Exact Solutions to KdV 6 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


Introduction
The sixth-order nonlinear wave equation 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 w 0 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.

The Method
In the search of the traveling wave solutions to nonlinear partial differential equation of the form To find solutions to 2.3 , we suppose that v ξ can be expressed as where H f ξ , g ξ is a rational function in the new variables f ξ , g ξ which are solutions to the system being ρ / 0 an arbitrary constant to be determinate and R f ξ a rational function in the variable f ξ .Taking where φ ξ / 0, and N / 0, then 2.5 reduces to

2.10
The following are solutions to 2.9 :

2.11
Therefore, solutions to 2.10 are given by

Exact Solutions to the Integrable KdV6 System
Using the traveling wave transformation u x, t v ξ , w x, t w ξ , ξ x λt ξ 0 , 3.1 the system 1.6 reduces to w ξ 4v ξ w ξ 2v ξ w ξ 0.

3.3
Integrating 3.2 with respect to ξ and setting the constant of integration to zero we obtain w ξ 4v ξ w ξ 2v ξ w ξ 0.

3.4
Using the idea of the projective Riccati equations method 19-22 , we seek solutions to 3.4 as follows: where f ξ and g ξ satisfy the system given by 2.10 with ρ 1 .Substituting 3.5 into 3.4 , after balancing we have that M 2, 3.6 and Nis an arbitrary positive constant.By simplicity we take N M. Therefore, 3.5 reduce to

3.7
Substituting this last two equations into 3.4 , using 2.10 we obtain an algebraic system in the unknowns a 0 , a

3.10
A soliton solution is given by where a 0 , α, γ are arbitrary constants and ξ x λt ξ 0 .

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 x , from 3.14 the following system is derived: s t s xxx 12s 2 s x − w x 0, w xxx 8s 2 w x 8s x z 0, z x − sw x 0.

3.17
We believe that traveling wave solutions to these systems can be obtained using the method used here.By reasons of space, we omit them.

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 .
xx 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 w x v 2x , then the new MKdV6 equation is equivalent to new system