I am trying to use the split-step Fourier (SSF) method for solving partial differential equation $$\frac{\partial u(x,t)}{\partial t} =u^*f(x,t)+\hat{H}\, u(x,t) \tag{1}$$ The first term causes problems. As you can see, the equation cannot be written in the form $$\frac{\partial u(x,t)}{\partial t} =\hat{L}\, u(x,t) \tag{2}$$
I can rewrite eq.(1) as $$\frac{\partial u(x,t)}{\partial t} =\left[f(x,t)\frac{u^*(x,t)}{u(x,t)}+\hat{H}\right] u(x,t) \tag{3}$$
Can I still use the SSF method? Will the method remain stable as before? Maybe there are more non-trivial ways to solve this problem?