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

Question

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

1. Split-Step Fourier Transform
2. Inverse Scattering Transform
3. Interpolation Approaches

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.

Answer 1

You won't find universal agreement among experts on a single best way to solve this equation. On the other hand, since it is a PDE in one spatial dimension there are many approaches that will give good accuracy in a reasonable amount of time.

A very simple place to start is a pseudospectral discretization in space with Runge-Kutta integration in time; you can learn about such methods in a lot of good books, or you can find a quick introduction with examples and code here (disclaimer: that links to something that I wrote).

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.

I won't attempt to comment on approaches 2-3 from your list, as I am not an expert on either.