Abstract: The explicit analytical solutions of nonlinear partial differential equations, in particular, the travelling wave solutions, contain rich information about the equations, and they are very important for describing the development of various phenomena. In the paper, many types of new explicit travelling wave solutions are presented for the KdV equation. First, many new explicit analytical solutions of the Riccati equation are given by using the trial function method and Matlab software. Second, many types of new explicit analytical solutions of the sine-Gordon equation are obtained by using the extended tanh-function method and new solutions of Riccati equation. Finally, as a new application, a great many new travelling wave solutions of the KdV equation are provided by using the sine-cosine function method, new solutions of the sine-Gordon equation and its simplified transformation forms. The obtained results extend and complement some relevant existing works. In particular, these methods and obtained new results can be applied to find explicit new travelling wave solutions of many nonlinear partial differential equations.

Key words: Riccati equation, one dimensional sine-Gordon equation, KdV equation; , extended Tanh-function method, sine-cosine function method, travelling wave solution