On a selection principle for multivalued semiclassical flows

We study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture

Uniform estimates for positive solutions of a class of semilinear elliptic equations and related Liouville and one-dimensional symmetry results

We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furthermore, we can generalize an old result, concerning semilinear elliptic nonlinear eigenvalue problems. Moreover, we study the boundary layer of global minimizers of the corresponding singular perturbation problem. For the above applications, our approach is based on a refinement of a useful result, concerning the behavior of global minimizers of the associated energy over large balls, subject to Dirichlet conditions. Combining this refinement with global bifurcation theory and the celebrated sliding method, we can prove uniform estimates for solutions away from their nodal set. Combining our approach with a-priori estimates that we obtain by blow-up, a doubling lemma, and known Liouville type theorems, we can give a new proof of a known Liouville type theorem without using boundary blow-up solutions. We can also provide an alternative proof, and a useful extension, of a Liouville theorem, involving the presence of an obstacle. Making use of the latter extension, we consider the singular perturbation problem with mixed boundary conditions. Moreover, we prove some new one-dimensional symmetry and rigidity properties of certain entire solutions to Allen-Cahn type equations, as well as in half spaces, convex cylindrical domains. In particular, we provide a new proof of Gibbons' conjecture

Asymptotic Solutions of the Phase Space Schrodinger Equation: Anisotropic Gaussian Approximation

We consider the singular semiclassical initial value problem for the phase space Schrodinger equation. We approximate semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit approximation; we accordingly construct a semiclassical approximation of the phase space propagator, semiclassical wave packet propagator, which admits WKBM semiclassical states as initial data. By the semiclassical propagator we construct asymptotic solutions of the phase space Schrodinger equation, noting the connection of this construction to the initial value repsresentations for the Schrodinger equation

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial

On periodic orbits in a slow-fast system with normally elliptic slow manifold

In this note we consider the bifurcation of a singular homoclinic orbit to periodic ones in a 4-dimensional slow-fast system of ordinary differential equations, having a 2-dimensional normally elliptic slow manifold, originally studied by Feckan and Rothos. Assuming an extra degree of differentiability on the system, we can refine their perturbation scheme, in particular the choice of approximate solution, and obtain improved estimates

Consistent Discretizations for Vanishing Regularization Solutions to Image Processing Problems

A model problem is used to represent a typical image processing problem of reconstructing an unknown in the face of incomplete data. A consistent discretization for a vanishing regularization solution is defined so that, in the absence of noise, limits first with respect to regularization and then with respect to grid refinement agree with a continuum counterpart defined in terms of a saddle point formulation. It is proved and demonstrated computationally for an artificial example and for a realistic example with magnetic resonance images that a mixed finite element discretization is consistent in the sense defined here. On the other hand, it is demonstrated computationally that a standard finite element discretization is not consistent, and the reason for the inconsistency is suggested in terms of theoretical and computational evidence

Magnetization oscillations by vortex-antivortex dipoles

A vortex-antivortex dipole can be generated due to current with in-plane spin-polarization, flowing into a magnetic element, which then behaves as a spin transfer oscillator. Its dynamics is analyzed using the Landau-Lifshitz equation including a Slonczewski spin-torque term. We establish that the vortex dipole is set in steady state rotational motion due to the interaction between the vortices, while an external in-plane magnetic field can tune the frequency of rotation. The rotational motion is linked to the nonzero skyrmion number of the dipole. The spin-torque acts to stabilize the vortex dipole at a definite vortex-antivortex separation distance. In contrast to a free vortex dipole, the rotating pair under spin-polarized current is an attractor of the motion, therefore a stable state. Three types of vortex-antivortex pairs are obtained as we vary the external field and spin-torque strength. We give a guide for the frequency of rotation based on analytical relations

Simulated acoustic emissions from coupled strings

We consider traveling transverse waves on two identical uniform taut strings that are elastically coupled through springs that gradually decrease their stiffness over a region of finite length. The wave system can be decomposed into two modes: an in-phase mode ( + ) that is transparent to the coupling springs, and an out-of-phase mode ( − ) that engages the coupling springs and can resonate at a particular location depending on the excitation frequency. The system exhibits linear mode conversion whereby an incoming ( + ) wave is reflected back from the resonance location both as a propagating ( + ) wave and an evanescent ( − ) wave, while both types emerge as propagating forward through the resonance location. We match a local transition layer expansion to the WKB expansion to obtain estimates of the reflection and transmission coefficients. The reflected waves may be an analog for stimulated emissions from the ear