饱和非线性光学介质中带折射率项的薛定谔方程的数值模拟

Numerical simulation of the Schrödinger equation with refractive index term in saturated nonlinear optical medium

首先将带折射率项的非线性薛定谔方程转化成无限维哈密尔顿系统,证明了方程的质量和能量守恒特性;再利用傅里叶拟谱方法和平均向量场方法离散方程,对离散格式中非积分项采用Boole离散进行线积分近似,得到了离散方程的能量守恒数值格式,同时给出了方程的辛格式;然后以不同振幅的入射双曲正割型光脉冲为初值条件,模拟了保能量格式和辛格式在不同参数条件下光孤子的演化过程.最后分析了不同初始光脉冲和参数对光孤子传输的影响和保方程质量和能量守恒特性.

Firstly,the nonlinear Schrödinger equation with refractive index term was transformed into an infinitedimensional Hamiltonian system,and the mass and energy conservation properties of the equation were proved.Secondly,the Fourier pseudo-spectral method and the average vector field method were performed to discretize the Schrödinger equation,the Boole discrete line integral approximation was used for the non-integral term in the discrete format,and the discrete energy conservation numerical format of the equation was obtained,and the symplectic scheme of the equation was proposed.Thirdly,the different hyperbolic secant pulses were used as the initial value conditions,the evolution of optical soliton under the different parameters of the energy preserving scheme and the symplectic scheme was simulated.Finally,the effects of different initial optical pulses and parameters on optical soliton transmission were analyzed,and the mass and energy preservation property of the equation were also investigated.