Finite difference methods for an AKNS eigenproblem Original Research Article

We consider the numerical solution of the AKNS eigenproblem associated with the nonlinear Schrödinger equation. Four finite difference methods are considered: two standard schemes (forward and central differences), a discretization introduced by Ablowitz and Ladik (1976), and a modified version of the latter scheme. By comparing these methods both numerically and theoretically we show that the modified Ablowitz-Ladik scheme has several desirable features. This includes the property that with a given number of gridpoints it approximates much larger sections of the spectrum than its rivals.