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I want to solve the following differential equation numerically:

\begin{equation} i\partial_{t}\psi(r,t)=\left[-\frac{\Delta}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r,t)\right]\psi(r,t) \end{equation}

The equation is of Schrödinger type, the difference being that it is made non-linear by the interaction term $g$. $\psi$ is a complex function in 2 spatial dimension $r=(x,y)$ and one temporal dimension. The potential $V_d(r)$ that I am interested in has the simple form $V_d(r,t)=g_V \delta(r-Vt)$. This equation models a Bose-Einstein condensate of atoms at $T=0$ in 2 dimensions, in the presence of a point-like impurity moving with velocity $V$.

The expected result is that when one plots the probability density $\left|\psi(r,t)\right|^{2}$ one would see waves like the ones created by a duck moving on the surface of a lake, as in the picture below.

modulus square of the wavefunction

While I do not expect an answer in the form of a full solution, I would like some advice on how to proceed implementing this interesting problem in Mathematica. I intend to post regular edits based on my progress.

Andrei
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  • I'd say the first step would be to read about NDSolve. – b.gates.you.know.what May 26 '13 at 10:35
  • First step is to define an appropriate function. Simply you can not define your function in terms of just r & t. But still it depends on what you want to do. If you just want the plot in the picture, there t seems to be constant, then you can define a function in terms of x and y. please post complete equations not just the differential equation. then we can help more – Raymond Ghaffarian Shirazi May 26 '13 at 13:33
  • The linked question shows how to do things for a two-state Hamiltonian. For your spatial problem, you can use the method of lines – Jens May 26 '13 at 23:18

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