Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit

We consider the stationary solutions for a class of Schroedinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the non-linearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.Comment: 46 pages, 4 figure

An instability criterion for nonlinear standing waves on nonzero backgrounds

A nonlinear Schr\"odinger equation with repulsive (defocusing) nonlinearity is considered. As an example, a system with a spatially varying coefficient of the nonlinear term is studied. The nonlinearity is chosen to be repelling except on a finite interval. Localized standing wave solutions on a non-zero background, e.g., dark solitons trapped by the inhomogeneity, are identified and studied. A novel instability criterion for such states is established through a topological argument. This allows instability to be determined quickly in many cases by considering simple geometric properties of the standing waves as viewed in the composite phase plane. Numerical calculations accompany the analytical results.Comment: 20 pages, 11 figure

Dark solitons in atomic Bose-Einstein condensates: from theory to experiments

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher-dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.Comment: 82 pages, 13 figures. To appear in J. Phys. A: Math. Theor

Nonlinear Waves in Bose-Einstein Condensates: Physical Relevance and Mathematical Techniques

The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.Comment: 69 pages, 10 figures, to appear in Nonlinearity, 2008. V2: new references added, fixed typo

Randomized trial of achieving healthy lifestyles in psychiatric rehabilitation: the ACHIEVE trial

<p>Abstract</p> <p>Background</p> <p>Overweight and obesity are highly prevalent among persons with serious mental illness. These conditions likely contribute to premature cardiovascular disease and a 20 to 30 percent shortened life expectancy in this vulnerable population. Persons with serious mental illness need effective, appropriately tailored behavioral interventions to achieve and maintain weight loss. Psychiatric rehabilitation day programs provide logical intervention settings because mental health consumers often attend regularly and exercise can take place on-site. This paper describes the Randomized Trial of Achieving Healthy Lifestyles in Psychiatric Rehabilitation (ACHIEVE). The goal of the study is to determine the effectiveness of a behavioral weight loss intervention among persons with serious mental illness that attend psychiatric rehabilitation programs. Participants randomized to the intervention arm of the study are hypothesized to have greater weight loss than the control group.</p> <p>Methods/Design</p> <p>A targeted 320 men and women with serious mental illness and overweight or obesity (body mass index ≥ 25.0 kg/m²) will be recruited from 10 psychiatric rehabilitation programs across Maryland. The core design is a randomized, two-arm, parallel, multi-site clinical trial to compare the effectiveness of an 18-month behavioral weight loss intervention to usual care. Active intervention participants receive weight management sessions and physical activity classes on-site led by study interventionists. The intervention incorporates cognitive adaptations for persons with serious mental illness attending psychiatric rehabilitation programs. The initial intensive intervention period is six months, followed by a twelve-month maintenance period in which trained rehabilitation program staff assume responsibility for delivering parts of the intervention. Primary outcomes are weight loss at six and 18 months.</p> <p>Discussion</p> <p>Evidence-based approaches to the high burden of obesity and cardiovascular disease risk in person with serious mental illness are urgently needed. The ACHIEVE Trial is tailored to persons with serious mental illness in community settings. This multi-site randomized clinical trial will provide a rigorous evaluation of a practical behavioral intervention designed to accomplish and sustain weight loss in persons with serious mental illness.</p> <p>Trial Registration</p> <p>Clinical Trials.gov NCT00902694</p

Numerical resolution of stochastic focusing NLS equations

AbstractIn this note, we numerically investigate a stochastic nonlinear Schrödinger equation derived as a perturbation of the deterministic NLS equation. The classical NLS equation with focusing nonlinearity of power law type is perturbed by a random term; it is a strong perturbation since we consider a space-time white noise. It acts either as a forcing term (additive noise) or as a potential (multiplicative noise). For simulations made on a uniform grid, we see that all trajectories blow-up in finite time, no matter how the initial data are chosen. Such a grid cannot represent a noise with zero correlation lengths, so that in these experiments, the noise is, in fact, spatially smooth. On the contrary, we simulate a noise with arbitrarily small scales using local refinement and show that in the multiplicative case, blow-up is prevented by a space-time white noise. We also present results on noise induced soliton diffusion

Numerical resolution with finite-differences of a coupled system derived from plasmas dynamics

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Solitons in quadratic media

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HADES code for numerical simulations of high-mach number astrophysical radiative flows

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