Adaptive artificial boundary conditions for Schrödinger equation taking into account the first order dispersion of laser pulse and diffraction of laser beam

Abstract

We consider 3D+1D nonlinear Schrödinger equation, which describes non-stationary propagation of laser beam. Optical radiation evolution in time is described by taking into account the first order dispersion. To enhance efficiency and accuracy of the computer simulation, we develop adaptive artificial boundary conditions those use information about the nonlinear Schrödinger equation solution near artificial boundaries. Consequently, our artificial boundary condition uses a local wave number that varies both in time and in corresponding spatial coordinates. As rule, the constant wave number was used in such kind of boundary conditions early. To construct the conservative finite-difference scheme for nonlinear Schrödinger equation with artificial boundary conditions we propose two-step iterative process, because widely used split-step method does not possess conservatism of finite-difference scheme with respect to Hamiltonian of the system. We make a short comparison of mentioned methods for the problem with zero-value boundary conditions.