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The integrating factor technique is widely used to solve numerically (inparticular) the Schr\"odinger equation in the context of spectral methods.Here, we present an improvement of this method exploiting the freedom providedby the gauge condition of the potential. Optimal gauge conditions are derivedconsidering the equation and the temporal numerical resolution with an adaptiveembedded scheme of arbitrary order. We illustrate this approach with thenonlinear Schr\"odinger (NLS) and with the Schr\"odinger-Newton (SN) equations.We show that this optimization increases significantly the overallcomputational speed, sometimes by a factor five or more. This gain is crucialfor long time simulations.

Optimized integrating factor technique for Schr\"odinger-like equations