Extensive Properties of the Complex Ginzburg–Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in R^d, d <3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube Q_L of side L. We cover this set by a (minimal) number N_QL(l) of balls of radius l in $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, H_\epsilon =\lim_{L\to\infty} L^{-d} \logtwo N_{Q_L}(\epsilon)< /FORMULA > exists. In particular, we bound < FORMULA FORM=ÏNLINE» exists. In particular, we bound H_\epsilon< /FORMULA > by < FORMULA FORM=ÏNLINE» by \OO\bigl(\logtwo(1/\epsilon )\bigr)< /FORMULA > , which shows that the attracting set is < SMALL > smaller < /SMALL > than the set of bounded analytic functions in a strip. We finally give a positive lower bound: < FORMULA FORM=ÏNLINE», which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H_\epsilon>\OO\bigl (\logtwo(1/\epsilon)\bigr)$.     

Inviscid Limits¶of the Complex Ginzburg–Landau Equation

In the inviscid limit the generalized complex Ginzburg–Landau equation reduces to the nonlinear Schr?dinger equation. This limit is proved rigorously with H ¹ data in the whole space for the Cauchy problem and in the torus with periodic boundary conditions. The results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of L ² in the focusing case, in any spatial dimension. Furthermore, optimal convergence rates are proved. The proofs are based on estimates of the Schr?dinger energy functional and on Gagliardo–Nirenberg inequalities. Received: 2 April 1999 / Accepted: 29 March 2000     

Block model for the XY-type Landau–Ginzburg–Wilson Hamiltonian with random temperature

The phase fluctuation near the saddle point solution of the XY-type Landau–Ginzburg–Wilson Hamiltonian with random temperature is studied. Through some examples, it is argued that the systems are self-organized into blocks, which are coupled as a XY model with random bond. The couplings obtained in this way agree with those by the domain wall method.     

Analysis of Energy Eigenvalue in Complex Ginzburg–Landau Equation

In this paper, we consider the two-dimensional complex Ginzburg–Landau equation(CGLE) as the spatiotemporal model, and an expression of energy eigenvalue is derived by using the phase-amplitude representation and the basic ideas from quantum mechanics. By numerical simulation, we find the energy eigenvalue in the CGLE system can be divided into two parts, corresponding to spiral wave and bulk oscillation. The energy eigenvalue of spiral wave is positive, which shows that it propagates outwardly; while the energy eigenvalue of spiral wave is negative, which shows that it propagates inwardly. There is a necessary condition for generating a spiral wave that the energy eigenvalue of spiral wave is greater than bulk oscillation. A wave with larger energy eigenvalue dominates when it competes with another wave with smaller energy eigenvalue in the space of the CGLE system. At the end of this study, a tentative discussion of the relationship between wave propagation and energy transmission is given.     

Spectral Analysis of Weakly Coupled Stochastic Lattice Ginzburg–Landau Models

We consider the relaxation to equilibrium of solutions , t>0, , of stochastic dynamical Langevin equations with white noise and weakly coupled Ginzburg–Landau interactions. Using a Feynman–Kac formula, which relates stochastic expectations to correlation functions of a spatially non-local imaginary time quantum field theory, we obtain results on the joint spectrum of H, , where H is the self-adjoint, positive, generator of the semi-group associated with the dynamics, and P ^j , j= 1, …, d are the self-adjoint generators of the group of lattice spatial translations. We show that the low-lying energy-momentum spectrum consists of an isolated one-particle dispersion curve and, for the mass spectrum (energy-momentum at zero-momentum), besides this isolated one-particle mass, we show, using a Bethe–Salpeter equation, the existence of an isolated two-particle bound state if the coefficient of the quartic term in the polynomial of the Ginzburg–Landau interaction is negative and d= 1, 2; otherwise, there is no two-particle bound state. Asymptotic values for the masses are obtained. Received: 27 September 2000 / Accepted: 16 January 2001     

Time-dependent Ginzburg–Landau equations for multi-gap superconductors

We numerically solve the time-dependent Ginzburg–Landau equations for two-gap superconductors using the finite-element technique. The real-time simulation shows that at low magnetic field, the vortices in small-size samples tend to form clusters or other disorder structures. When the sample size is large, stripes appear in the pattern. These results are in good agreement with the previous experimental observations of the intriguing anomalous vortex pattern, providing a reliable theoretical basis for the future applications of multi-gap superconductors.     

A stochastic Ginzburg–Landau equation with impulsive effects

In this paper we consider a stochastic Ginzburg–Landau equation with impulsive effects. We first prove the existence and uniqueness of the global solution which can be explicitly represented via the solution of a stochastic equation without impulses. Then, based on our obtained result, we study the qualitative properties of the solution, including the boundedness of moments, almost surely exponential convergence and pathwise estimations. Finally, we give a first attempt to study a fractional version of impulsive stochastic Ginzburg–Landau equations.     

