Extended trial equation method to generalized nonlinear partial differential equations

In this article, we give the extended trial equation method for solving nonlinear partial differential equations with higher order nonlinearity. By use of this method, the exact travel-ing wave solutions including soliton solution, singular soliton solutions, rational function solution and elliptic integral function solution to one-dimensional general improved KdV (GIKdV) equation and Rðm; nÞ equation are obtained. Also, a more general trial equation method is proposed. Constructing exact solutions to partial differential equations is an important problem in nonlinear science. Therefore, numerous methods of integrability have been presented in recent years. Some of these methods are Hirota method, tanh–coth method, the exponential function method, ðG 0 =GÞ-expansion method, ansatz method, mapping method, Weierst-rass elliptic function method, modified F-expansion method, extended Jacobian elliptic function expansion method [1–15]. Recently, Liu proposed trial equation method and its new versions to classify the traveling wave solutions to nonlinear physical problems and applied these methods to some nonlinear partial differential equations [16–20]. Furthermore, some authors studied Liu's trial equation method in Refs. [21–24]. In Ref. [25], we defined a new trial equation method to obtain 1-soliton, singular soliton, elliptic integral function and Jacobi elliptic function solutions and the others to the generalized nonlinear evolution equations. In this paper, we first give the extended trial equation method for nonlinear differential equations. In Section 3, as applications , we obtain some exact solutions to two nonlinear problems with higher nonlinear terms. First of all, we consider the one-dimensional general improved KdV equation [26,27] u t þ au n u x þ bu xxx À cu xxt ¼ 0; ð1Þ where n is a positive integer, and a; b; c are positive constants. Then, we solve a generalized version of the regularized long wave (RLW) equation, called as the Rðm; nÞ equation, is given by Biswas [28] and Biswas and Kara [29] u t þ au x þ bðu m Þ x þ cðu n Þ xxt ¼ 0: ð2Þ Here, x and t are the independent variables, u is the dependent variable, m and n are the power-law nonlinearity parameters. When we choose n ¼ 1, then Rðm; nÞ equation reduces to RLW equation with power law nonlinearity [29]. It is noted that for