I am studying literature on the nonlinear Schroedinger equation. More generally, I am wondering how to tackle an equation of motion such as


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!


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?

  • $\begingroup$ 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 $\endgroup$ – Abhijit Mar 25 '17 at 6:24
  • $\begingroup$ I added a comment to the question. Thanks anyways. $\endgroup$ – Hamurabi Mar 25 '17 at 21:17
  • 1
    $\begingroup$ @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? $\endgroup$ – udrv Apr 4 '17 at 3:15

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