Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations
The paper is devoted to the mathematical justification of the heuristic adiabatic principle proposed by Manton for hyperbolic analogues of the parabolic Ginzburg-Landau (GL) equations, arising in superconductivity theory. These hyperbolic GL equations are the Euler-Lagrange (EL) equations for the action functional given by the GL Lagrangian. Before Manton formulated the adiabatic principle stating that any dynamic solution of the GL equations, which are close to static ones, can be obtained as a perturbation of some adiabatic trajectory (geodesic of a Riemannian metric on the space of vortices -- static solutions of the GL equations). First, in the paper, the hyperbolic GL equations are introduced as the EL equations for the \((2+1)\)-dimensional Abelian Higgs model arising in gauge field theory. With this aim, the GL action functional is defined through kinetic energy and potential energy terms with variables of \(\mathrm{U}(1)\)-connection and Higgs field. The GL equations are the EL equations for the GL action being invariant under the gauge transformation. The solutions of the EL equations are dynamic solutions. Static solutions of the GL equations realize local minima of the potential energy and they have an integer-valued topological invariant, the vortex number \(N\). By considering dynamic solutions of the GL equations, the authors introduce an adiabatic limit with limiting adiabatic trajectory in the configuration space. Every point of the adiabatic trajectory is a static solution, at the same time the whole trajectory cannot be a dynamic solution. The approximate description of `slow' dynamic solutions in a term of moduli space of static solutions was proposed at a heuristic level by Manton. The authors give a rigorous formulation of the adiabatic principle by considering the tangent structure of the moduli space of \(N\)-vortices. Then a linearized vortex operator and infinitesimal gauge transformations are introduced. The main theorem of the paper is devoted to the existence of a dynamic solution to the GL equations satisfying the given estimate. In order to prove it, the authors replace the original set of GL equations by an auxiliary set of equations having only a finite-dimensional degeneration in contrast to the original set. To solve this auxiliary equation set, two theorems are used by defining a local existence and a slow-time a priori estimate of a solution. A strategy for proving the existence of a solution to the auxiliary set is explained. The proof proceeds by successively applying the local existence theorem, while the slow-time a priori estimate guarantees that the constructed solution extends in time to ranges of length of order \(1/\epsilon\), where \(\epsilon\) is a prescribed precision.
scientific article; zbMATH DE number 6345586