What are the limitations of Gross-Pitaevskij equation?

Question

I'm a mathematician and I'm studying as an aside something about superfluids. If I understood correctly in some kind of superfluids as Bose-Einstein Condensate a wave in the fluid must satisfy the Gross-Pitaevskij equation, i.e. $$\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+g\psi^{*}\psi-\mu_{0}\right)\psi=i\hbar\frac{\partial\psi}{\partial t}.$$

1. Is that right?

2. I've given to understand that this works only on BEC and at Low temperature, what are the limitations of this model?

3. Are there other more general formulas for superfluidity in general or is this one the standard to use?

Answer 1

1.) Not quite. The Gross-Pitaevskii equation describes the behavior of a particular wavefunction (i.e. a mathematical object encoding measurable information about an object), not waves in a BEC. In particular, it describes the ground state (i.e. lowest energy configuration) of an ensemble of identical bosons. This, in turn, approximates the state of a BEC at absolute zero. Any finite-temperature BEC (and indeed, any that we can actually create) exists as a superposition between the ground state described by the Gross-Pitaevskii equation and many other excited states, which can probably be obtained through the Hartree-Fock approximation. Anything resembling a wave (i.e. a traveling excitation in BEC) would exist as an excited state of the BEC, so they are not describable using the Gross-Pitaevskii equation. I should probably also add that this equation has little to do with superfluidity.

2.) Since the limits of this equation are determined mainly by the limits of the Hartree-Fock approximation and pseudopotential model used to derive it, I refer you to these sources:

https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method

https://en.wikipedia.org/wiki/Pseudopotential

3.) Not sure, as I am not a condensed matter physicist. In any case, the equation arises from a relatively standard approximation in quantum mechanics, so I'd be surprised if there was anything more straightforward. It's essentially just the original Schrodinger equation with an added interaction term. Again, though, this equation has little to do with superfluidity at face value.