I am new to the field of computational physics and have a couple of questions regarding solving the non-linear Schrödinger equation using Operator splitting.

1. If the hamiltonian is of the form $H=\frac{\partial^{2}}{\partial x^{2}}+\gamma\psi|\psi|^{2}$ then the standard procedure I understand is to exponentiate $-i(\gamma \psi|\psi|^{2})\Delta t/\hbar$ and operate it on the initial value of $\psi$, then take a fourier transform to convert it to momentum space and operate it with exponential of $-ip^{2}\Delta t/\hbar$ and convert the resultant back to position space. We repeat this for each time interval $\Delta t$. Instead, why can't we do everything in the momentum space to begin with? Why this back and forth shifting from position to momentum space?

2. Suppose now I have an additional term of $\frac{\partial^{2}}{\partial x^{2}}|\psi|^{2}$ in the Hamiltonian, then how do I accomodate this term in the scheme of split operator method?