Research On Lie Group Reduction Method For Fractional Nonlinear Schr (?) Dinger Equation

In recent years, the research about fractional order differential equations becomes a new hot spot. The fractional order nonlinear Schr?dinger equation is an important research object. To find the solitary wave solutions, the group invariant solutions and the power series solutions of the fractional order nonlinear Schr?dinger equation play a very important role in both theory and application for researching and analyzing the chaotic phenomena in quantum mechanics.In this paper, we apply the Lie group reduction method to research three kinds of the fractional order nonlinear Schr?dinger equation at the first time.Firstly, the Lie group reduction method is applied to a kind of time fractional order nonlinear Schr?dinger equation. Some new single parameter solutions and Lie symmetric reduction equations are obtained. By solving these equations, some group invariant solutions, elliptic function solutions, power series solutions and the solitary wave solutions are obtained for the time fractional order nonlinear Schr?dinger equation.Secondly, the Lie group reduction method is applied to a kind of space-fractional order nonlinear Schr?dinger equation. Single parameter solutions as well as Lie symmetric reduced equations are obtained. By solving these equations, some group invariant solutions and solitary wave solutions are obtained for the space-fractional order nonlinear Schr?dinger equation.Finally, the Lie group reduction method is applied to a kind of time and space fractional order nonlinear Schr?dinger equation(the space-time fractional order nonlinear Schr?dinger equation for short). We obtained The form of solutions depending on a single parameter and Lie symmetric reduced equations. By solving reduced equations,we looked for some group invariant solutions about the space-time fractional order nonlinear Schr?dinger equation.