Exact solutions to complex Ginzburg–Landau equation

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 Ginzburg–Landau equation. Both the Kerr and power laws of nonlinearity are considered. All the possible exact travelling wave solutions consisting of the rational function-type solutions, solitary wave solutions, triangle function-type periodic solutions and Jacobian elliptic functions solutions are obtained, and some of them are new solutions. In addition, concrete examples are presented to ensure the existence of obtained solutions. Moreover, four types of representative solutions are depicted to present the nature of the obtained solutions.     

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, $L^infin(Q_L). We show that the Kolmogorov l-entropy per unit length, H_\epsilon =\lim_{L\to\infty} L^{-d} \logtwo N_{Q_L}(\epsilon)< /FORMULA > exists. In particular, we bound < FORMULA FORM=ÏNLINE» exists. In particular, we bound H_\epsilon< /FORMULA > by < FORMULA FORM=ÏNLINE» by \OO\bigl(\logtwo(1/\epsilon )\bigr)< /FORMULA > , which shows that the attracting set is < SMALL > smaller < /SMALL > than the set of bounded analytic functions in a strip. We finally give a positive lower bound: < FORMULA FORM=ÏNLINE», which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H_\epsilon>\OO\bigl (\logtwo(1/\epsilon)\bigr)$.     

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.     

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     

Analysis of Thermodynamics of Liquid d— and f—Shell Metals with the Variational Approach

We use reently proposed potential to calculate internal energy (enthalpy), entropy and Helmholtz free energy of liquid d- and f-shell metals with the variational approach. The parameter of the potential is determined with the standard zero pressure condition along with well established Taylor screening function for exchange and correlation effects to the liquid d- and f-shell metals. Here we clearly mention that the parameter of the potential is independent of any fitting procedure either with any experimental data or any theoretical values of any physical properties. The structure factor derived by Percus-Yevick solution for hard sphere fluids, which is characterized by hard sphere diameter, is used. A good agreement between theoretical investigations and experimental findings has confirmed the ability of the model potential to the liquid d- and f-shell metals.     

Time-Dependent Variational Approach to the Phonon Dispersion Relation of the Commensurate Quantum Frenkel-Kontorova Model

The phonon dispersion relation of the commensurate quantum Frenkel-Kontorova model is studied by means of the time-dependent variational approach combined with a Hartree-type many-body trial wavefunction for the particles. The single-particle state is taken to be a frozen Jackiw-Kerman wavefunction. Under the condition of minimum uncertainty, equations of motion for the particle expectation values are derived to obtain the phonon dispersion relation. It is shown that the strength of the substrate potential and the phonon excitation gap are reduced due to the quantum fluctuations in comparison with those of the classical model. We also compare our results with those previously obtained by using the path-integral molecular dynamics.     

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     

Instability of a triangular Abrikosov lattice at values of the Ginzburg–Landau parameter κ close to unity

The “soft” transverse mode of gapless excitations related to the deformation of a triangular Abrikosov lattice with a single flux quantum per unit cell at an arbitrary value of the Ginzburg–Landau parameter κ is investigated. An Abrikosov lattice with the angle φ = π/3 between the unit cell vectors is shown to be unstable in a narrow range of values, 1 < κ < 1.000634. The excitation spectrum of the mode under consideration at low values of the momentum k (in the k² approximation) is isotropic at k lying in a plane perpendicular to the magnetic field.     

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.     

Properties of parallel upper critical field within continuous Ginzburg–Landau model

In this paper, we employ a continuous Ginzburg–Landau model to study the behaviors of the parallel upper critical field of an intrinsically layered superconductor. Near T_c where the order parameter is nearly homogeneous, the parallel upper critical field is found to vary as (1−T/T_c)^1/2. With a well-localized order parameter, the same field temperature dependence holds over the whole temperature range. The profile of the order parameter at the parallel upper critical field is of a Gaussian type, which is consistent with the usual Ginzburg–Landau theory. In addition, the influences of the unit cell dimension and the average effective masses on the parallel upper critical field and the associated order parameter are also addressed.     

Variational Approach to Study PT-Symmetric Solitons in a Bose–Einstein Condensate with Non-locality of Interactions

Considering the non-locality of interactions in a Bose–Einstein condensate, the existence and stability of solitons subject to a PT-symmetric potential are discussed. In the framework of the variational approach, we investigate how the non-locality of interactions affects the self-localization and stability of a condensate with attractive two-body interactions. The results reveal that the non-locality of interactions dramatically influences the shape,width, and chemical potential of the condensate. Analytically variational computation also predicts that there exists a critical negative non-local interaction strength(p_c 0) with each fixed two-body interaction(g_0 0),and there exists no bright soliton solution for p_0 p_c. Furthermore, we study the effect of the non-locality interactions on the stability of the solitons using the Vakhitov–Kolokolov stability criterion. It is shown that for a positive non-local interaction(p_0 0), there always exist stable bright solitons in some appropriate parameter regimes.     

