Two-Frequency Autowave Processes in the Complex Ginzburg–Landau Equation

We study the complex Ginzburg–Landau equation with zero Neumann boundary conditions on a finite interval and establish that this boundary problem (with suitably chosen parameters) has countably many stable two-dimensional self-similar tori. The case of periodic boundary conditions is also investigated.     

三维全空间上Ginzburg Landau方程的整体吸引子

作者在三维全空间中考虑研究复Ginzburg Landau方程(CGL)的解的长时间行为。通过引入权空间，应用内插不等式和在权空间的先验估计，获得复 Ginzburg Landau方程整体解的存在性，进一步建立了整体吸引子的存在性。     

复Ginzburg Landau方程的自相似解及其极限行为

该文证明了复Ginzburg Landau方程在非标准的函数空间X_{s,p}中整体解的存在唯一性；考察了其解在X_{0,α+2}中的极限行为，得到当参数ε→0++或a→0, ε→0++时，Ginzburg Landau方程的解关于时间一致收敛到相应极限方程的解     

广义Ginzburg-Landau方程和Rangwala Rao方程的显式精确解

该文通过适当代换并结合假设待定法，求出了具高阶非线性项的Liénard方程a″(ξ)+la(ξ)+ma\+\{2p+1\}(ξ)+na\+\{4p+1\}(ξ)=0的三类精确解. 据此求出了广义Ginzburg Landau方程、Rangwala Rao方程及若干 导数schr〖AKo¨D〗dinger型方程的孤波解和三角函数型周期波解.     

Controllability of the Ginzburg–Landau equation

This Note investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Null-controllability results are derived from a Carleman estimate and an analysis based upon the theory of sectorial operators. To cite this article: L. Rosier, B.-Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the equilibrium position of Ginzburg Landau vortices

We study a few problems related to superconducting vortices. Our main concern is the stable equilibrium distribution of them. Our starting point is the asymptotic form of the Ginzburg Landau energy functional with a large Ginzburg Landau parameter. We consider in particular the interaction of the vortices with an applied magnetic field, and the effects of impurities on the vortex locations.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

Magnetic Ginzburg–Landau functional with discontinuous constraint

This Note reports on results obtained for minimizers of a Ginzburg–Landau functional with discontinuous constraint. These results concern vortex-pinning and boundary conditions for inhomogeneous superconducting samples. To cite this article: A. Kachmar, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Dynamics of multiple degree Ginzburg–Landau vortices

For the two-dimensional complex parabolic Ginzburg–Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff–Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we describe carefully. The only assumption is a natural energy bound on the initial data. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Soliton-like solutions for the modified variable-coefficient Ginzburg–Landau equation

This paper is devoted to studying the solutions of the modified Ginzburg–Landau equation with variable coefficient. Based on computerized symbolic computation, several families of new exact dark- and bright-soliton-like solutions are presented. Moreover, we derive some similarity solutions, which can be illustrated in terms of the elliptic and the second kind of Painlevé transcendent equation.     

Limiting properties of the second order Ginzburg–Landau minimizers

The author studies the weak convergence for the gradient of the minimizers for a second order energy functional when the parameter tends to 0. And this paper is also concerned with the location of the zeros and the blow-up points of the gradient of the minimizers of this functional. Finally, the strong convergence of the gradient of the radial minimizers is obtained.     

On Ginzburg-Landau Vortices of Superconducting Thin Films

In this paper, we discuss the vortex structure of the superconducting thin films placed in a magnetic field. We show that the global minimizer of the functional modelling the superconducting thin films has a bounded number of vortices when the applied magnetic field hex 〈 Hc1 ＋ K log ｜ logε｜ where Hc1 is the lower critical field of the film obtained by Ding and Du in SIAM J. Math. Anal., 2002. The locations of the vortices are also given.     

The mixed boundary condition for the Ginzburg Landau model in thin films

The mixed boundary condition for the Ginzburg Landau model of superconductivity is considered in thin films. A simplified model is derived in the asymptotic limit of very small thickness. We also show that under certain conditions there is no nucleation of superconductivity at all.     

Lines of vortices for solutions of the Ginzburg–Landau equations

For disc domains and for periodic models, we construct solutions of the Ginzburg–Landau equations which verify in the limit of a large Ginzburg–Landau parameter specified qualitative properties: the limit density of the vortices concentrates on lines.     

Adiabatic limit in Ginzburg—Landau and Seiberg—Witten equations

Theoretical and Mathematical Physics - We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ?1+2 and Seiberg—Witten equations on four-dimensional...     

Nonconforming quadrilateral finite element method for Ginzburg–Landau equation

Nonconforming finite element method is studied for a linearized backward fully-discrete scheme of the Ginzburg–Landau equation with the quadrilateral element. The unconditional convergent result of order O(h + τ) in the broken H¹-norm is deduced rigorously based on a splitting technique, by which the ratio between the subdivision parameter h and the time step τ is removed. Furthermore, numerical results are provided to confirm the theoretical analysis. The analysis developed herein can be regarded as a framework to deal with the unconditional convergent analysis of the Ginzburg–Landau equation for other known low order nonconforming elements.     

Ginzburg–Landau equations on Riemann surfaces of higher genus

We study the Ginzburg–Landau equations on Riemann surfaces of arbitrary genus. In particular, we

• –construct explicitly the (local moduli space of gauge-equivalent) solutions in the neighborhood of the constant curvature ones;
• –classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel–Jacobi map, and symmetric products of the surface;
• –determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.