Travelling Wave Solutions of Kaup-Kupershmidt Equation Which Describes Pseudo Spherical Surfaces

In this paper I introduce the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudo-spherical surface for Kaup-Kupershmidt Equation with constant Gaussian curvature −1. I obtained new soliton solutions for Kaup-Kupershmidt Equation by using the modified sine-cosine method.


Introduction
Many partial differential equations which continue to be investigated due to their role in mathematics and physics exhibit interrelationships with the geometry of surfaces, or submanifolds, immersed in a three-dimensional space [1].In particular, it has been known for a while that there is a relationship between surfaces of a constant negative Gaussian curvature in Euclidean three-space, the Sine-Gordon Equation and Bäcklund transformations which are relevant to the given equation [2].Moreover, the original Bäcklund transformation for the Sine-Gordon Equation is also a simple geometric construction for pseudospherical surfaces [3]- [5].It is well known that nonlinear complex physical phenomena are related to nonlinear partial differential equations (NLPDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc.
As mathematical models of the phenomena, the investigation of exact solutions to the NLPDEs reveals to be very important for the understanding of these physical problems.Many mathematicians and physicists have well understood this importance when they decided to pay special attention to the development of sophisticated methods for constructing exact solutions to the NLPDEs.Thus, a number of powerful methods have been pre-sented.
Consider Kaup-Kupershmidt Equation, depending on u and its derivatives such that the one-forms satisfy the structure equations of a pss, i.e., I obtain that the Kaup-Kupershmidt Equation ( 1) describes pseudospherical surfaces, with associated one forms As a consequence, each solution of the DE provides a local metric on 2 M , whose Gaussian curvature is con- stant, equal to −1.Moreover, the above definition is equivalent to saying that DE for u is the integrability condition for the problem [19] [29]: where d denotes exterior differentiation, φ is a column vector and the 2 2

Exact Solution for Kaup-Kupershmidt Equation
With the rapid development of science and technology, the study kernel of modern science is changed from linear to nonlinear step by step.Many nonlinear science problems can simply and exactly be described by using the mathematical model of nonlinear equation.Up to now, many important physical nonlinear evolution equations are found, such as Sine-Gordon Equation, KdV Equations, Schrodinger Equation all possess solitary wave solutions.There exist many methods to seek for the solitary wave solutions, such as inverse scattering method, Hopf-Cole transformation, Miura transformations, Darboux transformation and Bäcklund transformation [7]- [10], but solving nonlinear equations is still an important task [27]- [30].In this paper, with the aid of Mathematica, a traveling wave solution for a class of Kaup-Kupershmidt Equation, In order to obtain the soliton solution of (1), I will use the modified sine-cosine to develop traveling wave solutions to this equation.The modified sine-cosine method admits the use of solutions [30] ( ) ( ) and where a is the soliton amplitude, µ is the width of the soliton, c is the soliton velocity and 0 b is constant to be determined later, the unknown index n will be determined during the course of derivation of the solution of Equation ( 8).From Equation ( 8), I obtain From Equation ( 9), I obtain With the aid of Mathematica or Maple, from ( 8) and ( 10), we can get Solving the above system by the aid of Wu elimination method [31], I obtain the three solutions Then the soliton solutions of the Kaup-Kupershmidt Equation is given by ( ) ( ) The double-kink solutions (19), (20), and ( 21) are characterized by the eigenvalue 1 µ = (see Figures 1-6).

Conclusions
The new types of exact traveling wave solution obtained in this paper for the Kaup-Kupershmidt Equation will