Nonminimal solutions to the Ginzburg-Landau equations

We use two different methods to prove the existence of novel, nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds. To our knowledge these are the first such examples on nontrivial line bundles, that is, with nonzero total magnetic flux. The first method works with the 2-dimensional, critically coupled Ginzburg-Landau theory and uses the topology of the moduli space. This method is nonconstructive, but works for generic values of the remaining coupling constant. We also prove the instability of these solutions. The second method uses bifurcation theory to construct solutions, and is applicable in higher dimensions and for noncritical couplings, but only when the remaining coupling constant is close to the "bifurcation points", which are characterized by the eigenvalues of a Laplace-type operator.Comment: 25 pages, no figures. Submitted version. Comments are still welcome

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator, these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to 3 dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a hyperk\"ahler manifold in 3 different ways, which contains the ordinary Bogomolny moduli space as a complex Lagrangian submanifold---an (ABA)-brane---with respect to any of these structures. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have showed a vanishing theorem in [12].Comment: 30 pages, no figures,title changes made by referee's reques

On the bifurcation theory of the Ginzburg-Landau equations

We construct nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles.Comment: 13 pages, no figures, submitted version, comments are welcome! Some overlap with older versions of arXiv:2103.05613, which is now separated into two paper

Flow-based Capacity Allocation in the CEE Electricity Market: Sensitivity Analysis, Multiple Optima, Total Revenue

The paper introduces the mechanism of the Flow-based Capacity Allocation (FBA) method on the electricity market of the Central-Eastern Europe (CEE) Region, proposed by the Central Allocation Office (CAO). The method is a coordinated heterogeneous multi-unit uniform price auction where the allocation is determined by the solution of a linear programming problem. On one hand, the properties of the underlying linear programming problem are discussed: the possibilities of multiple solutions are analysed, then a non-standard sensitivity analysis method of the market spread auction is developed. On the other hand, a global optimization problem is presented that yields uniform auction prices corresponding to higher total income than at the original allocation method. Several numerical examples and results of practical test problems are presented