Research On Applications Of Lie Transform To Integer And Time Fractional Order Schr(?)dinger Equation

Nonlinear Schr dinger equation is widely used in the fields of quantummechanics, physics, fluid mechanics and mathematics. It is well-known thatthe methods for solving this equation are inverse scattering transform,Backlund transformation, Darboux changes, Similarity reduction and so on.With scholars’ continuous study and exploration, Lie symmetry analysismethod is used to solve problems of the Nonlinear Schr dinger Equationsincreasingly.In this dissertation, we primarily aim to derive the exact solution of thenonlinear integer and fractional order Schr dinger equations by Lie symmetryanalysis method.The main contents are given as follows:1)For the integer-order nonlinear Schr dinger equation,we get thecorresponding partial differential equations by the transformation of the wavepacket.With application of the Lie Symmetry,we give one-parametertransformation group and second order prolongation. We find the Liesymmetry of the original equation by the standard transformation. Exactsolutions of the Schr dinger equation are obtained finally.2)For the time fractional nonlinear Schr dinger equation, we find theequivalent equation of the original equation with application of the Liesingle-parameter transformation and invariance of the symmetry operators. Wethen establish a relationship between the original and equivalent equations byusing the Limited conversion. We derive exact solutions associated with theoriginal equation by solving the equivalent equation.