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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 $n$?

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  • $\begingroup$ @CosmasZachos , excuse me, in fact, I'm not confusing $r$ with $\rho$. I just used one in place of the other. For me, $r$ remains the radius on the polar plane. $\endgroup$ – lucenalex May 13 at 16:06
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    $\begingroup$ OK, then, repeat exactly the polar plane representation of the equations of motion and flux calculation of Nielsen and Olesen, and confirm their results: a circularly periodic solution is the definition of a vortex, and angular single-valuedness is needed for consistency, so integer widning number n. $\endgroup$ – Cosmas Zachos May 13 at 16:09
  • $\begingroup$ What I imagine is that there must be a primary motion equation for the vortex static whose solution should lead to (A). However, I do not know such a theory. As authors eventually wish a vortex-related description, it is natural for them to choose (hence why they call ansatz) such a previously known solution. But, I turn back the question: what is the (theory) origin? $\endgroup$ – lucenalex May 13 at 16:16
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    $\begingroup$ An Ansatz is a wishful guess which is then confirmed, if lucky. $\endgroup$ – Cosmas Zachos May 13 at 16:21
  • $\begingroup$ So, according to the steps suggested by you, in your the previous comment, is it immediate that ansatz (A) emerges from Nielsen and Olesen's equations? Okay then. I'll try to do what you say. $\endgroup$ – lucenalex May 13 at 16:36

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