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Sir, I have been studying the Gross-Pitaevskii equation for weakly interacting Bose gas and I want to find out the Green's function for the equation: $$i\hbar\frac{\partial}{\partial t}\psi(r,t)=\Big[-\frac{\hbar^2}{2m}\nabla^2+\frac{1}{2}m\omega_0^2r^2+g\big|\psi(r,t)\big|^2\Big]\psi(r,t)$$

where, $V_{ext}=\frac{1}{2}m\omega_0^2r^2$ i.e. the gas is harmonically trapped and for simplicity it has been considered that $\omega_x=\omega_y=\omega_z=\omega_0$, $g$ is the coupling constant.

Further, in the absence of the term $V_{ext}$ and interaction term $g\big|\psi(r,t)\big|^2$ the above equation takes a simple form such as $$i\hbar\frac{\partial}{\partial t}\psi(r,t)=-\frac{\hbar^2}{2m}\nabla^2\psi(r,t)$$ which is nothing but Schrodinger equation and the derivation of the green's function of this equation has been shown in page no. 857 of "Methods of theoretical Physics, volume 1" by Morse and Feshbach.But, I can not find out the Green's function for the Gross-Pitavskii equation following the same approach.

Sir, would you kindly suggest me any reference or technique by which I can find out the Green's function of the Gross-Pitaevskii equation in presence of the interaction term.

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    $\begingroup$ Non linear equation do not have a Green function. $\endgroup$
    – mike stone
    Commented Sep 11, 2021 at 12:10

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You don't need a Green's function. Just assume a static system, and solve the ODE's!

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