This is the Gross-Pitaevskii equation:
$$\boxed{\mathrm i\hbar\frac{\partial{\psi(r,t)}}{\partial{t}}=\left(-~\frac{\hbar^2}{2m}\Delta-\mu\right)\psi(r,t)+g\psi^{*}(r,t)\psi(r,t)\psi(r,t)}$$ Where $\mu$ is the chemical potential and $g$ is a term that accounts for atomic collisions.
I am need for a step wise solution of how to normalise this wavefunction. I request the following in the answer:
a) Solution of normalizing the function,I.e.,
$$\boxed{Normalization=\int_{-\inf}^{\inf}\mid{\psi^2}\mid dV=1}$$
Here Volume is that of a cube of dimensions: $L_{x},L_{y},L_{z}$.
b) The equation of the small perturbations of the complex wavefunction, and then make a real wave equation from this complex equation, which will contain derivatives higher order than the previous complex one.
Assumptions: Let $c =\sqrt{\frac{gN_{c}}{mV}}$ denote a parameter which has the dimension of velocity and let $\xi =\frac{\hbar}{mc}$ denote another parameter, which is small, and which has the dimension of length.
My attempt:
If a harmonic trap potential is considered, the single particle equation becomes:
$$\boxed{i\hbar\frac{\partial{\psi(r,t)}}{\partial{t}}=-\frac{\hbar^2}{2m}\Delta(\psi)+V(r)\psi+NU_{0}\mid{\psi^2}\mid\psi}\tag{1}$$
Where $r=(x,y,z)^{T}$ is the spatial coordinate vector, $U_{0}=\frac{4(\pi)(\hbar^2)a}{m}$ with a,the s-wave scattering; and $$V(r)=\frac{m}{2}(\omega_{x}^{2}x^2+\omega_{y}y^2+\omega_{z}z^2)$$is the harmonic trap potential.
Multiplying $(1)$ by $\frac{1}{m\omega_{x}^2 a_{0}^{\frac{1}{2}}}$ and normalising the function I got:
$$\boxed{i\frac{\partial{\psi(r,t)}}{\partial{t}}=-\frac{1}{2}\Delta(\psi(r,t))+V(r)\psi+\beta\mid{\psi^2}\mid\psi(r,t)}\tag{2}$$
Hence, $(2)$ is the local density, I.e., $\frac{N_{c}}{V}=\frac{N_{c}}{L_{x}L_{y}L_{z}}$
Now, I am aware of the dark solition (oscillating dark solutions-arxiv)solutions, am I to use that along with introducing perturbations? How do I write down the small perturbations in terms of a complex wavefunction and then take a real value of it?
I'm asking the above question because I am completely aware about the physical concepts behind this phenomenon but lack the mathematical ability to comprehend and solve them. Any help will be deeply valued and appreciated.