Optimized integrating factor technique for Schr{ö}dinger-like equations

The integrating factor technique is widely used to solve numerically (in particular) the Schr{\"o}dinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge condition of the potential. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We illustrate this approach with the nonlinear Schr{\"o}dinger (NLS) and with the Schr{\"o}dinger-Newton (SN) equations. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. This gain is crucial for long time simulations.