Geodesic
Adiabatic limit for hyperbolic Ginzburg--Landau equations
Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 4, Tome 48 (2013), pp. 111-119

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We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are Euler–Lagrange equations for the Abelian Higgs model. Solutions of Ginzburg–Landau equations in this limit converge to geodesics on the moduli space of static solutions in the metric determined by the kinetic energy of the system. According to heuristic adiabatic principle, every solution of Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some geodesic. A rigorous proof of this result was proposed recently by Palvelev. 
@article{CMFD_2013_48_a8,
     author = {A. G. Sergeev},
     title = {Adiabatic limit for hyperbolic {Ginzburg--Landau} equations},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {111--119},
     publisher = {mathdoc},
     volume = {48},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2013_48_a8/}
}
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A. G. Sergeev. Adiabatic limit for hyperbolic Ginzburg--Landau equations. Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 4, Tome 48 (2013), pp. 111-119. http://geodesic.mathdoc.fr/item/CMFD_2013_48_a8/