A basic property of solutions to the ginzburglandau equation. Nonlinear stability of source defects in the complex. Exact solutions to complex ginzburglandau equation request pdf. Gls equation is found for a general class of nonlinear evolution problems including several classical. Imaginary parts of the approximate and the exactleft solutions of example 2 when m 6. In this paper, the trial equation method and the complete discrimination system for polynomial method are applied to retrieve the exact travelling wave solutions of complex ginzburglandau equation. This redundancy of information motivates the search for a phenomenological. Ginzburglandau equation article about ginzburglandau. Front solutions for the ginzburg landau equation eckmann, j. Introduction the 2d generalized ginzburglandau equation ut. Mid this article has been rated as midimportance on the. Hence, before one can derive a ginzburglandau equation it is necessary to determine the linear, or spectral, stability of the laminar solution. Ginzburglandau vortices progress in nonlinear differential equations and their applications book kindle edition by fabrice bethuel, haim brezis, frederic helein.
In its initial form, it was postulated as a phenomenological model which could describe typei superconductors without examining their microscopic properties. At a critical point, the magnetization is continuous as the parameters are tuned closer to the critical point. Pdf the ginzburglandau equation solved by the finite element. Numerical discretization and simulation of ginzburg landau. This paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the ginzburglandau theory as a unified language. The existence and uniqueness of a timeperiodic solution to this equation is proved. To be able to solve this equation correctly you need to have the number of mesh points such that the distance between them in the kink region does not exceed 1. The world of the complex ginzburglandau equation i. Numerical discretization and simulation of ginzburg landau models for superconductivity 1 alexandre ardelea, anand l.
The hole solution shows two types of instability and bifurcates to. The landau ginzburg free energy as a hamiltonian c 2012 by w. Pullback attractor for nonautonomous ginzburg landau equation with additive noise li, yangrong and cui, hongyong, abstract and applied analysis, 2014. Nguyeny bj orn sandstedez kevin zumbrunx february 12, 2014 abstract in an appropriate moving coordinate frame, source defects are timeperiodic solutions to reaction. Solving generalized quintic complex ginzburglandau equation by. Linearized ginzburglandau equation how is linearized ginzburglandau equation abbreviated. Existence and uniform boundedness of strong solutions of the timedependent ginzburglandau equations of superconductivity zaouch, fouzi, abstract and applied analysis, 2005. Solutions of the real valued ginzburglandau equation develop areas with values 1, which are separated by phase transition regions, i. In physics, ginzburglandau theory, often called landauginzburg theory, named after vitaly. The ginzburg landau functional appears in various parts of science.
Pdf dynamic bifurcation of the ginzburglandau equation. Just as the gl equations enabled researchers to explore the behaviour of charge superconductors, our ginzburglandautype equations. In this paper we study the notion of topological entropy by unit length for the dynamical system given by the complex ginzburg landau equation on the line cgl. Phase dynamics and localized solutions to the ginzburg. Available formats pdf please select a format to send. Use features like bookmarks, note taking and highlighting while reading ginzburglandau vortices progress in nonlinear differential equations and. Mathematical and numerical analysis of timedependent ginzburg landau superconductivity. In this paper we study the notion of topological entropy by unit length for the dynamical system given by the complex ginzburglandau equation on the line cgl.
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. But in your case the boundary condition u0 is fixed at a finite distance. On the validity of the ginzburglandau equation springerlink. Why does encrypting the same file with aes256 not give the same output. On nonlocally coupled complex ginzburglandau equation. Physica d 61 1992 253259 northholland ginzburglandau equation coupled to a concentration field in binarymixture convection hermann riecke department of engineering sciences and applied mathematics, northwestern university, evanston, il 60208, usa localized travellingwave trains are investigated as they arise in binarymixture convection. We study the asymptotic behavior of the positive solutions of the ginzburg. We are interested in the numerical approximation of the complex ginzburglandau. Ginzburglandautype theory of spin superconductivity nature. A phenomenological theory of superconductivity which accounts for the coherence length. The cubic complex ginzburglandau equation is one of the moststudied nonlinear equations in the physics community.
