On The Exact Solutions For A Type Of Nonlinear Schr?dinger Equations

This thesis studies a type of nonlinear Schrodinger equation with a harmonic poten-tial,i?t+?xx-?2/4x2?+k|?|?=0,where(?)represents imaginary unit,?=(?(t,x)is a complex function in(t,x)?R+×R,?,k ?R are all real parameters.The above equation is also known as the Gross-Pitaevskii(GP)equation which describes the famous Bose-Einstein Condensation(BEC).Because of the importance of Bose-Einstein Condensation in quantum mechanics,many scholars have studied this equation,especially its exact solutions.In view of the diversity and complexity of these problems,the improvement of solving solutions method produces more new exact solutions,and different exact solutions correspond to different physical phenomena.Therefore,this thesis will further improve the existing solution method to obtain the new exact solutions of the above equation.Firstly,we apply some new transformations to solve the exact solutions of this equa-tion by the low-order Sub-ODE method.In this thesis,the nonlinear Schrodinge equation with a harmonic potential is transformed into the classical nonlinear Schrodinge equa-tion by a new class of traveling wave transformation,then the problem of exact solutions is changed into the solutions of ordinary differential equation by the method of undeter-mined function.Then,we use the low-order Sub-ODE method to solve the exact solutions of this ordinary differential equation,and bring the solutions back in order to obtain the exact solutions of the nonlinear Schrodinger equation with harmonic potential.These exact solutions are new exact solutions because we added a new traveling wave transfor-mations.Finally,the obtained exact solutions are simulated and the simulation's results are analyzed in detail.Secondly,we apply the traveling wave transformations to solve the exact solutions of this equation by using the hyperbolic function method.In this thesis,the nonlinear Schrodinger equation with a harmonic potential is converted into a class of classical non-linear Schrodinger equation by using the traveling wave transformation.Then,the ex-act solutions of the GP equation is transformed into the solutions of ordinary differential equation by the method of undetermined function.The solutions of the ordinary differ-ential equation are discussed by introducing hyperbolic trigonometric function sinh and cosh into two groups,and then the exact solutions is brought back in sequence to get the exact solutions of the original equation.Since the new transformation is applied to the hyperbolic function method,the exact solutions are the new exact solutions.Finally,the exact solutions obtained by this method are numerically simulated and the advantages and disadvantages of the two methods are compared.