Phase dynamics in the complex Ginzburg-Landau equation

For αβ>−1, stable time periodic solutions A(X,T)=A_qe^iqX+iω_qT are the locally preferred planform for the complex Ginzburg-Landau equation

     

Well-posedness and dynamics for the fractional Ginzburg-Landau equation

This article studies the global well-posedness and long-time dynamics for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The global existence and some uniqueness criterion of weak solutions are given with compactness method. To study the strong solutions with the semigroup method, we generalize some pointwise estimates for the fractional Laplacian to the complex background and study carefully the linear evolution of the equation. Finally, the existence of global attractors is studied.     

Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation

We consider the dynamics of the Ginzburg-Landau equation in a small neighborhood of a known pulse solution by studying a Poincaré map,P: _{T T} , where _T is a section which is transverse to the pulse. Due to the fact that the Ginzburg-Landau equation possesses both a rotational symmetry and a spatial symmetry, we are able to conduct a detailed analytical study of this map in neighborhoods arbitrarily close to the pulse solution. Thus, we are able to complement the work of Holmes [8], who conducted an analytical study of the Poincaré map in a punctured neighborhood of the pulse. We find that the Poincaré map contains an invariant set _itT, where is not necessarily a Cantor set of points, such thatP: is homeomorphic to a shift map on (at least) two symbols. Furthermore, we find that for eachm 1 the mapP _itm possesses a fixed point. Since is not necessarily a Cantor set, this is not immediately clear. Finally, we find that when the pulse solution is broken, for eachm1 there exist parameter values such that pulses possessingm maxima appear.On leave at the University of Utah during 1993/94. Supported by the DFG, Habilitationsstipendium Ma 1587/1-1.     

The vortex dynamics of a Ginzburg-Landau system under pinning effect

It is proved that the vortices of a Ginzburg-Landau system are attracted by impurities or inhomo-geneities in the super-conducting materials. The strong H1-convergence for the system is also studied.     

The attractor of the stochastic generalized Ginzburg-Landau equation

The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system.Then we prove the random system possesses a global random attractor in H_0~1.     

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem w_t=ℵ₀Δw+ℵ₁w-ℵ₂w|w|²,w|∂Ω₀=0 in the domain Ω₀={(x,y):0 ≤ x ≤ l₁,0 ≤ y ≤ l₂}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵ_j, j=0,1,2, are complex constants withRe ℵ_j > 0. We show that under a rather general choice of the parameters l₁ and l₂, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ₀ → 0 andRe ℵ₀ → 0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 205–220, November, 2000.     

The inviscid limit for the complex Ginzburg-Landau equation

We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrödinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations.     

Ginzburg-Landau equation with magnetic effect in a thin domain

We study the Ginzburg-Landau equation with magnetic effect in a thin domain in , where the thickness of the domain is controlled by a parameter . This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in , we consider a reduced problem as and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in near the solution to the reduced equation if is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros. Received: 11 May 2001 / Accepted: 11 July 2001 /Published online: 19 October 2001     

An elliptic equation under the effect of two nonlocal terms

The aim of this note is to investigate the existence of signed and sign‐changing solutions to the Kirchhoff type problem (0.1) where Ω is a bounded smooth domain in (N = 1,2,3), a,b > 0 and 2 < p < 2^?, with 2^?=+∞ if N = 1,2 and 2^?=6 if N = 3. Using variational methods, we show that (0.1) possesses three solutions of mountain pass type (one positive, one negative and one sign‐changing) and infinitely many high‐energy sign‐changing solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

Spatiotemporal complexity of the cubic Ginzburg-Landau equation

The linear dispersive relation of the travelling-wave solution is investigated for cubic G-L equation. Moreover, the relation among the parameter c₀, the amplitude |μ_o| and the most unstable wave number q is discussed. Then convergence of an unconditionally stable, explicit pseudo-spectral scheme is proved by energy estimates. Finally, by using the proposed scheme, the chaotic attractor, bifurcation structure and asymptotic dynamics are obtained. The results show there exist two different types of chaotic attractors for the most unstable wave number q_o and was fixed the amplitude |μ_o| in the same one system.     

Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. 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. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

Stability of -vortices in the Ginzburg-Landau equation

We consider the class of -vortex solutions to the time-independent Ginzburg-Landau equation on . We prove an inequality governing the solutions of a particular boundary value problem. This inequality is crucial for an elementary proof by Ovchinnikov and Sigal that such -vortices are unstable in the case .     

Slow Time-periodic solutions of the cubic-quinitic Ginzburg-Landau equation

In this paper, the Ginzburg-Landau equation with small complex coefficients is considered. A translation is introduced to transform the Ginzburg-Landau equation into a dynamical system. Moreover, the existence and the properties of the equilibria are discussed. The spatial quasiperiodic solutions disappear due to the perturbation are proved. Finally, several types of heteroclinic orbits are proposed and numerical analysis are provided.     

Inviscid limit for the energy-critical complex Ginzburg-Landau equation

In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg-Landau equation in Rn, n?3. In lower dimension case (3?n?6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in , T>0, s=0,1, as a by-product, we get the regularity of solutions in H³ for the nonlinear Schrödinger equation. In higher dimension case (n>6), we get the similar convergent behavior in C(0,T,L²(Rn)). In both cases we obtain the optimal convergent rate.     

Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch

Generalizing previous results of M. Comte and P. Mironescu, it is shown that for degree d large enough (such that ), there is a bifurcation branch in the set of the solutions of the Ginzburg-Landau equation, emanating from the branch of radial solutions at the critical value _d of the parameter. Moreover, the solutions on the bifurcation branch admit exactly d zeroes, and the energy on the bifurcation branch is strictly smaller than the energy on the radial branch.     

Vortex-lines motion for the Ginzburg-Landau equation with impurity

In this paper,we study the asymptotic behavior of solutions of the Ginzburg-Landau equation with impurity.We prove that,asymptotically,the vortex-lines evolve according to the mean curvature flow with a forcing term in the sense of the weak formulation.