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korea_mania
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I am trying to solve for the density variation due to a single vortex. The order parameter looks like. [tex]\Psi_0(\vec{r},t)=\sqrt{n}f(\eta)e^{i(s\phi-\mu t)}[/tex] where f satisfy the ODE [tex]\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[/tex] 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.
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.
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