Abstract
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.
Highlights
The research and application of solitons have gone deep into various fields, such as mathematics, fluid dynamics, nonlinear electromagnetism, optical fiber communication, solid state physics and so on
Physicists believe that as long as the energy of the wave is limited and is distributed in a limited space or time range, even if the wave changes during the propagation process, such as the high order optical soliton in the optical fiber, they are called soliton
According to the formula (40), the form energy of the symplectic algorithm of (34) is the square of the modulus of wave function ψ (x,t ). It is in conformity with the physical meaning of the Schrödinger equation which is that the cumulative probability is a constant
Summary
The research and application of solitons have gone deep into various fields, such as mathematics, fluid dynamics, nonlinear electromagnetism, optical fiber communication, solid state physics and so on. Lai Lianyou and Xu Weijian: Motion and Interaction of Envelope Solitons in Schrödinger Equation Simulated by Symplectic Algorithm equation. Because of maintaining the symplectic structure, the symplectic algorithm makes the numerical calculation of each step be a symplectic transformation, eliminate the artificial dissipation introduced non symplectic algorithm in the calculation process. It has the long time tracking ability and high stability for the evolutionary computation [10,11,12,13,14,15]. The researching objects of this paper is the Gauss envelope soliton in Schrödinger equations, their motion and interaction process, using the method of combining the analytical method and symplectic algorithm.
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