Nonlilnear Partial Differential Equations in Heterogeneous Superconducting Systems and High Critical Temperature Cuprate Compounds

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GiorgiDMS-0604843 The investigator analyzes mathematical models ofheterogeneous superconducting systems and ofhigh-critical-temperature copper oxides that show superconductingproperties. Topics considered in heterogeneous superconductingsystems include the effect of surface cold working or of contactwith semiconductors on the critical parameters (such astransition temperature) that describe superconductivity inmesoscopic samples, the effect of interactions of mesoscopicmagnetic textures with superconductors on transition temperaturesand vortex pinning, and surface nucleation in superconductorssurrounded by normal materials. In high-temperaturesuperconductivity, the investigator extends mathematicaltechniques developed for low-temperature systems to the SO(5)model of high-temperature copper compounds. Superconductors are materials that show zero or almost zeroresistance to the flow of electrical current. Usually thishappens at extremely low temperatures. Some materials that aresuperconductors at higher temperatures are known, including somecopper compounds of recent interest. But the theory that woulddescribe their behavior is incomplete. The investigator studiesmathematical models of composite superconducting materials and ofsuperconductors based on copper compounds. A large body ofexperimental data are available, but definitive models forcuprate compounds do not exist. They are of special interestbecause materials based on these compounds show superconductingproperties at desirably high temperatures. Superconductors areof commercial interest for building electronic sensing devices,energy storage devices, magnets, and quantum computers. Progressin nanotechnologies has raised the possibility of industrialapplications of mesoscopic heterogeneous magnetic andsuperconducting systems.