Numerical discretization and simulation of ginzburg landau models for superconductivity 1 alexandre ardelea, anand l. In the usual ginzburg landau theory, it is r that is taken as the wave function or order parameter, as it is sometimes called in that theory, because, as we shall see, the internal angular part does not matter for symmetry reasons in the case of an in nite sample. Fractional complex ginzburg landau equation, dissipative solitons, effective diffusion 1. Then we will introduce the phenomenological ginzburglandau theory for neutral superfluids. We prove that a set of n not necessarily distinct points in the plane determine a unique, real analytic solution to the first order ginzburg landau equations with vortex number n. The complex ginzburglandau equation cgle was first derived in the studies of. Phase chaos and the transition to defect chaos 1 2. Onedimensional ginzburg landau equations with derivatives of noninteger order are considered. We shall start from the equations for the thermodynamic green functions3 4 which we obtained. Ginzburglandau equation encyclopedia of mathematics.
Chapter 2 ginzburg landau phenomenology the divergence of the correlation length in the vicinity of a secondorder phase transition indicates that the properties of the critical point are insensitive to microscopic details of the system. Chapter 2 ginzburglandau phenomenology the divergence of the correlation length in the vicinity of a secondorder phase transition indicates that the properties of the critical point are insensitive to microscopic details of the system. Traveling waves in the complex ginzburglandau equation. In the context of pattern formation the real ginzburg landau. Microscopic derivation of the ginzburglandau equations in. Within the phase equation, domain structures are only stable if the total phase, i. Dynamic bifurcation of the ginzburg landau equation. Landau gordon and breach, new york traveling hole solutions of the complex ginzburglandau equation. Xin 2 1 department of mathematics, university of michigan, ann arbor, mi 48109, usa 2 department of mathematics, university of arizona, tucson, az 85721, usa. H,t of a typeii superconductor in the mixed state is a universal function of hh c2t. Gls equation is found for a general class of nonlinear evolution problems. In the context of pattern formation the real ginzburglandau. Numerical approximations of the ginzburglandau models for.
The boundary of defect chaos towards nozakibekki holes 115 iv. I am trying to do some numerical calculation with ginzburg landau gl equation for a superconductor. We present an approximate solution of the twocomponent ginzburglandau equation for a broad class of initial conditions. Complex ginzburglandau equations with dynamic boundary.
Gorkov, microscopic derivation of the ginzburglandau equations in the theory. Much research exists on longtime dynamics of the cubic ginzburglandau equation in one spatial dimension 1wx wx3, 2325 and in two spatial dimensions 1, 3, 4, 7 uy. We will start with the landau ginzburg free energy and a derivation of the equations of motion 1. Approximate solutions of the twocomponent ginzburglandau. Nonlinear stability of source defects in the complex ginzburg. E, 7 where the prime on the summation in this context means sum only over ksuch that f ginzburg landau theory, it is r that is taken as the wave function or order parameter, as it is sometimes called in that theory, because, as we shall see, the internal angular part does not matter for symmetry reasons in the case of an in nite sample.
Pdf dynamic bifurcation of the ginzburglandau equation. The ginzburglandau equation is an amplitude equation and has applications in many. Three forms of localized solutions of the quintic complex ginzburg. Landau gordon and breach, new york traveling hole solutions of the complex ginzburg landau equation. A quantum theory london theory is classical introduction of pseudo wave function. Therefore, the heteroclinic cycles merge into the stationary homoclinic. Dynamic stability of vortex solutions of ginzburglandau and. We regard this equation as a perturbation of the real ginzburglandau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. Dynamic stability of vortex solutions of ginzburglandau and nonlinear schr6dinger equations m. Gls equation is found for a general class of nonlinear evolution problems including several classical. Find out information about ginzburg landau equation. Psiseriessolutionoffractionalginzburglandau equation. In physics, ginzburglandau theory, often called landauginzburg theory, named after vitaly lazarevich ginzburg and lev landau, is a mathematical physical theory used to describe superconductivity.
R1 about the minimum critical value of the control parameter r for small values of. The complex ginzburglandau equation for beginners w. Ginzburglandau theory some of the key ideas for the landau mean. Dynamic stability of vortex solutions of ginzburglandau. The general objective of the work is to study dynamics of dissipative solitons in the framework of a onedimensional complex ginzburg landau equation cgle of a fractional order. Poiseuille flow 53 and reactiondiffusion systems 26.
