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?

• From the look of it I guess that V(x,t) would give you a term of the form $\psi(x,t)^{n}$. I have not come across a n order nonlinearity. However, there exists enormous literature on the Gross Pitaevskii nonlinear equation, a general information about which can be found on the following wikipedia page : en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation – Abhijit Mar 25 '17 at 6:24
• I added a comment to the question. Thanks anyways. – Hamurabi Mar 25 '17 at 21:17
• @Hamurabi Are you sure the form of $V(x,t)$ is correct? It looks like $$V(x, t) = \Big[ \int{d{\vec y}\;\psi({\vec x} - {\vec y}, t)}\Big]^n$$but then if the integral extends over the entire space a change of variable shows that it's simply independent of ${\vec x}$. Is this what you have in mind? – udrv Apr 4 '17 at 3:15