Mathematical Problems in Engineering

Volume 2010, Article ID 524567, 7 pages

http://dx.doi.org/10.1155/2010/524567

## Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation

^{1}Universidad de Caldas, Calle 65 No. 26-10, P.O. Box: Apartado Aéreo 275, Manizales, Caldas, Colombia^{2}Department of Mathematics, Universidad Nacional de Colombia, Carrera 27 No. 64-60, P.O. Box: Apartado Aéreo 275, Manizales, Colombia^{3}Department of Mathematics, Universidad Nacional de Colombia, Calle 45, Carrera 30. P.O. Box: Apartado Aéreo 52465, Bogota, Colombia^{4}Department of Mathematics, Universidad Nacional de Colombia, P.O. Box: Apartado Aéreo 127, Manizales, Colombia

Received 24 December 2009; Revised 5 May 2010; Accepted 4 August 2010

Academic Editor: Katica R. (Stevanovic) Hedrih

Copyright © 2010 Alvaro H. Salas 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 Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7).

#### 1. Introduction

Many direct and computational methods have been used to handle nonlinear partial differential equations (NLPDE’s). Some methods used in a satisfactory way to obtain exact solutions to NLPDE’s are inverse scattering method [1], Hirota bilinear method [2, 3], Backlund transformations [4], Painlevé analysis [5], Lie groups [6], the tanh method [7], the generalized tanh method [8, 9], the extended tanh method [10–12], the improved tanh-coth method [13, 14], the Exp-function method [15–17], the projective Riccati equation method [18], the generalized projective Riccati equations method [19–24], the extended hyperbolic function method [25], variational iteration method [26, 27], He’s polynomials [28], homotopy perturbation method [29], and many other methods [30]. However, there is not a unified method that could be used to handle all NLPDE’s; in this sense, the implementation of new methods or variants of the some well-known methods is relevant. The principal objective of this paper consists in obtaining exact traveling wave solutions which include periodic and soliton solutions to a particular case of the general seventh-order which is a generalization of the seventh-order Sawada-Kotera-Ito (SKI-KdV7) equation, by using a variant of the exp-function method. The general seventh-order KdV (KdV7) equation [31] reads The was introduced initially by Pomeau et al. [32] for discussing the structural stability of equation under a singular perturbation. Some particular cases of (1.1) are seventh-order Lax equation [1, 6] (): seventh-order Sawada-Kotera-Ito equation [1, 8–10] (): seventh-order Kaup-Kupershmidt equation [1, 7] ():

#### 2. Generalization of the Lax KdV7 and the Sawada-Kotera-Ito KdV7

Observe that (1.2) and (1.3) satisfy the relation For this reason we will study equation We seek solutions to (2.2) in the Cole-Hopf form where is some constant to be determined later and Substituting (2.3) into (2.2), we obtain a polynomial equation in the variable . Equating the coefficients of the different powers of to zero, we obtain following algebraic system: Eliminating and from system (2.5) gives It is easy to verify that (1.2) and (1.3) are particular cases of general equation (1.1) subject to (2.1) and (2.6). This motivates us to define the generalized Lax-Sawada-Kotera-Ito seventh-order equation (LSKI KdV7) as follows:

#### 3. Solutions to Generalized LSKI KdV7

In order to look for solutions to (2.7), we will use the exp ansatz where , , and are some constants. Substituting (3.1) into (2.7) gives an algebraic system. Solving it, we obtain From (2.4), (3.1), and (3.2), we obtain following solution to (2.7) subject: In particular, if , equation (3.3) gives Replacing with gives the following periodic solutions: On the other hand, if , equation (3.3) gives Replacing with gives the following periodic solutions:

#### 4. Solutions to Sawada-Kotera-Ito KdV7 Equation

From (3.3)–(3.7) with and we obtain the following analytic solutions to equation (1.3):

#### 5. Conclusions

We exhibited an equation that generalizes both seventh-order Lax equation and seventh-order Sawada-Kotera-Ito equation. At the same time, we obtained exact solutions to these equations with the aid of a Cole-Hopf ansatz. These same ideas are suitable for the seventh-order Kaup-Kupershmidt equation. We think that some of the solutions in this work are new in the open literature. We may apply other methods to find exact solutions to a variety of nonlinear PDE’s. See [3, 12–52].

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