arXiv subjects

Stan Alama

Publications and source records attributed to Stan Alama.

27 records · Page 2Linked to original sources

arXiv2014

Weak Anchoring for a Two-Dimensional Liquid Crystal

We study the weak anchoring condition for nematic liquid crystals in the context of the Landau-De Gennes model. We restrict our attention to two dimensional samples and to nematic director fields lying in the plane, for which the Landau-De Gennes energy reduces to the Ginzburg--Landau functional, and the weak anchoring condition is realized via a penalized boundary term in the energy. We study the singular limit as the length scale parameter $\varepsilon\to 0$, assuming the weak anchoring parameter $λ=λ(\varepsilon)\to\infty$ at a prescribed rate. We also consider a specific example of a bulk nematic liquid crystal with an included oil droplet and derive a precise description of the defect locations for this situation, for $λ(\varepsilon)=K\varepsilon^{-α}$ with $α\in (0,1]$. We show that defects lie on the weak anchoring boundary for $α\in (0,\frac12)$, or for $α=\frac12$ and $K$ small, but they occur inside the bulk domain $Ω$ for $α>\frac12$ or $α=\frac12$ with $K$ large.

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arXiv2013

Domain walls in the coupled Gross-Pitaevskii equations

A thorough study of domain wall solutions in coupled Gross-Pitaevskii equations on the real line is carried out including existence of these solutions; their spectral and nonlinear stability; their persistence and stability under a small localized potential. The proof of existence is variational and is presented in a general framework: we show that the domain wall solutions are energy minimizing within a class of vector-valued functions with nontrivial conditions at infinity. The admissible energy functionals include those corresponding to coupled Gross--Pitaevskii equations, arising in modeling of Bose-Einstein condensates. The results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall. The methods apply to many systems of interest and integrability is not germane to our analysis. Finally, sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential, thus giving rigorous justification to earlier results in the physics literature.

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arXiv2013

Stability of symmetric vortices for two-component Ginzburg-Landau systems

We study Ginzburg-Landau equations for a complex vector order parameter. We consider the Dirichlet problem in the disk in the plane with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on both the Ginzburg-Landau parameter as well as the sign of the interaction term in the energy.

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arXiv2013

Symmetric vortices for two-component Ginzburg-Landau systems

We study Ginzburg--Landau equations for a complex vector order parameter Psi=(psi_+,psi_-). We consider symmetric (equivariant) vortex solutions in the plane R^2 with given degrees n_\pm, and prove existence, uniqueness, and asymptotic behavior of solutions for large r. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles |psi_+|, |psi_i| are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and supersolution construction and a comparison theorem for elliptic systems.

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arXiv2012

On compound vortices in a two-component Ginzburg-Landau functional

We study the structure of vortex solutions in a Ginzburg-Landau system for two complex valued order parameters. We consider the Dirichlet problem in the disk in R^2 with symmetric, degree-one boundary condition, as well as the associated degree-one entire solutions in all of R^2. Each problem has degree-one equivariant solutions with radially symmetric profile vanishing at the origin, of the same form as the unique (complex scalar) Ginzburg-Landau minimizer. We find that there is a range of parameters for which these equivariant solutions are the unique locally energy minimizing solutions for the coupled system. Surprisingly, there is also a parameter regime in which the equivariant solutions are unstable, and minimizers must vanish separately in each component of the order parameter.

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arXiv2012

Singular Limits for Thin Film Superconductors in Strong Magnetic Fields - Maan Field Model for Thin Films

We consider singular limits of the three-dimensional Ginzburg-Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain has characteristic thickness on the scale $\eps>0$, and we consider the simultaneous limit as the thickness $\eps\rightarrow 0$ and the Ginzburg-Landau parameter $κ\rightarrow\infty$. We assume that the applied field is strong (on the order of $\eps^{-1}$ in magnitude) in its components tangential to the film domain, and of order $\logκ$ in its dependence on $κ$. We prove that the Ginzburg-Landau energy $Γ$-converges to an energy associated with a two-obstacle problem, posed on the planar domain which supports the thin film. The same limit is obtained regardless of the relationship between $\eps$ and $κ$ in the limit. Two illustrative examples are presented, each of which demonstrating how the curvature of the film can induce the presence of both (positively oriented) vortices and (negatively oriented) antivortices coexisting in a global minimizer of the energy.

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arXiv2010

Minimizers of the Lawrence-Doniach Functional with Oblique Magnetic Fields

We study minimizers of the Lawrence--Doniach energy, which describes equilibrium states of superconductors with layered structure, assuming Floquet-periodic boundary conditions. Specifically, we consider the effect of a constant magnetic field applied obliquely to the superconducting planes in the limit as both the layer spacing $s\to 0$ and the Ginzburg--Landau parameter $κ=\eps^{-1}\to\infty$, under the hypotheses that $s=\eps^α$ with $0<α<1$. By deriving sharp matching upper and lower bounds on the energy of minimizers, we determine the lower critical field and the orientation of the flux lattice, to leading order in the parameter $\eps$. To leading order, the induced field is characterized by a convex minimization problem in $\RR^3$. We observe a ``flux lock-in transition'', in which flux lines are pinned to the horizontal direction for applied fields of small inclination, and which is not present in minimizers of the anisotropic Ginzburg--Landau model. The energy profile we obtain suggests the presence of ``staircase vortices'', which have been described qualitatively in the physics literature.

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arXiv2009

Thin film limits for Ginzburg--Landau with strong applied magnetic fields

In this work, we study thin-film limits of the full three-dimensional Ginzburg-Landau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain Gamma-convergence results in several regimes, determined by the asymptotic ratio between the magnitude of the parallel applied magnetic field and the thickness of the film. Depending on the regime, we show that there may be a decrease in the density of Cooper pairs. We also show that in the case of variable thickness of the film, its geometry will affect the effective applied magnetic field, thus influencing the position of vortices.

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arXiv2009

Gamma-convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of two-dimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that, suitably normalized, the energy functionals Gamma-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

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