A classification of all periodic self-dual static vortex solutions of the Jackiw-Pi model is given. Physically acceptable solutions of the Liouville equation are related to a class of functions, which we term Ω-quasi-elliptic. This class includes, in particular, the elliptic functions and also contains a function previously investigated by Olesen. Some examples of solutions are studied numerically and we point out a peculiar phenomenon of lost vortex charge in the limit where the period lengths tend to infinity, that is, in the planar limit.

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29.
On a surface of constant curvature the conformal factor of the metric in isothermal coordinates satisfies the Liouville equation; that is, if the metric is ds2 = ρ (dx2 + dy2) with ρ > 0, then ρ satisfies Eq. (13) and e2 is equal to the Gaussian curvature K of the surface. In this situation, the case K < 0 is, of course, not excluded and corresponds to solution ((14), II). It is known that Eq. ((13), I) has no nowhere vanishing solution on the torus (Ref. 17). Thus, by necessity, all our torus solutions given below have zeros.
30.
Indeed, we may conjecture that if f has a non-isolated singularity, then its associated density ρf is unbounded.
31.
In the plane case the integral extends over
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32.
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33.
For
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and any complex function f we define
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.
34.
For completeness, we mention the elementary rule T1γ2) = T1)T2) for all γ1, γ2 ∈ U(2).
35.
It turns out to be immaterial which branches we choose for the square roots. In this sense, the choice of parameters is essentially unique.
36.
“Drempel” is a Dutch word which, amongst other things, denotes a speed bump.
37.
On an unpunctured torus, there are no such metrics, compare (Ref. 29).
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