The direct symmetry method and its application in variable coefficients Schrödinger equation

By using a direct transformation, the variable coefficients Schrödinger equation is changed to two nonlinear partial differential equations with variable coefficients. We consider the two differential equations and derive different forms of symmetry by the direct symmetry method. At last some symmetry reductions and explicit solutions of the variable coefficients Schrödinger equation are obtained in terms of similarity solutions of the two differential equations.