On the numerical solution of the cubic Schrödinger equation in one space variable
The authors study some basic issues involving the time and spatial discretizations of the cubic nonlinear Schrödinger equation defined over the entire real line. Two types of spatial discretizations are considered: the \(L^ 2\)-Galerkin method with product approximation and the integrable finite difference scheme of \textit{M. J. Ablowitz} and \textit{J. F. Ladik} [Stud. Appl. Math. 55, 213-229 (1976; Zbl 0338.35002)]. By numerical studies using NAG library routines, it is shown that the Galerkin method suffers from an instability during long time integrations with features similar to those observed for the periodic problem. It is further demonstrated that the instability may be avoided (at the response of phase errors) using the integrable discretization of Ablowitz and Ladik. The authors conclude the paper by stressing that since integrable schemes are designed specifically for individual problems, more general numerical techniques remain as important as ever.
scientific article; zbMATH DE number 167070