Exponential time integration of solitary waves of cubic Schrödinger equation

The aim of the present paper is to study the suitability of using exponential methods for the time integration of cubic Schrödinger equation till long times. We center on second-order methods, for which we prove a higher order of accuracy on the main invariants when integrating solitary waves. Some geometric implicit exponential methods are considered as well as some explicit suitably projected ones. The comparison in terms of efficiency is performed.