The Ginzburg–Landau theory and the surface energy of a colour superconductor

We apply the Ginzburg–Landau theory to the colour superconducting phase of a lump of dense quark matter. We calculate the surface energy of a domain wall separating the normal phase from the super phase with the bulk equilibrium maintained by a critical external magnetic field. Because of the symmetry of the problem, we are able to simplify the Ginzburg–Landau equations and express them in terms of two components of the di-quark condensate and one component of the gauge potential. The equations also contain two dimensionless parameters: the Ginzburg–Landau parameter κ and ρ. The main result of this paper is a set of inequalities obeyed by the critical value of the Ginzburg–Landau parameter—the value of κ for which the surface energy changes sign—and its derivative with respect to ρ. In addition we prove a number of inequalities of the functional dependence of the surface energy on the parameters of the problem and obtain a numerical solution of the Ginzburg–Landau equations. Finally a criterion for the types of colour superconductivity (type I or type II) is established in the weak coupling approximation.     

Ginzburg–Landau description of laminar-turbulent oblique band formation in transitional plane Couette flow

Plane Couette flow, the flow between two parallel planes moving in opposite directions, is an example of wall-bounded flow experiencing a transition to turbulence with an ordered coexistence of turbulent and laminar domains in some range of Reynolds numbers [R _g, R _t] . When the aspect-ratio is sufficiently large, this coexistence occurs in the form of alternately turbulent and laminar oblique bands. As R goes up trough the upper threshold R _t, the bands disappear progressively to leave room to a uniform regime of featureless turbulence. This continuous transition is studied here by means of under-resolved numerical simulations understood as a modelling approach adapted to the long time, large aspect-ratio limit. The state of the system is quantitatively characterised using standard observables (turbulent fraction and turbulence intensity inside the bands). A pair of complex order parameters is defined for the pattern which is further analysed within a standard Ginzburg–Landau formalism. Coefficients of the model turn out to be comparable to those experimentally determined for cylindrical Couette flow.     

Some Features of Critical Parameters Calculations of Nonhomogeneous Superconducting Films Using the Ginzburg–Landau Theory

Bulletin of the Lebedev Physics Institute - A method for calculating the critical state of inhomogeneous superconducting films using the Ginzburg–Landau (GL) theory is proposed. From the...     

Quantum critical properties in the topological Ginzburg–Landau theory of self-dual Josephson junction arrays

This paper examines the multicritical behavior of a generalized U(_N1)×U(_N2)$U\left(N_1\right)\times U\left(N_2\right)$ Ginzburg–Landau theory containing two multicomponent complex fields which couple differently to two gauge fields described by two Maxwell terms and one mixed-Chern–Simons term. This model is relevant to the dynamics of Cooper pairs and vortices in a self-dual Josephson junction array system near its superconductor–insulator transition. We develop a renormalization group flow at fixed dimension and obtain the beta functions at one loop when both disorder fields are critical. Two sets of infrared-stable charged fixed points solutions are found for N>_Nc$N>N_c$: partially charged solutions with respect to the gauge fields exist with _Nc=35.6$N_c=35.6$, and fully charged solutions exist with _Nc=12.16$N_c=12.16$. We show that fine tuning the ratio of the two energy scales in the model has the effect of reducing the critical number _Nc$N_c$ and thus enlarges the region where the quantum phase transition is continuous. It is also found that the decoupled fixed point which is stable in the neutral case is no longer attainable in the presence of fluctuating gauge fields. We probe the conductivity at the critical point and show that it has a universal character determined by the renormalization group infrared-stable fixed-point values of the gauge couplings.     

Double Hopf bifurcation induces coexistence of periodic oscillations in a diffusive Ginzburg–Landau model

We perform bifurcation analysis in a complex Ginzburg–Landau system with delayed feedback under the homogeneous Neumann boundary condition. We calculate the amplitude death region, and it turns out that the boundary of the amplitude death region consists of two Hopf bifurcation curves with wave number zero. The existence conditions for double Hopf bifurcations are established. Taking the feedback strength and time delay as bifurcation parameters, normal forms truncated to the third order at double Hopf singularity are derived, and the unfolding near the critical points is given. The bifurcation diagram near the double Hopf bifurcation is drawn in the two-parameter plane. The phenomena of amplitude death, the existence of stable bifurcating periodic solutions, and the coexistence of two stable periodic solutions with fast oscillation and slow oscillation respectively are simulated.