Numerical approximations of the ginzburglandau models. Nonlinear stability of source defects in the complex ginzburglandau equation margaret beck toan t. Their aim is to study the relevant solutions in order to gain insight into nonequilibrium phenomena in spatially extended systems. What is the boundary condition for ginzburg landau equation.
It means we set j rj to a xed constant, but still allow r to vary. The ginzburglandau free energy functional of color superconductivity at weak coupling ioannis giannakisa and haicang rena y aphysics department, the rockefeller university 1230 york avenue, new york, ny 100216399 abstract we derive the ginzburglandau free energy functional of color su. We prove clearingout results and an orthogonal anchoring condition of the vortex on the boundary for the ginzburglandau equation with magnetic eld. A phenomenological theory of superconductivity which accounts for the coherence length. The onedimensional complex ginzburglandau equation. The dynamic stability of vortex solutions to the ginzburglandau and. Introduction based on the pathintegral approach, a fractional generalization of the. Chapter 5 the ginzburglandau equation ginzburg landauequations have been used to model a wide variety of physical systems see, e.
Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real ginzburglandau equation in the limit in which the imaginary terms. A complete bifurcation diagram of the ginzburg landau equation with periodic boundary conditions. More recent papers 1,2 combine numerical simulations with an analytical study, which allows. We provide a study at the boundary for a class of equation including the ginzburg landau equation as well as the equation of travelling waves for the grosspitaevskii model. At a critical point, the magnetization is continuous as the parameters are tuned closer to the critical point, it gets smaller, becoming zero at the critical point. Micromagnetic modelling the current state of the art. Pdf soliton dynamics in a fractional complex ginzburg. The starting point is a microscopic hamiltonian including a. Asymptotics for the generalized twodimensional ginzburg. The coherent length should be included in a new theory.
Triggered fronts in the complex ginzburg landau equation ryan goh and arnd scheel august 31, 20 abstract we study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex ginzburglandau equation. Aranson materials science division, argonne national laboratory 9700 s. Complex ginzburglandau equations with dynamic boundary conditions wellington jose corr ea and t urker ozsari,y abstract. Oscillatory systems with nonlinear global coupling. Ginzburglandau equation article about ginzburglandau. Pdf derivation of the ginzburglandau equations of a. The ginzburglandau type modulation equations are such simplified mathematical models for nonlinear systems in mechanics, physics, and other areas. This redundancy of information motivates the search for a phenomenological.
Topological entropy by unit ssoyeur for the ginzburg landau equation on the alaih. It is shown that the phenomenological ginzburg landau equations follow from the theory of superconductivity in the london temperature region in the neighborhood of tc. We present an approximate solution of the twocomponent ginzburg landau equation for a broad class of initial conditions. In this chapter we will present the formal derivation process by which the ginzburg landau equation can be derived.
Numerical discretization and simulation of ginzburg landau. Nonlinear stability of source defects in the complex. In the context of pattern formation the real ginzburglandau equation rgle was. In the usual ginzburglandau theory, it is r that is taken as the wave function or order parameter, as it is sometimes called in that theory, because, as we shall see, the internal angular part does not matter for symmetry reasons in the case of an in nite sample. Dynamic stability of vortex solutions of ginzburg landau and nonlinear schr6dinger equations m. The ginzburg landau free energy functional of color superconductivity at weak coupling ioannis giannakisa and haicang rena y aphysics department, the rockefeller university 1230 york avenue, new york, ny 100216399 abstract we derive the ginzburg landau free energy functional of color su. Richardson texas institute for computational and applied mathematics, the university of texas at austin austin, texas, 78712 usa. It turns out that for conventional lowtc superconductors, mean. Triggered fronts in the complex ginzburg landau equation. Nguyeny bj orn sandstedez kevin zumbrunx february 12, 2014 abstract in an appropriate moving coordinate frame, source defects are timeperiodic solutions to reaction. On the validity of the ginzburglandau equation springerlink. The name ginzburglandau, both of the equation and of the functional, comes from a paper on superconductivity. E, 7 where the prime on the summation in this context means sum only over ksuch that f merge into single pulses, with the respective merger distance also determined by li. Box 9506, 2300 ra leiden the netherlands this article appeared in spatiotemporal patterns in nonequilibrium complex systems, eds.
