Study of travelling wave solutions for some special-type nonlinear evolution equations

The tanh-function expansion method has been improved and used to construct travelling wave solutions of the form $U={\sum }_{j=0}^n{a}_j{\tanh }^j\xi$ for some special-type nonlinear evolution equations, which have a variety of physical applications. The positive integer n can be determined by balancing the highest order linear term with the nonlinear term in the evolution equations. We improve the tanh-function expansion method with n = 0 by introducing a new transform $U=-W^{\prime }\left(\xi \right)/W^2$. A nonlinear wave equation with source terms, and mKdV-type equations, are considered in order to show the effectiveness of the improved scheme. We also propose the tanh-function expansion method of implicit function form, and apply it to a Harry Dym-type equation as an example. (paper)