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An improved algorithm is devised for using the (G /G)-expansion method to solve nonlinear differential-difference equations. With the aid of symbolic computation, we choose a discrete nonlinear Schrödinger equation to illustrate the validity and advantages of the improved algorithm. As a result, hyperbolic function solutions, trigonometric function solutions and rational solutions with parameters are obtained, from which some special solutions including the known solitary wave solution are derived by setting the parameters as appropriate values. It is shown that the improved algorithm is effective and can be used for many other nonlinear differential-difference equations in mathematical physics.

In this paper, a generalized (G'/G)-expansion method, combined with suitable transformations, is used to construct exact solutions of the nonlinear Schrödinger equation with variable coefficients. As a result, hyperbolic function solutions, trigonometric function solutions, and rational solutions with parameters are obtained. When the parameters are taken as special values, some solutions including the known kink-type solitary wave solution and the singular travelling wave solution are derived from these obtained solutions. It is shown that the generalized (G'/G)-expansion method is direct, effective, and can be used for many other nonlinear evolution equations with variable coefficients in mathematical physics.

A modified (G /G)-expansion method is proposed to construct exact solutions of nonlinear evolution equations. To illustrate the validity and advantages of the method, the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is considered and more general travelling wave solutions are obtained. Some of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions, and rational solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Communicated by J. CashBased on a Riccati equation and one of its new generalized solitary solutions constructed by the Exp-function method, new analytic solutions with free parameters and arbitrary functions of a (2+1)-dimensional variable-coefficient BroerKaup system are obtained. These free parameters and arbitrary functions reveal that the (2+1)-dimensional variablecoefficient Broer-Kaup system has rich spatial structures. As an illustrative example, two new spatial structures are shown by setting the arbitrary functions as different Jacobi elliptic functions. Compared with tanh-function method and its extensions, the method proposed in this paper is more powerful and it can be applied to other nonlinear evolution equations.