We prove that a set of n not necessarily distinct points in the plane determine a unique, real analytic solution to the first order ginzburglandau equations with vortex number n. Complex ginzburg landau equations with dynamic boundary conditions wellington jose corr ea and t urker ozsari,y abstract. The fractional ginzburg landau equation with distributional initial data. Pdf traveling waves in the complex ginzburglandau equation. Irreducible ginzburglandau fields in dimension 2 arxiv. Ginzburglandau equations either in their original or modified form describe a wealth of phenomena and systems including superconductivity and superfluidity, nonlinear waves in active media, pat. On the initialvalue problem for the generalized two. Introduction in 1950 landau and ginzburg proposed a theory which phenomenologically describes much of the behavior seen in superconductors. Since equations of this type do not appear generically from the derivation. This equation is the complex ginzburg landau equation with coefficients with small imaginary parts.
In these equations there occurs, however, twice the electronic charge. The ginzburglandau functional appears in various parts of science. In this paper, the authors study a complex generalized ginzburglandau equation with two spatial dimensions 2d. Historically this was introduced when only type i supercon. However, i am confused about the boundary condition of the gl equation.
Request pdf on may 12, 2017, wellington jose correa and others published complex ginzburglandau equations with dynamic boundary conditions find, read and cite all the research you need on. Specifically, we consider the formation of vortex antivortex pairs, from an initial condition consisting of small random fluctuations about zero. In section 5 we combine results from the previous sections and a. Iv we therefore study the ginzburglandau equation with a spatially ramped control parameter which allows the total phase to change. Using psiseries with fractional powers, the solution of the fractional ginzburg landau fgl equation is derived. Our method of solution is based on a novel singular perturbation expansion. The initialdynamic boundary value problem idbvp for the complex ginzburg landau equation cgle on bounded domains of rn is studied by converting the given mathematical model into a wentzell initialboundary value. Landau gordon and breach, new york on the problem of turbulence l. We prove clearingout results and an orthogonal anchoring condition of the vortex on the boundary for the ginzburg landau equation with magnetic eld. Free energy london approximation the london approximation is analogous to the xedlength spin approximation in statistical mechanics of lattice models.
The initialdynamic boundary value problem idbvp for the complex ginzburglandau equation cgle on bounded domains of rn is studied by converting the given mathematical model into a wentzell initialboundary value. The bound state equation becomes uk v p k0 0u k0 2 k. Traveling waves in the complex ginzburglandau equation article pdf available in journal of nonlinear science 3. We provide a study at the boundary for a class of equation including the ginzburglandau equation as well as the equation of travelling waves for the grosspitaevskii model. Iv we therefore study the ginzburg landau equation with a spatially ramped control parameter which allows the total phase to change. The famous ginzburglandau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterr lies in the unstable regiono. Chapter 5 the ginzburglandau equation ginzburglandauequations have been used to model a wide variety of physical systems see, e. Ginzburglandau equation cgle, as this constitutes the normal form in the vicinity of a hopf. Triggered fronts in the complex ginzburg landau equation ryan goh and arnd scheel august 31, 20 abstract we study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex ginzburg landau equation. Combining both coupling mechanisms, linear and nonlinear global couplings, ren. In this particular case, the magnetic susceptibility. In its initial form, it was postulated as a phenomenological model which could describe typei superconductors without examining their microscopic properties. I am trying to do some numerical calculation with ginzburglandau gl equation for a superconductor.
Lieb and loss 19 showed that, restricted to functions satis. We combine energy methods with monotonicity properties. The leadingorder behaviours of solutions about an arbitrary singularity, as well as their resonance structures, have been obtained. We regard this equation as a perturbation of the real ginzburg landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation.
Sufficient conditions for the existence and uniqueness of global solutions for the initialvalue problem of the generalized 2d ginzburglandau equation are obtained. Pdf we derive a ginzburglandau free energy for a pwave ferromagnetic superconductor. Arbitrary nvortex solutions to the first order ginzburg. We model the trigger by a spatial inhomogeneity moving with constant speed. Find out information about ginzburglandau equation. This equation is the complex ginzburglandau equation with coefficients with small imaginary parts. In this chapter we will present the formal derivation process. Nonlinear stability of source defects in the complex ginzburg landau equation margaret beck toan t. Charged superfluid 589 so lets assume the interacting ground state is still bosecondensed and approximate it by placing all nb bosons in the same state r, to be optimized variationally.
1425 1090 650 1188 836 1090 1200 1518 1500 1340 301 1040 1365 547 1623 1306 459 1008 618 703 191 953 949 1527 1247 1221 1063 715 1137 1138 1351 1170 178 823 308 453 1592 194 1357 180 996 1123 365 1494 264 755 1170