What's the origin of the vortex's ansatz $\phi\big(\vec{x}\big)=f\big(r\big)e^{-in\theta}$ in the de Vega and Schaposnik paper?
In their article Classical vortex solution of the Abelian Higgs model, de Vega and Schaposnik state that the Maxwell-Higgs model has classic solutions of the vortex type given by ansatz $$\phi\big(\vec{x}\big)=f\big(r\big)e^{-in\theta}. \tag{A}$$ However, they did not give any clue as to how we can see that this ansatz actually leads to the vortex-like solution. Of course, they reference the articles of Nielsen-Olesen, and Ginzburg-Landau, but even so, it is still not obvious to me how this ansatz can emerge as a vortex-like solution.
An other question, which is connected with winding number $n$, is: How and why this winding number arise? How we should be to interpret the winding number $n$?
