Construction of non-travelling wave solutions for the generalized variable-coefficient Gardner equation

With the aid of two first-order nonlinear ordinary differential equations, the new generalized algebraic method is used to study the generalized variable-coefficient Gardner equation. The main idea of the method is to take full advantage of solutions to the first-order ordinary differential equations. More new exact non-travelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions, for the Gardner equation are obtained. The method used here can be also extended to many other nonlinear partial differential equations.