Non trivial solutions for a system of coupled Ginzburg-Landau equations

Abstract

This article addresses both the existence and properties of non-trivial solutions for a system of coupled Ginzburg-Landau equations derived from nematic-superconducting models. Its main goal is to provide a thorough numerical description of the region in the parameter space containing solutions that behave as a mixed (non trivial) nematic-superconducting state along with a rigorous proof for the existence of this region. More precisely, the rigorous approach establishes that the parameter space is divided into two regions with qualitatively different properties, according to the magnitude of the coupling constant: for small values (weak coupling), there is a unique non-trivial solution, and for large values (strong coupling), only trivial solutions exist. In addition, using a shooting method-based numerical approach, the profiles for the nematic and superconducting components of the non trivial solution are given, together with an algorithm computing the transition values representing the boundaries for the weak coupling region: from superconducting to mixed, and from mixed to nematic. Finally, numerical evidence is given for the existence of a third region, related to neither a small nor a strong coupling parameter (medium coupling) for which multiple non trivial solutions exist.