Reduction of field equations and saturated bounds for vortex-line theories

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Abstract
The Euler-Lagrange equations following from the two-dimensional Higgs-type Lagrangian and from the chirally nonlinear Lagrangian are shown to be reducible to one second-order differential equation. This simplification is possible for an arbitrary configuration of static parallel vortex lines but only for a specific potential. In both cases the result is connected with a saturated bound for the energy and also with conditions arising from the scale variation.