Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm

The expression of Gaussian envelope soliton in Schrodinger equations are given and proved in this paper. According to the characteristics of the Gauss envelope soliton, further proposed that the interaction between Gaussian envelope solitons exists in Schrodinger equation. The symplectic algorithm for solving Schrodinger equation is proposed after analysis characteristics of Schrodinger equation. First, the Schrodinger equation is transformed into the standard Hamiltonian canonical equation by separating the real and imaginary parts of wave function. Secondly, the symplectic algorithm is implemented by using the Euler center difference method for the canonical equation. The conserved quantity of symplectic algorithm is given, and the stability of symplectic algorithm is proved. The numerical simulation experiment was carried out on Schrodinger equation in Gauss envelope soliton motion and multi solitons interaction. The experimental results show that the proposed method is correct and the symplectic algorithm is effective.