Consider Schrodinger equation with harmonic potential: $$i\partial u_t + \Delta u - |x|^2u +g(u)=0, u(x,0)=u_0$$ where $u:\mathbb R \times \mathbb R^2 \to \mathbb C$, $g(u)= uG'(|u|^2)$ for some positive real function $G\in C^3(\mathbb R_+)$ satisfying $G(0)=G'(0)=G''(0)=0.$
Author claims that above equation models Bose-Einstein condensates with attractive interparticle interactions under a magnetic trap.
Question: (1) If we take $g(u)=(K\ast |u|^2)u$ (typically appears in Hartree equation), then still above equation models Bose-Einstein condensates with attractive interparticle interactions under a magnetic trap. ($K$ is some function like $|x|^{-\gamma} \ (\gamma>0)$) (2) Any known applications of the above equation with $g(u)= (K\ast |u|^2)u$ in physics or so?
Side Thoughts: (1) If we take $G'(|u|^2 (x))= K \ast |u|^2(x),$ then $G(|u|^2(x))= \int K \ast |u|^2(x) dx.$ Roughly speaking $G$ satisfies the conditions $G(0)=G'(0)=G''(0)=0$ (without knowing explicitly $G$!) (2) I do not know models Bose-Einstein condensates with attractive interparticle interactions under a magnetic trap (I'm a pure mathematician).
Any suggestions and comments would be helpful to me.