Suppose I want to solve a non-linear Schrödinger equation using imaginary time propagation to get the ground state solution. I choose $t = - i \tau$, and then solve the equation using the split-step Crank-Nicholson method. All the excited states will decay faster than the ground state and eventually, only the ground state remains.
Suppose, I would like to check whether the obtained solution is stable or not. To that end, I would add a small perturbation to the obtained solutions and evolve it. If the solution is stable, it should converge back to its unperturbed state, if not, it will diverge.
My question is whether it is possible to do the second part using imaginary time propagation? Can somebody tell me where to use and where not to use the real time and imaginary time propagation, respectively?