The homogeneous balance method, which is simple and straightforward, is extended to seek for Backlund transformation, exact bell shape soliton solutions and similarity reduction of the Boussinesq equation.

A class of nonlinear evolution equations is reduced by the three kinds of similarity reductions in this paper first,and then the ordinary differential equations and the similarity solutions are obtained.

Four families of similarity reductions are obtained for the nonlinear evolution equation arising in the Fermi-Pasta-Ulam problem via using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou.

In this paper, the similarity reductions of the Fisher-type equation with variable coefficient is given.

Using the direct method introduced by Clarkson and Kruskal,we obtain thenew similarity reductions of the combined KdV equation.

By using the direct symmetry method, we get the symmetry of the equation, five types of similarity reductions and many new exact solutions of the equation, including rational solutions, hyperbolic function solutions, Jacobi elliptic function solutions and triangular periodic solutions.

The white noise functional solutions of Wick-type stochastic Fisher equation are given by using similarity reduction method and the Hermite transform in Kondratiev distribution space (S)_(-1).