A Study On Some Asymptotic For Solving Nonlinear Differential Equations

Based on studying various asymptotic methods for solving linear and nonlinear equations, we apply them to solve nonlinear differential equations. Most traditional study for solving nonlinear problems is limited to the weakly nonlinear problems, but not too much to the strongly nonlinear problems. According to the recent developments with different techniques in this field, with different equations, we analyze their advantages and disadvantages, further, we give many sinificant solutions to some equations with suitable asymptotic method, and we can get their new solutions not obtained before. So we may give some united and valid methods for different style nonlinear differential equations. Otherwise, according to analyzing the character of obtained solutions, we can discuss its physical sense and give relative error analysis.By means of theory of asymptotic techniques, including Euler-Lagrange equation, variational principle, Ritz method and so on, and use for reference Prof. He JH's methods, applying similar methods, combining actual examples, meanwhile, consulting other methods, we propose significant solutions of extensive nonlinear partial differential equations.Take advantage of Prof. He JH's variational approximate approach, variational iteration method, semi-inverse method, exp-function method, firstly we solve some important differential equations, such as Duffing equation, Riccati equation, Lambert equation, Kdv equation and so on. Then we can obtain some new solutions for some important nonlinear differential equations discussed in this paper, such as nonlinear Schr?dinger equation with variable coefficient, the coupled Klein-Gordon-Schr?dinger equations, (2+1)-dimensio- nal breaking soliton equation, and other some Schr?dinger equations.