Complex Ginzburg–Landau Equation with Absorption: Existence,Uniqueness and Localization Properties

In this paper we study the time-dependent complex Ginzburg–Landau equation with a nonlinear absorbing term in ${\Omega \times(0,T),\, \Omega }$ open bounded set in ${\mathbb{R}^{n}}$ . We prove global existence and uniqueness of solutions for the initial and boundary-value problem and study the properties of localization and extinction of solutions in some special cases.     

Optical vortices in the Ginzburg–Landau equation with cubic–quintic nonlinearity

In a dissipative system with cubic–quintic nonlinearity, the curious evolution of optical vortex beams characterized by different topological charges (TCs) is simulated numerically and presented their evolution profiles. We find that new vortices will be induced during propagation, and the behavior of vortices, as affected by the TC and the number of beads of the incident beam, as well as its size, is also discussed. Common rules associated with the initial conditions coming from various incident beams are developed to determine the number of induced vortices and the corresponding rotation direction. Attributed to the nonlinearity, during propagation we see the beams slowly expand to induce new vortices, which commonly appear in oppositely charged pairs, while the net topological charge of the vortex is conserved. Our results not only deepen the understanding of optical vortices, but also widen their potential applications.     

Boundary Behavior of the Ginzburg–Landau Order Parameter in the Surface Superconductivity Regime

We study the 2D Ginzburg--Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order (in L ² norm) by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address the question of whether this approximation can hold uniformly in the boundary region. We prove that this is indeed the case as a corollary of a refined, second order energy expansion including contributions due to the curvature of the sample. Local variations of the GL order parameter are controlled by the second order term of this energy expansion, which allows us to prove the desired uniformity of the surface superconductivity layer.     

Convergence of Ginzburg–Landau Functionals in Three-Dimensional Superconductivity

In this paper we consider the asymptotic behavior of the Ginzburg–Landau model for superconductivity in three dimensions, in various energy regimes. Through an analysis via Γ-convergence, we rigorously derive a reduced model for the vortex density and deduce a curvature equation for the vortex lines. In the companion paper (Baldo et al. Commun. Math. Phys. 2012, to appear) we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field H _c1, and the critical angular velocity of rotating Bose–Einstein condensates.     

The Ginzburg–Landau Functional with Vanishing Magnetic Field

Archive for Rational Mechanics and Analysis - We study the infimum of the Ginzburg–Landau functional in a two dimensional simply connected domain and with an external magnetic field allowed...     

A nonlocal phase-field model of Ginzburg–Landau–Korteweg fluids

A thermodynamic model of Korteweg fluids undergoing phase transition and/or phase separation is developed within the framework of weakly nonlocal thermodynamics. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure recently introduced in the literature. Possible forms of the free energy and of the stress tensor, which generalize some earlier ones proposed by several authors in the last decades, are carried out. Owing to the new procedure applied for exploiting the entropy principle, the thermodynamic potentials are allowed to depend on the whole set of variables spanning the state space, including the gradients of the unknown fields, without postulating neither the presence of an energy or entropy extra-flux, nor an additional balance law for microforce.     

Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers

Nonlinear Dynamics - This paper presents a study on the energy exchange taking place on articulated helicopter main rotor blades. The blades are hinged, and the flap/lag modes are highly coupled....     

Multi-Vortex Solutions to Ginzburg–Landau Equations with External Potential

We consider the existence of multi-vortex solutions to the Ginzburg–Landau equations with external potential on \mathbbR²{\mathbb{R}^2} . These equations model equilibrium states of superconductors and stationary states of the U(1) Higgs model of particle physics. In the former case, the external potential models impurities and defects. We show that if the external potential is small enough and the magnetic vortices are widely spaced, then one can pin one or an arbitrary number of vortices in the vicinity of a critical point of the potential. In addition, one can pin an arbitrary number of vortices near infinity if the potential is radially symmetric and of an algebraic order near infinity.     

Existence of 3-Dimensional Tori for 1D Complex Ginzburg–Landau Equation Via a Degenerate KAM Theorem

In this paper, we consider complex Ginzburg–Landau equation in one space dimension and rigorously show the existence of 3-dimensional tori. The proof is based on degenerate infinite-dimensional KAM theory and normal form technique.     

On the Shape of Meissner Solutions to a Limiting Form of Ginzburg–Landau Systems

In this paper we study a semilinear system involving the curl operator, which is a limiting form of the Ginzburg–Landau model for superconductors in \({{\mathbb{R}}^3}\) for a large value of the Ginzburg–Landau parameter. We consider the locations of the maximum points of the magnitude of solutions, which are associated with the nucleation of instability of the Meissner state for superconductors when the applied magnetic field is increased in the transition between the Meissner state and the vortex state. For small penetration depth, we prove that the location is not only determined by the tangential component of the applied magnetic field, but also by the normal curvatures of the boundary in some directions. This improves the result obtained by Bates and Pan in Commun. Math. Phys. 276, 571–610 (2007). We also show that the solutions decay exponentially in the normal direction away from the boundary if the penetration depth is small.     

Ginzburg–Landau modelling of precursor nanoscale textures in ferroelastic materials

A Ginzburg–Landau free-energy model is proposed to study spatially inhomogeneous states that often occur as precursors of ferroelastic/martensitic transitions. Disorder is included in the harmonic coefficient of the free-energy density which gives rise to a spatial distribution of transition temperatures, and lattice integrity is imposed through Saint-Vénant compatibility conditions which lead to a long-range anisotropic elastic interaction. We show that precursor textures are a result of the competition between elastic anisotropy and disorder. Cross-hatched modulations (tweed patterns) take place for temperatures above the martensitic phase in the limit of high anisotropy and/or low disorder while a nano-cluster phase-separated state occurs at low anisotropies or high disorder. In the latter case, nanoscale inhomogeneities give rise to glassy behaviour while the structural transition is inhibited. Interestingly, in this case, the ferroelastic system also displays a large thermo-mechanical response so that the low-symmetry structure can be easily induced by the application of relatively small stresses within a broad temperature range.     

On Asymptotic Stability of Kink for Relativistic Ginzburg–Landau Equations

We prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the Ginzburg–Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of the kink and dispersive part described by the free Klein–Gordon equation. The remainder converges to zero in a global norm.     

On the First Critical Field in Ginzburg–Landau Theory for Thin Shells and Manifolds

In this article, we investigate the response of a thin superconducting shell to an arbitrary external magnetic field. We identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H _c1 in Ginzburg–Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a simply connected surface of revolution and vertical and constant field, we further determine the exact number of vortices in the sample as the intensity of the applied field is raised just above H _c1. Finally, we derive via Γ-convergence similar statements for three-dimensional domains of small thickness, where in this setting point vortices are replaced by vortex lines.