Dimensional Ginzburg-Landau Equation Research Articles

Entire solutions to 4 dimensional Ginzburg–Landau equations and codimension 2 minimal submanifolds

We consider the magnetic Ginzburg–Landau equations in R4{−ε2(∇−iA)2u=12(1−|u|2)u,ε2d⁎dA=〈(∇−iA)u,iu〉, formally corresponding to the Euler–Lagrange equations for the energy functionalE(u,A)=12∫R4|(∇−iA)u|2+ε2|dA|2+14ε2(1−|u|2)2. Here u:R4→C, A:R4→R4 and d denotes the exterior derivative acting on the one-form dual to A. Given a minimal surface M2 in R3 with finite total curvature and non-degenerate, we construct a solution (uε,Aε) which has a zero set consisting of a smooth surface close to M×{0}⊂R4. Away from the latter surface we have |uε|→1 anduε(x)→z|z|,Aε(x)→1|z|2(−z2ν(y)+z1e4),x=y+z1ν(y)+z2e4 for all sufficiently small z≠0. Here y∈M and ν(y) is a unit normal vector field to M in R3.

Higher dimensional Ginzburg–Landau equations under weak anchoring boundary conditions

For n≥3 and 0<ϵ≤1, let Ω⊂Rn be a bounded, simply connected, smooth domain, and uϵ:Ω⊂Rn→R2 solve the Ginzburg–Landau equation under the weak anchoring boundary condition:{−Δuϵ=1ϵ2(1−|uϵ|2)uϵinΩ,∂uϵ∂ν+λϵ(uϵ−gϵ)=0on∂Ω, where the anchoring strength parameter λϵ=Kϵ−α for some K>0 and α∈[0,1), and gϵ∈C2(∂Ω,S1). Motivated by the connection with the Landau–De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the asymptotic behavior of uϵ as ϵ goes to zero under the condition that the total modified Ginzburg–Landau energy satisfies Fϵ(uϵ,Ω)≤M|log⁡ϵ| for some M>0.

Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics**Project supported by the National Natural Science Foundation of China (Grant No. 11674036), the Beijing Youth Top-notch Talent Support Program, China (Grant No. 2017000026833ZK08), and the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) (Grant Nos. IPOC2016ZT04 and

In this paper, the (1+1)-dimensional variable-coefficient complex Ginzburg–Landau (CGL) equation with a parity-time (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.

New Exact Solutions of the (2+1)-Dimensional Ginzburg-Landau Equation

A novel identical reforming of differential equation and the high order auxiliary methods are used to construct solitary solutions and periodic solutions of (2 + 1)-Dimensional Ginzburg-Landau equation. It is shown that the high order auxiliary method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.

Dynamics of spiral rings in the three dimensional Ginzburg-Landau equation

In this paper, we show that the spiral rings of the 3D Ginzburg-Landau equation shrink out and disappear in finite time. In contrast with the 2D case in which the interaction between two spirals of opposite charge is exponential. We show analytically and numerically that the spiral rings collapse with a power law due to the continuity of the defect line.

Transitions between Different Stable States in One Dimensional Ginzburg-Landau Equation

The smallest perturbation, which induced the transition between two given stable solutions of one dimensional Ginzburg-Landau equation differing by its wavenumbers, is found in the explicit form

Transitions between Different Stable States in One Dimensional Ginzburg-Landau Equation

The smallest perturbation, which induced the transition between two given stable solutions of one dimensional Ginzburg-Landau equation differing by its wavenumbers, is found in the explicit form.

Dynamics of spiral waves in non-equilibrium media

In this paper the spiral waves in two-dimensional nonequilibrium media are studied on the basis of the two dimensional Ginzburg-Landau equation. In the general case, the perturbation of origin spiral waves leads to the spiral rotation and to the motion of spiral cores. Motion in media with hard and soft excitation is analysed separately.

Critical fields H c2 and H c3 of thin film type II superconductors

One dimensional Ginzburg-Landau equations are solved, using a Laplace transform method, and approximate analytical solutions are obtained. Critical fields H c2 and H c3 of thin film type-II superconductors are calculated.