Stability of the higher-order splitting methods for the nonlinear Schrödinger equation with an arbitrary dispersion operator

Abstract

The numerical solution of the generalized nonlinear Schrödinger equation by simple splitting methods can be disturbed by so-called spurious instabilities. We analyze these numerical instabilities for an arbitrary splitting method and apply our results to several well-known higher-order splittings. We find that the spurious instabilities can be suppressed to a large extent. However, they never disappear completely if one keeps the integration step above a certain limit and applies what is considered to be a more accurate higher-order method. The latter can be used to make calculations more accurate with the same numerically stable step, but not to make calculations faster with a much larger step.

Keywords

• nonlinear optics
• nonlinear fibers
• nonlinear Schrödinger equation
• generalized nonlinear Schrödinger equation (GNLSE)
• modulation instability (MI)
• four-wave mixing