Later, a version of ginzburglandau theory was derived from the. Oct 27, 2014 this paper presents an introduction to phase transitions and critical phenomena on the one hand, and nonequilibrium patterns on the other, using the ginzburg landau theory as a unified language. Numerical approximations of the ginzburglandau models. 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. 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. Numerical approximations of the ginzburglandau models for. Ginzburglandau equation coupled to a concentration field. In the context of pattern formation the real ginzburglandau. The purpose of this work is to study superfluid properties of a neutral fermion system with pairing in a nonzero angular momentum state, in particul.
The name ginzburglandau, both of the equation and of the functional, comes from a paper on superconductivity. A ginzburg landau type equation with nonlocal coupling is derived systematically as a reduced form of a universal class of reactiondiffusion systems near the hopf bifurcation point and in the presence of another small parameter. Chapter 4 the landauginzburg free energy as a hamiltonian c 2012 by w. Nonlinear stability of source defects in the complex ginzburglandau equation margaret beck toan t.
Ginzburglandau equation for 3p2 pairing progress of. 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. Nonlinear stability of source defects in the complex ginzburg. 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. Aug 29, 2015 a basic property of solutions to the ginzburg landau equation. Klein, harvey gould, and jan tobochnik 1 october 2012 4. Pseudospectral solver for the complex ginzburg landau equation cgle. Fractional ginzburglandau equation for fractal media.
For freeslippermeable boundary conditions a complex ginzburg landau equation is derived which is coupled to a mean concentration field. Landau proposed a phenomenological theory for phase transitions. Limiting vorticities for the ginzburglandau equations sandier, etienne and serfaty, sylvia, duke mathematical journal, 2003. Microscopic derivation of the ginzburg landau equations in the theory of. Find out information about ginzburglandau equation.
If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. Ginzburglandau vortices progress in nonlinear differential. This equation implies an energy theorem, according to which the energy can be. Localized travellingwave trains are investigated as they arise in binarymixture convection. Pullback attractor for nonautonomous ginzburglandau equation with additive noise li, yangrong and cui, hongyong. The world of the complex ginzburg landau equation i.
A time dependent modification of the ginzburglandau equation is given which is based on the assumption that the functional derivative of the ginzburglandau free energy expression with respect to the wave function is a generalized force in the sense of irreversible thermodynamics acting on the wave function. Fractional generalization of the ginzburglandau equation. The ginzburglandau equation with nonzero neumann boundary. In the context of pattern formation the real ginzburg landau. This revealed that new types of instability which can never arise in the ordinary complex ginzburglandau equation are possible, and their physical implication is briefly discussed. The complex ginzburglandau equation for beginners w. 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. Phase dynamics and localized solutions to the ginzburglandau type amplitude equations 0 department of physics, college of general education kyushu university, fukuoka 810 we study different types of long wavelength phase modulation and localized modes in the dissipative media described by the ginzburglandau type amplitude equations. The cubic complex ginzburglandau equation cgle is one of the moststudied nonlinear equations in the physics community. The ginzburglandau functional appears in various parts of science.
The timeperiodic solution to a 2d generalized ginzburg. An unconventional approach to critical phenomena in complex media. Bistability ginzburg landau equation mathematica stack exchange. The name ginzburg landau, both of the equation and of the functional, comes from a paper on superconductivity. Below the superconducting transition temperature, the above equation is. In the context of pattern formation the real ginzburglandau equation rgle was. Historically this was introduced when only type i supercon. Ginzburglandau equation encyclopedia of mathematics. On some sufficient conditions for the blowup solutions of the nonlinear ginzburglandauschrodinger evolution equation nasibov, sh.
We derive the fractional generalization of the ginzburg landau equation from the variational eulerlagrange equation for fractal media. 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. We derive the fractional generalization of the ginzburglandau equation from the variational eulerlagrange equation for fractal media. Ginzburglandau equation with degennes boundary condition. A time dependent ginzburglandau equation and its application. 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. Ginzburglandau equation coupled to a concentration field in.
