The paper gives a method for combining the auxiliary equation with hyperbolic function by function transition. And then the method is applied to construct the new solitary wave solutions and the trigonomical function wave solutions to the (2+1)-dimensional Hybrid-Lattice and discrete mKdV equation with the help of the symbolic computation system Mathematica.

Based on the auxiliary equation method and the trial function method, a method for combining function transformation with auxiliary equation is proposed. And the method is applied to construct new exact solitary wave solutions and triangle function solutions of Volterra and KdV difference-differential equations with the help of symbolic computation system Mathematica. Our method can also be applied to other nonlinear difference-differential equations.

A function transformation method for constructing exact solutions of the variable coefficient nonlinear evolution equations is proposed. The method together with the the second kind of elliptic equation and the symbolic computation system Mathematica is used to construct the new exact Jacobi elliptic function solutions, the degenerated soliton-like solutions and trigonometric function solutions of the composed KdV equation with forced variable coefficients.

Based on tanh-function method, homogeneous balance method and auxiliary equation, a new auxiliary equation was introduced in the paper, meanwhile, a new invariance solution was obtained, and a new exact solitary wave solution for Benjamin-Bona-Mahoney(BBM) equation and modified BBM equation was constructed using the symbolic calculation system of Mathematica. The method introduced in the paper has general significance in searching for exact solutions to the nonlinear developing equation.

In this paper, several new exact solitary wave solutions to the BBM and the m BBM equations are constructed explicitly by combined use of a hyperbolic functio n assumption and a new auxiliary ordinary differential equation. This method also can be used to find new solitary wave solutions to other nonlinear evolution equ ations.