Study Of Energy Preserving Method And Symplectic Method Of Time-dependent Schrodinger Equation

In1984, Feng kang and his study team proposed the symplectic method of the Hamilton system, which has a long accurately computing capability and approximately preserves the energy-preserving property of the system. Recently, R.I.McLachlan and G.R.W.Quispel et al proposed the average vector field method of the Hamiltonian system based on the idea of the discrete gradient method. In this paper, we mainly investigate the theory of the average vector field method and its application in the high order nonlinear Schrddinger equation and apply the symplectic method to solve the two dimensional nonlinear Schrodinger equation,which describe the spatial soliton evolution in the three level, gaseous atomic EIT media,In chapter1, we first introduce the discrete gradent method of the Hamiltonian system and some properties of the energy conserving differential dynamical system. Based on the basis,the discrete gradi-ent method of the ordinary differential equation and the partial differential equation are introduced. At last, we investigate the accuracy of the discrete gradient method by the B series theory and the energy conservation properties of the classical numerical methods,such as the middle scheme and the Runge-Kutta method, of the Hamilton system.In chapter2, we propose to apply a new method known as the discrete gradient method to solve the high order nonlinear Schrodinger equation. First, the high order nonlinear Schrddinger equation is discretizated by the discrete gradient method. The discrete gradient scheme of the high order nonlinear Schrodinger equation is obtained. The soliton behaviors of the high order nonlinear Schrodinger equation are simulated by the discrete gradient scheme and the corresponding symplectic scheme with the different saturated nonlinear effects and the different amplitudes. Numerical results show that the discrete gradient scheme can well simulate the solitons behaviors of the high order nonlinear Schrodinger equation and prererve the energy conservation of the Hamiltonian system better than the symplectic scheme.In chapter3, Optical solitons in gaseous atomic media display many striking features under elec-tromagnetically induced transparency (EFT), Study of theoretical model, which describes these features of optical solitons, has important meaning in optical informational process and propagation. Two dimen-sional saturated nonlinear Schrodinger equation, which describes the spatial soliton evolution in the three level, gaseous atomic EIT media, is transformed into the Hamilton system with the symplectic structure. The Hamilton system is discretizated by the symplectic method. The corresponding symplectic scheme is obtained. Evolution behaviors of two and four spatial solitons with the same amplitude in three level, gaseous atomic EIT media are simulated by the symplectic scheme. Numerical results further show that the phase difference and the direction of the entering gauss beams have obvious effect to the interactions of multi-solitons The entering gausss beam can form the stable optical solitons in gaseous atomic media.