Exact Travelling Wave Solutions of Nonlinear Wave equations Using Tanh-function Method

Abstract

This paper is concerned with the exact traveling wave solutions of nonlinear wave equations. Using the tanh function method, we can obtain the accurate expression of the solutions. Further, according to the images of the solutions, we can get the variation depends on the velocity v . Introduction It is well known that nonlinear phenomena are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, capillary-gravity waves, and chemical physics. Most of these phenomena are described by nonlinear partial differential equations. Analytical solutions of this problems are usually not available, especially when the nonlinear terms are involved. Therefore, finding its travelling solutions is of practical importance. The methods of looking for exact traveling wave solutions of nonlinear evolution equations, has been tremendous development in recent decades, such as inverse scattering method [2], HI Rota’s bilinear technique [5], the Painlve expansion method [13]. In the early nineties of last century, Huibin and Kelin [7] proposed a new method. The main idea of this method is taking hyperbolic tangent function of the power series as possible traveling wave solutions of the nonlinear evolution equations. Then they substituted the power series directly to KdV equation, and obtained the coefficients of the power series. However this method involved very complicated algebra computation. In order to reduce the complex algebra computation, Malfiety [9-11] proposed the tanh-function method. Since all the derivatives of hyperbolic tangent can be expressed by the hyperbolic tangent in itself, this simple translation makes the method can be applied to more nonlinear evolution equations. Fan et al. [3] proposed the extended hyperbolic tangent method, which replace the tanh-function by the solutions of Riccati equation. In [1, 4, 14, 16, and 17], using the tanh function method, they got the exact form of traveling wave solutions of various types of evolution equations. In recent years, the G ′ /G function method [15], the auxiliary function method [6] is based on tanh-function method. This shows that the hyperbolic tangent function method is very effective and direct method when looking for the exact traveling wave solutions of nonlinear evolution equations. The Tanh-function Method Let’s consider the nonlinear partial differential equations ( ) 0 t =   , , , , , xxx xx x u u u u u N (2.1) Where ( ) t x u , is the real function on 2 R ? At first, we assume the traveling wave solutions of (2.1) are the form of (x, t) U( ) U(c(x )), u t ω u = = − (2.2) With the velocity v , and the constant c. Submitted (2.2) into (2.1), we can get the ODEs About ω (U, U , U , U ,...) 0. N ′ ′′ ′′′ = (2.3) Second, we assume the possibly traveling wave solutions can be written International Conference on Information Sciences, Machinery, Materials and Energy (ICISMME 2015) © 2015. The authors Published by Atlantis Press 1325 0 (x, t) U( ) H(Y) , K i i i u a Y ω