Stable largescale solver for ginzburglandau equations for. This should lead to the step with u0 at the left boundary and making a step with the width 1 coming to the horizontal asymptote u3. As is well known, the meanfield approximation breaks down below four. Chapter 5 the ginzburglandau equation ginzburg landauequations have been used to model a wide variety of physical systems see, e.
This article is within the scope of wikiproject physics, a collaborative effort to improve the coverage of physics on wikipedia. 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. A time dependent modification of the ginzburg landau equation is given which is based on the assumption that the functional derivative of the ginzburg landau free energy expression with respect to the wave function is a generalized force in the sense of irreversible thermodynamics acting on the wave function. Physica d 61 1992 253259 northholland ginzburg landau equation coupled to a concentration field in binarymixture convection hermann riecke department of engineering sciences and applied mathematics, northwestern university, evanston, il 60208, usa localized travellingwave trains are investigated as they arise in binarymixture convection. The complex ginzburglandau equation has a hole solution as a localized structure. Dynamic bifurcation of the ginzburg landau equation. Analysis of iterative methods for solving a ginzburglandau equation. 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.
Linear stability analysis of the reduced equation about the uniform oscillation is also carried out. Ginzburglandau equations were one of the first nonlinear theories to demonstrate solutions in the form of topological singularities. Namely, we will use the landau ginzburg free energy as a hamiltonian or action. In the first part, meanfield theory is presented, for both statics and dynamics, and its validity tested selfconsistently. Nonlinear stability of source defects in the complex ginzburg landau equation margaret beck toan t. The bound state equation becomes uk v p k0 0u k0 2 k. Topological entropy by unit length for the ginzburglandau. Linearized ginzburglandau equation how is linearized. Introduction in 1950 landau and ginzburg proposed a theory which phenomenologically describes much of the behavior seen in superconductors. Richardson texas institute for computational and applied mathematics, the university of texas at austin austin, texas, 78712 usa. It means we set j rj to a xed constant, but still allow r to vary.
Download it once and read it on your kindle device, pc, phones or tablets. Free energy london approximation the london approximation is analogous to the xedlength spin approximation in statistical mechanics of lattice models. The cubic complex ginzburglandau equation is one of the most studied nonlinear equations in the physics community. The complex ginzburg landau model for an oscillatory. Multiple time scale dynamics in coupled ginzburglandau equations lin, fangua and lin, taichia, communications in mathematical sciences, 2003. Bistability ginzburg landau equation mathematica stack. For freeslippermeable boundary conditions a complex ginzburglandau equation is derived which is coupled to a mean concentration field. A nonautonomous complex ginzburglandau equation for the finite amplitude of convection is derived based on a small perturbed parameter. The timeperiodic solution to a 2d generalized ginzburglandau equation. Note that the minimizers are not unique because of the fact that the ginzburg landau functional is.
It describes a vast variety of phenomena from nonlinear waves to secondorder phase transitions, from superconductivity, superfluidity and boseeinstein condensation to liquid crystals and strings in field theory. The complex ginzburg landau model for an oscillatory convection in a rotating fluid layer. Exact solutions to complex ginzburglandau equation. Find out information about ginzburg landau equation. Landau equation with the degennes boundary condition. Start this article has been rated as startclass on the projects quality scale. Instability of the hole solution in the complex ginzburg. Arbitrary nvortex solutions to the first order ginzburg. The ginzburglandau equation is a classical equation in the. E, 7 where the prime on the summation in this context means sum only over ksuch that f ginzburg landau 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. Ginzburglandau equations suitable for investigations of typeii superconductors on massively parallel architectures.
608 278 395 775 179 1053 887 1618 789 1038 180 415 531 340 786 1621 1323 129 1551 1174 694 748 115 1096 1010 1147 1351 927 648 891 609 89 1157 173 412