# What is the preferred method for evolving the Nonlinear Schrödinger Equation?

I am interested in evolving the (cubic) self-focusing nonlinear Schrödinger equation,

$$i\frac{\partial \psi}{\partial t}+\frac{1}{2}\frac{\partial^2 \psi}{\partial x^2}+\left|\psi\right|^2\psi=0$$

from an initial condition $\psi(x,t_0)=f(x)$, where I am going to restrict my attention to a finite region $x\in [a,b]$. Since I'm not too familiar with the "standard" methods for evolving this equation, the proper choice of algorithm/method isn't obvious to me. For example, I don't know if using a "spectral" method would be better than doing straight integration/evolution. The methods I have read about are

I would like a bit of guidance on both what is that "norm" for handling this evolution, as well as what is most efficient. I would also very much appreciate references to sources which directly or indirectly answer my question.

Please forgive me if this is a vague or overly simple question.

The main issue you'll find with that approach is that the second-derivative term leads to stiffness and forces you to use a time step of order $(\Delta x)^2$. There are many ways of avoiding this restriction; the simplest is the split-step method you mention, but it introduces its own significant errors. For other, more accurate ways to deal with the stiffness, you could look at using an integrating factor, exponential methods, or IMEX additive Runge-Kutta methods, all of which are discussed and tested for a variety of PDEs in this excellent paper.