The Dynamic Property Of The Nonlinear Schr(?)dinger Equation

The nonlinear Schr?dinger equation as a ubiquitous nonlinear evolution equation plays an important role in nonlinear physics, and it arises as an asymptotic limit of a slowly varying dispersive wave envelope in a nonlinear medium and as such has significant applications, e. g. nonlinear optics, plasma physics, laser fusion and condensed physics.Nonlinear Schr?dinger equation is an infinite-dimensional Hamiltonian system. But the Hamiltonian system canonical equation in symplectic transformation under the form is invariable. The time-evolution of Hamiltonian system is the evolution of symplectic transformation. Therefore, Ruth and Feng Kang presented the symplectic algorithm for solving the Hamiltonian system. Symplectic algorithm is the difference method that preserves the symplectic structure, and it is a better method in the calculation of long-time many-step and preserving the structure of system.In this thesis we investigate the dynamic properties of one-dimensional cubic nonlinear Schr?dinger equation with different nonlinear parameters in the perturbation initial condition. In the numerical calculations, we first discretize the spatial derivative of the nonlinear Schr?dinger equation: substitute the space difference approximation for spatial derivative, transform the nonlinear Schr?dinger equation to Hamiltonian canonical equations. We find the discrete Hamiltonian, and then transform the nonlinear Schr?dinger equation to Hamiltonian canonical equations. Hamiltonian system has a symplectic structure. So we solve the equation with implicit symplectic scheme of 2-step. The numerical result demonstration, the nonlinear parameter increases from 0.01 to 0.85, the phase trajectories of in the phase space by the elliptic orbit would be similar to the homoclinic orbit (HMO). The phase trajectories turns to the accurate cyclical circulation by the precise cycle circulating motion, and the phase trajectories gradually increases in the horizontal axis value. With the nonlinear parameter increasing, drifting velocity of the solution pattern becomes faster at the same time of evolution.In section IV, we solve the one-dimension Schr?dinger equation and the nonlinear Schr?dinger equation with absorbing boundary conditions. We solve the Schr?dinger equation by the initial wave function of a Gaussian wave packet. In the nonlinear Schr?dinger equation, we have studied the evolution of one soliton and two solitons, respectively. All the computational results demonstrate that we can simulate the evolution of wave function with absorbing boundary conditions effectively.