88 research outputs found
Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity
The bifurcation of asymmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg--Landau equations by the methods of formal asymptotics. The behavior of the bifurcating branch depends on the parameters d, the size of the superconducting slab, and , the Ginzburg--Landau parameter. The secondary bifurcation in which the asymmetric solution branches reconnect with the symmetric solution branch is studied for values of for which it is close to the primary bifurcation from the normal state. These values of form a curve in the -plane, which is determined. At one point on this curve, called the quintuple point, the primary bifurcations switch from being subcritical to supercritical, requiring a separate analysis. The results answer some of the conjectures of [A. Aftalion and W. C. Troy, Phys. D, 132 (1999), pp. 214--232]
The bifurcation diagrams for the Ginzburg-Landau system for superconductivity
In this paper, we provide the different types of bifurcation diagrams for a
superconducting cylinder placed in a magnetic field along the direction of the
axis of the cylinder. The computation is based on the numerical solutions of
the
Ginzburg-Landau model by the finite element method. The response of the
material depends on the values of the exterior field, the Ginzburg-Landau
parameter and the size of the domain.
The solution branches in the different regions of the bifurcation diagrams
are analyzed and open mathematical problems are mentioned.Comment: 16 page
Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation : a computer-assisted proof
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
Numerical bifurcation study of superconducting patterns on a square
This paper considers the extreme type-II Ginzburg-Landau equations that model
vortex patterns in superconductors. The nonlinear PDEs are solved using
Newton's method, and properties of the Jacobian operator are highlighted.
Specifically, it is illustrated how the operator can be regularized using an
appropriate phase condition. For a two-dimensional square sample, the numerical
results are based on a finite-difference discretization with link variables
that preserves the gauge invariance. For two exemplary sample sizes, a thorough
bifurcation analysis is performed using the strength of the applied magnetic
field as a bifurcation parameter and focusing on the symmetries of this system.
The analysis gives new insight in the transitions between stable and unstable
states, as well as the connections between stable solution branches.Comment: 31 page
Numerical bifurcation study of superconducting patterns on a square
This paper considers the extreme type-II Ginzburg-Landau equations that model
vortex patterns in superconductors. The nonlinear PDEs are solved using
Newton's method, and properties of the Jacobian operator are highlighted.
Specifically, it is illustrated how the operator can be regularized using an
appropriate phase condition. For a two-dimensional square sample, the numerical
results are based on a finite-difference discretization with link variables
that preserves the gauge invariance. For two exemplary sample sizes, a thorough
bifurcation analysis is performed using the strength of the applied magnetic
field as a bifurcation parameter and focusing on the symmetries of this system.
The analysis gives new insight in the transitions between stable and unstable
states, as well as the connections between stable solution branches.Comment: 31 page
Complex Patterns in Extended Oscillatory Systems
Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenĂŒber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und StabilitĂ€tsanalyse werden InstabilitĂ€ten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der GrenzĂŒbergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklĂ€rt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der KalziumsignalĂŒbertragung in Zellen identifiziert
Continuum theory of partially fluidized granular flows
A continuum theory of partially fluidized granular flows is developed. The
theory is based on a combination of the equations for the flow velocity and
shear stresses coupled with the order parameter equation which describes the
transition between flowing and static components of the granular system. We
apply this theory to several important granular problems: avalanche flow in
deep and shallow inclined layers, rotating drums and shear granular flows
between two plates. We carry out quantitative comparisons between the theory
and experiment.Comment: 28 pages, 23 figures, submitted to Phys. Rev.
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