Eigenstates of the linearlized Ginzburg–Landau equation for mesoscopic multiply-connected superconductor

Multiply-connected mesoscopic superconductors have rich structures of vortex systems that result from interference of order parameter. We studied magnetic field dependence of transition temperatures and vortex arrangements of finite sized honeycomb superconducting networks with 6-fold rotational symmetries. Near and above the lower critical field, vortices locate at center of the network. As increasing the field, vortices form a hexagon or hexagonal multi-shell structure. In higher field, order parameter damps exponentially from the central point of the network to the edge of the network.     

New Exact Solutions for the Wick-Type Stochastic Kudryashov–Sinelshchikov Equation

In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.     

Time-dependent Ginzburg–Landau equation in the Nambu–Jona-Lasinio model

We apply the closed time-path Green function formalism in the Nambu–Jona-Lasinio model. First of all, we use this formalism to obtain the well-known gap equation for the quark condensate in a stationary homogeneous system. We have also used this formalism to obtain the Ginzburg–Landau (GL) equation and the time-dependent Ginzburg–Landau (TDGL) equation for the chiral order parameter in an inhomogeneous system. In our derived GL and TDGL equations, there is no other parameters except for those in the original NJL model.     

Electron Paramagnetic Resonance and the Modified Landau–Lifshitz Equation in Strongly Correlated Electronic Systems with Quantum Fluctuations of the Magnetic Moment

Doklady Physics - The Landau–Lifshitz equation modified by quantum fluctuations of the magnetic moment is proposed, on the basis of which several new effects in electron paramagnetic...     

Optical solitons with complex Ginzburg–Landau equation for two nonlinear forms using F-expansion

This paper implements F-expansion scheme to obtain Jacobi’s elliptic function to complex Ginzburg–Landau equation with two nonlinear forms. In the limiting case of the modulus of ellipticity, bright and dark soliton solutions emerge.     

Landau–Ginzburg theory for ‘star’-shaped droplets

ABSTRACT

Molecular simulations have shown that when a nano-drop comprising a single spherical central ion and a dielectric solvent is charged above a well-defined threshold, it acquires a stable star morphology. A linear continuum model of the ‘star’-shapes comprised electrostatic and surface energy is not sufficient to describe these shapes. We employ combined molecular dynamics, continuum electrostatics and macroscopic modelling in order to construct a unified free energy functional that describes the observed star-shaped droplets. We demonstrate that the Landau free energy coupled to the third-order Steinhardt invariant mimics the shapes of droplets detected in molecular simulations. Using the maximum likelihood technique we build a universal free energy functional that describes droplets for a range of Rayleigh fissility parameter. The analysis of the macroscopic free energy demonstrates the origin of the finite amplitude perturbations just above the Rayleigh limit. We argue that the presence of the finite amplitude perturbations precludes the use of the small parameter perturbation method for the analysis of the shapes above the Rayleigh limit of the corresponding spherical shape.     

On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

We present new estimates on the two-dimensional Ginzburg–Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg–Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg–Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg–Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, which is related to a conjecture by Xing-Bin Pan.     

A New Lattice Bhatnagar-Gross-Krook Model for the Convection-Diffusion Equation with a Source Term

A new lattice Bhatnagar-Gross-Krook (LBGK) model for the convection-diffusion equation with a source term is proposed. Unlike the models proposed previously, the present model does not require any additional assumptionon the source term. Numerical results are found to be in excellent agreement with the analytical solutions. It is also found that the numerical accuracy of the model is much better than that of the existing models.     

Optical solitons with complex Ginzburg–Landau equation having three nonlinear forms

This paper secures, dark, singular and dark–singular combo optical soliton solutions to complex Ginzburg–Landau equation that is considered with three nonlinear forms. Two forms of integration architectures provide these solutions.     

On the nature of the phase transition triggered by vortex-like defects in the 2D Ginzburg–Landau model

The two-dimensional lattice Ginzburg–Landau Hamiltonian is simulated numerically for different values of the coherence length ξ in units of the lattice spacing a, a parameter which controls amplitude fluctuations. The phase diagram on the plane T–ξ is measured. Amplitude fluctuations change dramatically the nature of the phase transition: for values of ξ/a≃1, instead of the smooth Kosterlitz–Thouless transition there is a first-order transition with a discontinuity in the vortex density v and a sharper drop in the helicity modulus Γ. Both observables v and Γ are analyzed in detail at the crossover region between first and second order which occurs for intermediate values of ξ/a.     

Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics

In this paper, the(1+1)-dimensional variable-coefficient complex Ginzburg–Landau(CGL) equation with a paritytime(PT) symmetric potential U(x) is investigated. Although the CGL equations with a PT-symmetric potential are less reported analytically, the analytic solutions for the CGL equation are obtained with the bilinear method in this paper. Via the derived solutions, some soliton structures are presented with corresponding parameters, and the influences of them are analyzed and studied. The single-soliton structure is numerically verified, and its stability is analyzed against additive and multiplicative noises. In particular, we study the soliton dynamics under the impact of the PT-symmetric potential. Results show that the PT-symmetric potential plays an important role for obtaining soliton structures in ultrafast optics, and we can design fiber lasers and all-optical switches depending on the different amplitudes of soliton-like structures.