 
Summary: ABSORBING BOUNDARY CONDITIONS FOR
GENERAL NONLINEAR SCHR¨ODINGER
EQUATIONS
XAVIER ANTOINE, CHRISTOPHE BESSE, AND PAULINE KLEIN
Abstract. This paper addresses the construction of different families of absorbing boundary
conditions for the one and twodimensional Schr¨odinger equation with a general variable nonlinear
potential. Various semidiscrete time schemes are built for the associated initial boundary value
problems. Finally, some numerical simulations give a comparison of the various absorbing boundary
conditions and associated schemes to analyze their accuracy and efficiency.
Key words. absorbing boundary conditions, pseudodifferential operators, nonlinear Schr¨odinger
equation with potential, stable semidiscrete schemes, fixed point algorithm, relaxation scheme.
AMS subject classifications. 35Q41, 35Q55, 47G30, 26A33, 65M12, 65M60,
1. Introduction. Schr¨odinger equations have many applications in physics (see
e.g. [22, 25, 18, 15, 21, 8]). They can include repulsive or attractive variable potentials
and nonlinear versions are also often met in practice. All these situations represent
challenging applications where numerical methods are of upmost importance for pre
dicting the system behavior. These problems are physically set in an unbounded
domain which requires the introduction of a fictitious boundary for an efficient nu
merical simulation. On this boundary must be set an admissible boundary condition
such that the restriction of the exact solution to the initial problem coincides with
