Differential equation with nonlinear nonlocal interaction I am studying literature on the nonlinear Schroedinger equation. More generally,
I am wondering how to tackle an equation of motion such as
$\bigl(i\partial_t-\Delta\bigr)\psi(\vec{x},t)+V(\vec{x},t)=0$
where the potential is of the form  
$V(\vec{x},t)=\int d\vec{x}_1...d\vec{x}_n~ \psi(\vec{x}-\vec{x}_1,t)...\psi(\vec{x}-\vec{x}_n,t)$,
that means the potential is nonlocal and nonlinear in the field $\psi$.
Any hints towards literature, also numerical strategies would be really welcome!
Thanks!
Comment: 
The question refers in particular to the issue of how to deal with the integral in the interaction potential. Is there e.g. a nice way to discretize this object?
 A: Usually, a spatial nonlocality arises when there is a long-range interaction in a system. (Basically, an interaction, that decreases non-exponentally, is considered as a long-range interaction, for example, the interaction of electric or magnetic dipoles). The long-range interaction in a system results in a nonlocal response. This means that a field at point $x$ "feels" fields in other (remote) points. Mathematically, nonlocality is expressed, usually, by the convolution integral (an integral in the 3rd term of the equation below), so that the nonlocal nonlinear Schroedinger (NLS) equation has the form (in 1D):
$$ 
i\psi_{t} + \beta \psi_{xx} + g\, \psi \int_{-\infty}^{\infty} R(x-x') |\psi(x',t)|^2\, dx' = 0,
$$
where $\beta$ and $g$ are constant parameters, and $R(x)$ is the response function of a medium. Notice that when $R(x)$ is the $\delta$-function, the (self-)interaction becomes local ($g\, |\psi|^2 \psi$), and the model is reduced to the standard NLS equation. 
An example of nonlocal media is a Bose-Einstein condensate (BEC) made of atoms with large dipole moments (a dipolar BEC).
In optics, the nonlocal NLS equation describes also the propagation of light in media with stimulated Raman scattering, and in liquid crystals, see e.g. book by Y. Kivshar, G. Agrawal, Optical solitons, Ch.3 and 14 (2003). 
The convolution integral can easily be calculated numerically using the FFT method, see e.g. W. Press et al, Numerical Recipes
in Fortran 77, Ch.12 and 13 (1997).
