Conservative finite-difference scheme for the problem of laser pulse propagation in a medium with third-order dispersion

We develop the conservative finite-difference scheme for linear and nonlinear 1D Schrödinger equation taking into account the third-order dispersion. To illustrate the efficiency of developed finite-difference scheme we compare the results of computer modeling for linear equation with well-known analytical solution of this problem. Various statements of the problem are considered to show the essential influence of formulated boundary conditions on stability of the finite-difference scheme. To increase the efficiency of computer simulation we propose adaptive artificial boundary conditions for considered problem.