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

Firstly， the nonlinear Schrödinger equation with refractive index term was transformed into an infinite-dimensional 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.