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.

In the present work, we study minimizers of the Landau-de Gennes free energy in a bounded domain Ω ⊂ R 3 . We prove that at low temperature minimizers do not vanish, even for topologically non-trivial boundary conditions. This is in contrast with a simplified Ginzburg-Landau model for superconductivity studied by Bethuel, Brezis and Hélein. Merging this with an observation of Canevari we obtain, as a corollary, the occurence of biaxial escape: the tensorial order parameter must become strongly biaxial at some point in Ω. In particular, while it is known that minimizers cannot be purely uniaxial, we prove the much stronger and physically relevant fact that they lie in a different homotopy class.

We introduce a framework to study the occurrence of vortex filament concentration in 3D Ginzburg-Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical 3-dimensional domain, in certain scaling limits; it is shown to arise as the Γ-limit of a sequence of scaled Ginzburg-Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg-Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg-Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg-Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.

We prove L 2 orbital stability of Dirac solitons in the massive Thirring model. Our analysis uses local well posedness of the massive Thirring model in L 2 , conservation of the charge functional, and the auto-Bäcklund transformation. The latter transformation exists because the massive Thirring model is integrable via the inverse scattering transform method.

Abstract. This work presents a method to efficiently determine the dominant Karhunen-Loève (KL) modes of a random process with known covariance function. The truncated KL expansion is one of the most common techniques for the approximation of random processes, primarily because it is an optimal representation, in the mean squared error sense, with respect to the number of random variables in the representation. However, finding the KL expansion involves solving integral problems, which tends to be computationally demanding. This work addresses this issue by means of a work-subdivision strategy based on a domain decomposition approach, enabling the efficient computation of a possibly large number of dominant KL modes. Specifically, the computational domain is partitioned into smaller nonoverlapping subdomains, over which independent local KL decompositions are performed to generate local bases which are subsequently used to discretize the global modes over the entire domain. The latter are determined by means of a Galerkin projection. The procedure leads to the resolution of a reduced Galerkin problem, whose size is not related to the dimension of the underlying discretization space but is actually determined by the desired accuracy and the number of subdomains. It can also be easily implemented in parallel. Extensive numerical tests are used to validate the methodology and assess its serial and parallel performance. The resulting expansion is exploited in Part B to accelerate the solution of the stochastic partial differential equations using a Monte Carlo approach.

No abstract

We analyze the Ginzburg-Landau energy in the presence of an applied magnetic field when the superconducting sample occupies a thin neighborhood of a bounded, closed manifold in R 3 . We establish -convergence to a reduced Ginzburg-Landau model posed on the manifold in which the magnetic potential is replaced in the limit by the tangential component of the applied magnetic potential. We then study the limiting problem, constructing two-vortex critical points when the manifold M is a simply connected surface of revolution and the applied field is constant and vertical. Finally, we calculate that the exact asymptotic value of the first critical field H c1 is simply (4π/(area of M)) ln κ for large values of the Ginzburg-Landau parameter κ. Merging this with the -convergence result, we also obtain the same asymptotic value for H c1 in 3d valid for large κ and sufficiently thin shells.

We analyze a nonstandard isoperimetric problem in the plane associated with a metric having degenerate conformal factor at two points. Under certain assumptions on the conformal factor, we establish the existence of curves of least length under a constraint associated with enclosed euclidean area. As a motivation for and application of this isoperimetric problem, we identify these isoperimetric curves, appropriately parametrized, as traveling wave solutions to a bistable Hamiltonian system of PDEs. We also determine the existence of a maximal propagation speed for these traveling waves through an explicit upper bound depending on the conformal factor.