scholarly journals On the global bifurcation diagram for the one-dimensional Ginzburg–Landau model of superconductivity

2000 ◽  
Vol 11 (3) ◽  
pp. 271-291 ◽  
Author(s):  
E. N. DANCER ◽  
S. P. HASTINGS
Keyword(s):  
Boundary Value Problem ◽  
Normal State ◽  
Linear Equations ◽  
Global Bifurcation ◽  
Local Bifurcation ◽  
Lagrange Equations ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional ◽  
The One

Some new global results are given about solutions to the boundary value problem for the Euler–Lagrange equations for the Ginzburg–Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range there is a branch of asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also, simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These proofs require no detailed asymptotics for solution of the linear equations. Finally, an error in Odeh's work on this problem is discussed.

On the minimizers of the Ginzburg–Landau energy for high kappa: the one-dimensional case

1997 ◽  
Vol 8 (4) ◽  
pp. 331-345 ◽  
Author(s):  
AMANDINE AFTALION
Keyword(s):  
Magnetic Field ◽  
Asymptotic Behaviour ◽  
Dimensional Case ◽  
Numerical Computations ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional ◽  
Landau Parameter ◽  
Convergence Results ◽  
The One

The Ginzburg–Landau model for superconductivity is examined in the one-dimensional case. First, putting the Ginzburg–Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg–Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg–Landau energy is analysed and different convergence results are obtained, according to the exterior magnetic field. Numerical computations illustrate the various behaviours.

Global bifurcation structure of a one-dimensional Ginzburg–Landau model

2005 ◽  
Vol 46 (9) ◽  
pp. 095111 ◽  
Author(s):  
Satoshi Kosugi ◽  
Yoshihisa Morita ◽  
Shoji Yotsutani
Keyword(s):  
Global Bifurcation ◽  
Bifurcation Structure ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional

scholarly journals Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof

2014 ◽  
Vol 26 (1) ◽  
pp. 33-60 ◽  
Author(s):  
ANAÏS CORREC ◽  
JEAN-PHILIPPE LESSARD
Keyword(s):  
Contraction Mapping ◽  
Landau Equation ◽  
Computer Assisted ◽  
Nontrivial Solutions ◽  
Lagrange Equations ◽  
Contraction Mapping Theorem ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Computer Assisted Proof ◽  
The One

In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of solutions near numerical approximations. Coexistence of as many as seven nontrivial solutions is proved.

Another look at recent results concerning bifurcation in one-dimensional models of superconductivity

2000 ◽  
Vol 11 (1) ◽  
pp. 121-128 ◽  
Author(s):  
S. P. HASTINGS
Keyword(s):  
Preceding Paper ◽  
Ginzburg Landau ◽  
Landau Model ◽  
One Dimensional ◽  
Dimensional Models ◽  
The One

In the preceding paper of this issue of EJAM, Bolley & Helffer [5] add to their extensive set of results on bifurcation in the one-dimensional Ginzburg–Landau model of superconductivity. One of their new results concerns the direction of bifurcation of symmetric solutions. We give another approach to this problem.

scholarly journals Order parameter profiles in a system with Neumann – Neumann boundary conditions

2018 ◽  
Vol 145 ◽  
pp. 01009 ◽  
Author(s):  
Vassil M. Vassilev ◽  
Daniel M. Dantchev ◽  
Peter A. Djondjorov
Keyword(s):  
Boundary Conditions ◽  
Order Parameter ◽  
Neumann Boundary ◽  
Elliptic Functions ◽  
Thermodynamic System ◽  
Neumann Boundary Conditions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
The One ◽  
Field Plane

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems

2021 ◽  
Vol 31 (01) ◽  
pp. 2150005
Author(s):  
Ziyatkhan S. Aliyev ◽  
Nazim A. Neymatov ◽  
Humay Sh. Rzayeva
Keyword(s):  
Dirac Equation ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Bifurcation From Infinity ◽  
The One ◽  
Nodal Properties

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation

1992 ◽  
Vol 57 (3-4) ◽  
pp. 241-248 ◽  
Author(s):  
B.I. Shraiman ◽  
A. Pumir ◽  
W. van Saarloos ◽  
P.C. Hohenberg ◽  
H. Chaté ◽  
...  
Keyword(s):  
Landau Equation ◽  
Spatiotemporal Chaos ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Ginzburg Landau Equation ◽  
The One
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