Energy of a single vortex in BEC

[korea_mania]

I am trying to solve for the density variation due to a single vortex. The order parameter looks like. $$\Psi_0(\vec{r},t)=\sqrt{n}f(\eta)e^{i(s\phi-\mu t)}$$ where f satisfy the ODE $$\frac{1}{\eta}\frac{d}{d\eta}\left(\eta\frac{df}{d\eta}\right)+\left(1-\frac{s^2}{\eta^2}\right)-f^3=0$$ I realized that this is just the Bessel's differential equation except that there is an additional term $-f^3$. Since $f\to 0$ as $\eta\to 0$, I expect that the solution looks like Bessel function at least for small $\eta$.

Therefore, I try to solve the ODE by simply using scipy.integrate.odeint in python and plot the graph. It turns out that the graph do looks similar to the Bessel function at the beginning. The problem is, for a little bit larger $\eta$ the function go below zero and the function does not even tends to one at large $\eta$. How did Pitaevskii solve this equation numerically and got the graph that is printed on every textbook?

Moderator's note: post edited to fix the LaTeX. Please enclose the LaTeX with ## ## to get inlined equations.

 

[eljose79]

Thank you for sharing your progress and concerns with solving the density variation due to a single vortex. It seems like you are on the right track by using the Bessel's differential equation as a starting point. However, as you have observed, the additional term $-f^3$ may affect the behavior of the solution.

I would suggest trying to solve the ODE numerically using a different method, such as the Runge-Kutta method. This method is more flexible and can handle non-linear terms like $-f^3$. You can also try adjusting the initial conditions and parameters to see if that affects the behavior of the solution.

Additionally, it is important to consider the physical interpretation of the solution. Is it physically meaningful for the density to go below zero? It may be helpful to consult with other experts in the field or refer to other literature to see how they have solved this problem.

I hope this helps and good luck with your research. Keep us updated on your progress.