A-twisted Landau-Ginzburg models

金兹堡朗道理论

Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。

材料科学中的相场模拟方法

材料科学中的相场模拟方法随着材料科学领域的发展和应用需求不断增加，材料模拟成为一种重要的研究手段。

其中，相场模拟方法是一种有效的计算工具。

本文将介绍相场模拟方法的基本理论和应用领域，并分析其优缺点。

一、相场模拟方法的基本理论相场模拟方法可以视为一种用数学方法来模拟材料微观结构和变化的方法。

其基本思想是将材料视为时空上的一个相场系统，用数学模型描述相场在空间中的演化过程。

相场的变化可以由受力平衡条件来描述，其中能量泛函可以用Ginzburg-Landau方程来表征。

Ginzburg-Landau方程可以描述相场随时间和空间的演化，从而模拟材料的微观结构变化。

二、相场模拟方法的应用领域相场模拟方法可以用于各种材料的研究和应用。

主要应用领域包括晶体生长、固化过程、气固相变、材料挤出、相变润湿、电极电化学等。

相场模拟方法可用于计算各种材料的物理和化学性质，并通常与实验结合使用，对材料微观结构、热力学性质、机械性能等方面进行了深入探究。

三、相场模拟方法的优缺点分析相场模拟方法具有以下优点：1. 可模拟复杂微观结构：相场模拟方法可以模拟复杂的材料微观结构，包括晶体形态、孪晶等细节。

2. 全域性：相场模拟方法可以实现区域内整体的优化。

3. 数值计算效率：相场模拟方法可以借助计算机计算出复杂的物理现象。

4. 可变形性：相场模拟可以模拟变形行为。

但同时也有以下缺点：1. 计算复杂度高：相场模拟方法需要很高的计算复杂度。

2. 信息量大：相场模拟需要包含大量的材料参数，而参数不确定性会导致计算不可靠。

3. 实验验证难度大：相场模拟计算的参数需要实验验证，实验难度较大。

四、结语综合以上的分析，相场模拟方法是一种可以模拟材料微观结构变化的有效工具，虽然有一定缺点，但已广泛应用于各个领域。

相信随着计算机技术的不断进步，相场模拟方法可以应用地更加广泛和可靠，对材料科学研究和应用发展会产生深远的影响。

激光ginzbrg-landau方程

激光ginzbrg-landau方程
激光Ginzburg-Landau方程是用来描述激光和其它非线性波动的物理现象的方程。

这个方程最初由苏联的物理学家Emmanuel Ginzburg 和Lev Landau在1950年提出。

激光Ginzburg-Landau方程描述了激光的演化过程。

它与类似的方程不同，因为它包括了非线性项，这意味着激光的演化是不可预测的。

但这个方程也包含了一些关于激光的特殊性质。

方程的形式与量子力学中的Ginzburg-Landau方程相似。

它可以用来描述激光的相干性、频谱、群速度和非线性动力学。

激光Ginzburg-Landau方程的一个关键特征是它包括了两个相位。

这两个相位描述了激光波和介质中的非线性响应之间的相互作用。

激光Ginzburg-Landau方程的求解是一个非常复杂的问题，因为它是一个高度非线性的方程。

同时，它也涉及到许多理论和实验方面的问题，比如：激光的色散、自聚焦和自相位调制等。

在现代光学和激光技术中，激光Ginzburg-Landau方程被广泛应用。

它被用来研究一系列的现象，比如：激光脉冲的形成和发展、激光发生自聚焦的机制、激光在非线性介质中的传输和调制等。

总之，激光Ginzburg-Landau方程是一个重要的物理方程，它为我们理解和掌握激光和非线性波动的物理现象提供了宝贵的工具和手段。

用临界点附近的涨落讨论Landau相变理论适用范围

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根据热力学理论，偏离平均值的涨落可以用相对几率Ｗ来描述
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在物理学中，相变开始作为唯象理论，用以描述众多原子或离子在宏观尺度上的现象．表征相变的状态方程最早由ｖａｎｄｅｒＷａａｌｓ于１８７３年提出．１９３７年Ｌａｎｄａｕ建立了著名的二级相变理论．这是一个平均场理论，它在研究晶体相变方面有着广泛的应用．晶体的各种结构相变都牵涉到晶体的改变，这种对称性的改变可以用Ｌａｎｄａｕ的结构相变理论来研究，并且这种理论研究晶体的连续二级相变尤为方便．在远离临界点时，宏观体系的涨落很小，Ｌａｎｄａｕ的结构相变理论在定性上能合宜地描述二级相变，诸如铁电和铁磁中的很多现象．但在临界点附近某些序参量的涨落异常的大，此时Ｌａｎｄａｕ相变理论被Ｗｉｌｓｏｎ的重整化群理论代替．
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ginzburg-landau方程

ginzburg-landau方程
Ginzburg–Landau方程是一种在物理学中常见的方程，用于描述多体力学系统的时变特性，通常应用于液体、固体和气体中的物理过程。

Ginzburg–Landau方程是一种非线性椭圆方程，它研究了一个有自由参数的系统极限后的演化规律，可以比较准确地描述这种细微力学系统的变化状态，它可以用来模拟细微和复
杂的物理系统。

