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Suppose I want to solve Nonlinear Schrodinger equation using imaginary time propagation to get the ground state solution. I choose $t = - i t$, and then solve the equation using split step Crank Nicholson method. All the excited states will decay faster than the ground state and leave finally the ground state of the system. Suppose, I would like to check whether the obtain solution is stable or not. In this regard, I would add a small perturbation to the obtained solutions and evolve it. If the solution is stable, it should converge else it diverges. My question is that is it possible to do the second part using imaginary time propagation? or I have to do it using the real time propagation. Can somebody tell me where to use the real time propagation and imaginary time propagation and where not to use?

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2 Answers 2

You pretty much answered it for yourself. If you're really concerned with getting a stable solution with imaginary time for the ground state, you shouldn't be perturbing it a little. Instead, you might want to start out with a completely different initial wavefunction than the one you used to begin with.

Basically, imaginary time is used to find the ground state, as you are doing. Real time propagation is for studying dynamics. You can test your initial state solution by propagating in real time and checking that it doesn't change. You can try the little perturbation test, and you should get back to the right solution (although I don't know how much this really tells you).

Note that I've only done this in the context of Bose-Einstein condensation, so there might be other factors to consider if you're studying something different, although I can't think of any. If you have a "good" potential (no numerical singularities) and a sensible nonlinear term (you're not pushing the limits of your solver with huge or hugely small numbers) then you shouldn't have any problems.

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Agreed, if you are using the imaginary time evolution method (ITEM), this will give you a ground state for a given frequency. As far as stability goes, the ITEM only converges to linearly stable solutions, so if you get a ground state from the ITEM it is linearly stable. See, for example, the accelerated time evolution paper of Jianke Yang's

http://www.emba.uvm.edu/~jxyang/AITEM.pdf

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