  This paper describes an updated exponential Fourier based split-step method
that can be applied to a greater class of partial differential equations than
previous methods would allow. These equations arise in physics and engineering,
a notable example being the generalized derivative non-linear Schr\"odinger
equation that arises in non-linear optics with self-steepening terms. These
differential equations feature terms that were previously inaccessible to model
accurately with low computational resources. The new method maintains a 3rd
order error even with these additional terms and models the equation in all
three spatial dimensions and time. The class of non-linear differential
equations that this method applies to is shown. The method is fully derived and
implementation of the method in the split-step architecture is shown. This
paper lays the mathematical ground work for an upcoming paper employing this
method in white-light generation simulations in bulk material.
