Geodesic
Justification of the adiabatic principle for hyperbolic Ginzburg–Landau equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 199-214
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We study the adiabatic limit in hyperbolic Ginzburg–Landau equations which are the Euler–Lagrange equations for the Abelian Higgs model. By passing to the adiabatic limit in these equations, we establish a correspondence between the solutions of the Ginzburg–Landau equations and adiabatic trajectories in the moduli space of static solutions, called vortices. Manton proposed a heuristic adiabatic principle stating that every solution of the Ginzburg–Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some adiabatic trajectory. A rigorous proof of this result has been found recently by the first author. 
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R. V. Palvelev; A. G. Sergeev. Justification of the adiabatic principle for hyperbolic Ginzburg–Landau equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 199-214. http://geodesic.mathdoc.fr/item/TM_2012_277_a13/

[1] Jaffe A., Taubes C., Vortices and monopoles: Structure of static gauge theories, Birkhäuser, Boston, 1980 | MR | Zbl

[2] Manton N.S., “A remark on the scattering of BPS monopoles”, Phys. Lett. B, 110 (1982), 54–56 | DOI | MR | Zbl

[3] Palvelev R.V., “Obosnovanie adiabaticheskogo printsipa v abelevoi modeli Khiggsa”, Tr. Mosk. mat. o-va, 72, no. 2, 2011, 281–314

[4] Stuart D., “Dynamics of Abelian Higgs vortices in the near Bogomolny regime”, Commun. Math. Phys., 159 (1994), 51–91 | DOI | MR | Zbl