Local Absorbing Boundary Conditions for a Finite Element Discretization of the Cubic Nonlinear Schrödinger Equation

We consider in this work the initial value problem for the one dimensional cubic nonlinear Schrödinger equation. In order to integrate it numerically, one option frequently used, is to impose local absorbing boundary conditions. A finite element discretization in space of the cubic nonlinear Schrödinger equation is considered along with the absorbing boundary conditions obtained for an analogous discretization of the linear equation. For the implementation of these boundary conditions, an adaptive strategy is proposed, so that the boundary conditions change at each time step, depending on the numerical solution that is arriving to the boundary at that moment. The numerical experiments are satisfactory, obtaining a good absorption at the boundary.