Computing Linear Matrix Representations of Helton-Vinnikov Curves

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Base

Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation

The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations. Here we consider stability of the larger class of solutions whose phase is dependent on the spatial dimension of the one-dimensional wave form. We study the spectral stability of such nontrivial-phase solutions numerically, using Hill's method. We present evidence which suggests that all such nontrivial-phase solutions are unstable with respect to both one- and two-dimensional perturbations. Instability occurs in all cases: for both the elliptic and hyperbolic nonlinear Schrodinger equations, and in the focusing and defocusing case.Comment: Submitted: 13 pages, 3 figure

Qubit State Discrimination

We show how one can solve the problem of discriminating between qubit states. We use the quantum state discrimination duality theorem and the Bloch sphere representation of qubits which allows for an easy geometric and analytical representation of the optimal guessing strategies.Comment: 6 pages, 4 figures. v2 has small corrections and changes in reference

Generation Expansion Models including Technical Constraints and Demand Uncertainty

This article presents a Generation Expansion Model of the power system taking into account the operational constraints and the uncertainty of long-term electricity demand projections. The model is based on a discretization of the load duration curve and explicitly considers that power plant ramping capabilities must meet demand variations. A model predictive control method is used to improve the long-term planning decisions while considering the uncertainty of demand projections. The model presented in this paper allows integrating technical constraints and uncertainty in the simulations, improving the accuracy of the results, while maintaining feasible computational time. Results are tested over three scenarios based on load data of an energy retailer in Colombia

Pole dynamics for the Flierl-Petviashvili equation and zonal flow

We use a systematic method which allows us to identify a class of exact solutions of the Flierl-Petvishvili equation. The solutions are periodic and have one dimensional geometry. We examine the physical properties and find that these structures can have a significant effect on the zonal flow generation.Comment: Latex 40 pages, seven figures eps included. Effect of variation of g_3 is studied. New references adde

Exact Floquet states of a driven condensate and their stabilities

We investigate the Gross-Pitaevskii equation for a classically chaotic system, which describes an atomic Bose-Einstein condensate confined in an optical lattice and driven by a spatiotemporal periodic laser field. It is demonstrated that the exact Floquet states appear when the external time-dependent potential is balanced by the nonlinear mean-field interaction. The balance region of parameters is divided into a phase-continuing region and a phase-jumping one. In the latter region, the Floquet states are spatiotemporal vortices of nontrivial phase structures and zero-density cores. Due to the velocity singularities of vortex cores and the blowing-up of perturbed solutions, the spatiotemporal vortices are unstable periodic states embedded in chaos. The stability and instability of these Floquet states are numerically explored by the time evolution of fidelity between the exact and numerical solutions. It is numerically illustrated that the stable Floquet states could be prepared from the uniformly initial states by slow growth of the external potential.Comment: 14 pages, 3 eps figures, final version accepted for publication in J. Phys.

Bose-Einstein condensates in standing waves: The cubic nonlinear Schroedinger equation with a periodic potential

We present a new family of stationary solutions to the cubic nonlinear Schroedinger equation with a Jacobian elliptic function potential. In the limit of a sinusoidal potential our solutions model a dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided the ratio of the height of the variations of the condensate to its DC offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Numerical simulations suggest such stationary states are experimentally observable.Comment: 4 pages, 4 figure

Vortices in Bose-Einstein Condensates: Some Recent Developments

In this brief review we summarize a number of recent developments in the study of vortices in Bose-Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices we also consider the existence of stable dipole clusters for two-component condensates. Last but not least, we discuss mesoscopic patterns formed by vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted.Comment: 24 pages, 8 figs, ws-mplb.cls, to appear in Modern Physics Letters B (2005

Modulated Amplitude Waves in Bose-Einstein Condensates

We analyze spatio-temporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt's method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits (``modulated amplitude waves'') on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.Comment: 5 pages, 4 figs, revtex, final form of paper, to appear in PRE (forgot to include \bibliography command in last update, so this is a correction of that; the bibliography is hence present again

Stability of Repulsive Bose-Einstein Condensates in a Periodic Potential

The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schr\"odinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.Comment: 12 pages, 17 figure