

In this paper we have derived two new soliton solutions to KdV6 system (1.2) by using a new approach of the improved projective Riccati equations method. The results show that the method is reliable and can be used to handle other NLPDE's. Other methods such as tanh, tanh-coth, and exp-function methods can be derived from the new version of the projective Riccati equation method. Moreover, new methods can be obtained using the exposed ideas in the present paper. Other methods related to the problem of solving nonlinear PDEs exactly may be found in [39, 40].

#### References

- A. Karasu-Kalkanlı, A. Karasu, A. Sakovich, S. Sakovich, and R. Turhan, “A new integrable generalization of the Korteweg-de Vries equation,”
*Journal of Mathematical Physics*, vol. 49, no. 7, Article ID 073516, 10 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. A. Kupershmidt, “KdV6: an integrable system,”
*Physics Letters. A*, vol. 372, no. 15, pp. 2634–2639, 2008. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. H. Salas, “Exact solutions for a new integrable system (KdV6),”
*Journal of Mathematical Sciences. Advances and Applications*, vol. 1, no. 2, pp. 401–413, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Yao and Y. Zeng, “The bi-Hamiltonian structure and new solutions of KdV6 equation,”
*Letters in Mathematical Physics*, vol. 86, no. 2-3, pp. 193–208, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. K. Menlikov, “Integration of the Korteweg-de Vries equation with a source,”
*Inverse Problems*, vol. 6, no. 2, pp. 233–246, 1990. View at Google Scholar · View at MathSciNet - E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,”
*Zeitschrift für Naturforschung A*, vol. 57, no. 8, pp. 692–700, 2002. View at Google Scholar · View at Scopus - C. A. Gómez, “Exact solutions for a new fifth-order integrable system,”
*Revista Colombiana de Matemáticas*, vol. 40, no. 2, pp. 119–125, 2006. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. H. Salas, “Exact solutions for a reaction diffusion equation by using the generalized tanh method,”
*Scientia et Technica*, vol. 13, no. 35, pp. 409–410, 2007. View at Google Scholar - A.-M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,”
*Applied Mathematics and Computation*, vol. 184, no. 2, pp. 1002–1014, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez, “Special forms of the fifth-order KdV equation with new periodic and soliton solutions,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1066–1077, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez, “New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6),”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 357–360, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez and A. H. Salas, “The generalized tanh-coth method to special types of the fifth-order KdV equation,”
*Applied Mathematics and Computation*, vol. 203, no. 2, pp. 873–880, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Salas and C. A. Gómez, “Computing exact solutions for some fifth KdV equations with forcing term,”
*Applied Mathematics and Computation*, vol. 204, no. 1, pp. 257–260, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Zhang, “Exp-function method exactly solving the KdV equation with forcing term,”
*Applied Mathematics and Computation*, vol. 197, no. 1, pp. 128–134, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He and L.-N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,”
*Physics Letters. A*, vol. 372, no. 7, pp. 1044–1047, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,”
*International Journal of Modern Physics B*, vol. 22, no. 21, pp. 3487–3578, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - R. Conte and M. Musette, “Link between solitary waves and projective Riccati equations,”
*Journal of Physics. A. Mathematical and General*, vol. 25, no. 21, pp. 5609–5623, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Yan, “The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations,”
*MMRC, AMSS, Academis Sinica*, vol. 22, pp. 275–284, 2003. View at Google Scholar - E. Yomba, “The general projective Riccati equations method and exact solutions for a class of nonlinear partial differential equations,”
*Chinese Journal of Physics*, vol. 43, no. 6, pp. 991–1003, 2005. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. Salas, “Exact solutions for the generalized shallow water wave equation by the general projective Riccati equations method,”
*Boletín de Matemáticas. Nueva Serie*, vol. 13, no. 1, pp. 50–56, 2006. View at Google Scholar · View at MathSciNet - C. A. Gómez and A. Salas, “New exact solutions for the combined sinh-cosh-Gordon equation,”
*Lecturas Matemáticas*, vol. 27, pp. 87–93, 2006. View at Google Scholar · View at MathSciNet - C. A. Gómez, “New exact solutions of the Mikhailov—Novikov—Wang system,”
*International Journal of Computer, Mathematical Sciences and Applications*, vol. 1, pp. 137–143, 2007. View at Google Scholar - C. A. Gómez, “New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 241–250, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Shang, Y. Huang, and W. Yuan, “New exact traveling wave solutions for the Klein-Gordon-Zakharov equations,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1441–1450, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. A. Gómez and A. H. Salas, “The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation,”
*International Journal of Nonlinear Sciences and Numerical Simulation*. In press. - M. A. Noor and S.T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 9, no. 2, pp. 141–157, 2008. View at Google Scholar · View at Scopus - J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,”
*Chaos, Solitons and Fractals*, vol. 29, no. 1, pp. 108–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Travelling wave solutions of seventh-order generalized KdV equations using He's polynomials,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 2, pp. 223–229, 2009. View at Google Scholar - S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving partial differential equations,”
*Zeitschrift für Naturforschung A*, vol. 64, no. 3-4, pp. 157–170, 2009. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Mirgolbabaei, D. D. Ganji, and H. Taherian, “Soliton solution of the Kadomtse-Petviashvili equation by homotopy perturbation method,”
*World Journal of Modelling and Simulation*, vol. 5, no. 1, pp. 38–44, 2009. View at Google Scholar · View at Scopus - H. Mirgolbabaei and D. D. Ganji, “Application of homotopy perturbation method to solve combined Korteweg de Vries-Modified Korteweg de Vries equation,”
*Journal of Applied Sciences*, vol. 9, no. 19, pp. 3587–3592, 2009. View at Publisher · View at Google Scholar · View at Scopus - H. Mirgolbabai, A. Barari, and G. Domiri, “Analytical solition of forced-convective boundary-layer flow over a flat plate,”
*Archive of Civil and Mechanical Engineering*. In press. - S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Some relatively new techniques for nonlinear problems,”
*Mathematical Problems in Engineering*, vol. 2008, Article ID 234849, 25 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Travelling wave solutions of seventh-order generalized KdV equations by variational iteration method using Adomian's polynomials,”
*International Journal of Modern Physics B*, vol. 23, no. 15, pp. 3265–3277, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. H. He, “Some asymptotics methods for strongly nonlinear equation,”
*International Journal of Modern Physics*, vol. 20, no. 10, pp. 1144–1199, 2006. View at Google Scholar - C. A. Gómez and A. H. Salas, “The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system (KdV6),”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 957–962, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A.-M. Wazwaz, “The integrable KdV6 equations: multiple soliton solutions and multiple singular soliton solutions,”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 963–972, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhang, X.-N. Cai, and H.-X. Xu, “A note on “The integrable KdV6 equation: multiple soliton solutions and multiple singular soliton solutions”,”
*Applied Mathematics and Computation*, vol. 214, no. 1, pp. 1–3, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. H. Salas, “Symbolic computation of solutions for a forced Burgers equation,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 18–26, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Salas, “Symbolic computation of exact solutions to KdV equation,”
*Canadian Applied Mathematics Quarterly*, vol. 16, no. 4, 2008. View at Google Scholar