它主要通过允许有随机变化的非线性参数来模拟真实世界中的细微动态，
再通过乱序的散度等方法，无缝地合成出物理宏观现象中的演化过程。

Ginzburg–Landau方程在研究多体动力学物理方面有着显著的成就。

它在物理上的应用非常广泛包括熔融凝固耦合的物理过程，液体「拐角」问题，以及量子统计力学中粒子间相
互作用等。

这个方程被用在许多领域，如超流体动力学中的超流体撞击，复杂系统动力学
中的细微结构演化，统计物理学中的费米能级及磁化性等等。

因此，Ginzburg–Landau方程可以精确描述一个细微的物理系统的时变特性，可以帮助我们更好地理解多体动力学系统中复杂的结构。

它还可以作为一个模型，用于预测物理系统
中复杂行为的演化过程，为深入研究物理过程带来很大的可能性。

金兹堡朗道理论

Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。

Ginzburg-Landau方程的椭圆函数解与包络孤立波解

ａ一２ｌ＋ａ一（。ｓ＋ｉｐ＋（ａｋｙ＋卢卢是２ｄｋ＋口）［２ — ）ｚ
令方程（）的实部和虚部为零，６得
口一２忌＋ａ一（１＋口）一０１ｌ２口ｋ３，
一
（ ∞＋卢ｋ＋卢）］一０。ｓ．
圆方程。得ＧｉｚｕｇＬｎａ方程的多种椭圆函数解，极限情形可以还原为经典的包络孤立波解．求ｎｂｒ—ａｄｕ其关键词：ｎｂｒ— ａｄｕ程；椭圆函数解；络孤立波解ＧｉｚｕｇＬｎａ方包中图分类号：７．９０１５２文献标识码：Ａ文章编号：１０－７５２１）１００－４０１８３（０００ — ０７０
ｉ８ｕ
。
＋－ｉ券＋２ｉｌ“（ｉｏ（ｆ（－）Ｉ口Ｂ一，口ｌ＋ａｆ一ｓ。）ｌｕｑ２＋）。
Ａｅｚ，
（１）
（）２
其中ａ，（一１２３是实常数．献［］同时指出，程（）有如下形式的单波解：，，）文２方１具
基金项目：国家自然科学基金资助项目（０６０５；内蒙古自然科学基金资助项目（０９００）１７１０）２０ＭＳ１２
作者简介：杨丽英（９７）女，１６一，内蒙古包头市人，内蒙古农业大学酎教授，要从事复变函数和非线性发展方程研究．主
第３９卷第ｌ期

高维各向异性Ginzburg—Landau方程的渐近行为

（淮海工学院理学院，苏连云港江２２０）２０５
摘要：究了高维各向异性ＧｉｚｕｇＬｎａ研ｎｂｒ— ａｄｕ方程在有界单连通区域上的光滑解，能量在Ｅ（。 ≤ Ｍ０ＪｏＪ）ｌｇｅ的假设务件下当ｅ于零时解的渐近行为，画了其涡旋集的结构，出了内趋刻给部能量单调公式．在此基础上利用Ｈｏｇ－ｅＲａ分解证明了椭圆性．ｄｅｄｈｍ关键词：能量单调公式；矿椭圆性；涡旋；ｉｚｕｇＬｎａＧｎｂｒ — ａｄｕ方程中图分类号：７．０１５２文献标识码：Ａ
第ｌ卷第４期７２００８年ｌ２月
淮海工学院学报（自然科学版）
ＪｕｎｌｆＨｕｉａｎｔｕｅｏｃｎｌｇ（ｔｒＩｃｅｃｄｔｎｏｒａａｈｉｓｉｔｆＴｅｈｏｏｙＮａｕａＳｉｅＥｉｏ）ｏＩｔｎｉ
全局极小解且满足如下Ｇｎｂｒ— ａｄｕ方程：ｉｚｕｇＬｎａ
－
表示切向微分，０Ｃ为不依赖于￡常数．的
如果ｎ是三维空间时，（）是各向异性超导式１ＧｉｚｕｇＬｎａｎｂｒ — ａｄｕ模型的简化形式，实际的物理有意义．时，ＩＩ表示超导Ｃｏｅ电子对的密度，此ｏｐｒＩｌ。≈ １对应于超导状态，１ｌ ≈ ０应于普通对
ｓｓｅｏｈｕｅｃｎｕｔｉ．Ｗｅｐｏｅｔｅｍｏｏｏｉｉｙｆｒｌｓｏｎｒｙａｄｔｅｒｅｌｔｃｙｔｍｆｔｅｓｐｒｏｄｃｉｔｖｙｒｖｈｎｔｎｃｔｏｍｕａｆｅｅｇｎｈ－ｌｐｉ— ／ｉｉｙｂｈｌｓｉａｏｇｈｏｙｔｙｔｅｃａｓｃｌＨｄｅｔｅｒ．Ｋｅｒｓｙｗｏｄ：ｍｏｏｏｉｉｙｆｒｌｓＦｌｐｉｉｙ；ｖｒｉｅ；ＧｉｚｕｇＬｎａｑａｉｎｎｔｎｃｔｏｍｕａ；ｒｅｌｔｃｔｉｏｔｓｃｎｂｒ — ａｄｕｅｕｔｏ