A numerical study of vortex nucleation in 2D rotating Bose-Einstein condensates

This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature

A 24-month updated analysis of the comparative effectiveness of ZUMA-5 (axi-cel) vs. SCHOLAR-5 external control in relapsed/refractory follicular lymphoma

Follicular lymphoma; Axicabtagene ciloleucel; Comparative effectivenessLimfoma fol·licular; Axicabtagene ciloleucel; Eficàcia comparativaLinfoma folicular; Axicabtagene ciloleucel; Eficacia comparativaBackground In the ZUMA-5 trial (Clinical trials identification: NCT03105336), axicabtagene ciloleucel (axi-cel; a chimeric antigen receptor T-cell therapy) demonstrated high rates of durable response in relapsed/refractory (r/r) follicular lymphoma (FL) patients and clear superiority relative to the SCHOLAR-5 external control cohort. We update this comparison using the ZUMA-5 24-month data. Research design and methods The SCHOLAR-5 cohort is comprised of r/r FL patients who initiated ≥3rd line of therapy after July 2014 and meeting ZUMA-5 eligibility criteria. Groups were balanced for patient characteristics through propensity scoring on prespecified prognostic factors using standardized mortality ratio (SMR) weighting. The overall response rate was compared using a weighted logistic regression. Time-to-event outcomes were evaluated using a Cox regression. Results For SCHOLAR-5, the sum of weights for the 143 patients was 85 after SMR weighting, versus 86 patients in ZUMA-5. The median follow-up was 29.4 months and 25.4 months for ZUMA-5 and SCHOLAR-5, respectively. The hazard ratios for overall survival and progression-free survival were 0.52 (95% confidence interval (CI): 0.28–0.95) and 0.28 (95% CI: 0.17–0.45), favoring axi-cel. Conclusion This updated analysis, using a longer minimum follow-up than a previously published analysis, shows that the improved efficacy of axi-cel, relative to available therapies, in r/r FL is durable.This manuscript was funded by Kite, a Gilead Company

Treatment patterns and outcomes in relapsed/refractory follicular lymphoma: results from the international SCHOLAR-5 study

The SCHOLAR-5 study examines treatment patterns and outcomes of real-world follicular lymphoma (FL) patients on 3rd line of treatment (LoT) or higher, for whom existing data are limited. SCHOLAR-5 is a retrospective cohort study using data from adults (≥ 18 years) with grade 1-3a FL, initiating ≥3rd LoT after June 2014 at major lymphoma centers in the United States (US) and Europe. Objective response rate (ORR), complete response (CR), progression free survival (PFS) and overall survival (OS) were analyzed by LoT. Time-to-event outcomes were assessed using Kaplan-Meier methods. Of 128 patients, 87 initiated 3rd LoT, 63 initiated 4th LoT, and 47 initiated 5th LoT. At 1st eligible LoT, 31% progressed within 24-months of 1st LoT anti-CD20 combination therapy, 28% had prior autologous stem-cell transplantation, and 31% were refractory to the previous LoT. The most common regimen in each LoT was chemoimmunotherapy; however, experimental drugs were increasingly used at later LoTs. In the US, anti-CD20 monotherapy was more common at ≥3rd LoT compared to Europe, where stem cell transplants were more common. ORR at 3rd LoT was 68% (CR 44%), but decreased after each LoT to 37% (CR 22%) in ≥5 LoT. Median OS and PFS at 3rd LoT were 68 and 11 months, respectively, and reduced to 43 and 4 months at ≥5 LoT. Treatments were heterogenous at each LoT in both the US and Europe. Few FL patients achieved complete response in later LoT, and duration of response and survival diminished with each subsequent line

Global burden of 369 diseases and injuries in 204 countries and territories, 1990–2019: a systematic analysis for the Global Burden of Disease Study 2019

Background: In an era of shifting global agendas and expanded emphasis on non-communicable diseases and injuries along with communicable diseases, sound evidence on trends by cause at the national level is essential. The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) provides a systematic scientific assessment of published, publicly available, and contributed data on incidence, prevalence, and mortality for a mutually exclusive and collectively exhaustive list of diseases and injuries. Methods: GBD estimates incidence, prevalence, mortality, years of life lost (YLLs), years lived with disability (YLDs), and disability-adjusted life-years (DALYs) due to 369 diseases and injuries, for two sexes, and for 204 countries and territories. Input data were extracted from censuses, household surveys, civil registration and vital statistics, disease registries, health service use, air pollution monitors, satellite imaging, disease notifications, and other sources. Cause-specific death rates and cause fractions were calculated using the Cause of Death Ensemble model and spatiotemporal Gaussian process regression. Cause-specific deaths were adjusted to match the total all-cause deaths calculated as part of the GBD population, fertility, and mortality estimates. Deaths were multiplied by standard life expectancy at each age to calculate YLLs. A Bayesian meta-regression modelling tool, DisMod-MR 2.1, was used to ensure consistency between incidence, prevalence, remission, excess mortality, and cause-specific mortality for most causes. Prevalence estimates were multiplied by disability weights for mutually exclusive sequelae of diseases and injuries to calculate YLDs. We considered results in the context of the Socio-demographic Index (SDI), a composite indicator of income per capita, years of schooling, and fertility rate in females younger than 25 years. Uncertainty intervals (UIs) were generated for every metric using the 25th and 975th ordered 1000 draw values of the posterior distribution. Findings: Global health has steadily improved over the past 30 years as measured by age-standardised DALY rates. After taking into account population growth and ageing, the absolute number of DALYs has remained stable. Since 2010, the pace of decline in global age-standardised DALY rates has accelerated in age groups younger than 50 years compared with the 1990–2010 time period, with the greatest annualised rate of decline occurring in the 0–9-year age group. Six infectious diseases were among the top ten causes of DALYs in children younger than 10 years in 2019: lower respiratory infections (ranked second), diarrhoeal diseases (third), malaria (fifth), meningitis (sixth), whooping cough (ninth), and sexually transmitted infections (which, in this age group, is fully accounted for by congenital syphilis; ranked tenth). In adolescents aged 10–24 years, three injury causes were among the top causes of DALYs: road injuries (ranked first), self-harm (third), and interpersonal violence (fifth). Five of the causes that were in the top ten for ages 10–24 years were also in the top ten in the 25–49-year age group: road injuries (ranked first), HIV/AIDS (second), low back pain (fourth), headache disorders (fifth), and depressive disorders (sixth). In 2019, ischaemic heart disease and stroke were the top-ranked causes of DALYs in both the 50–74-year and 75-years-and-older age groups. Since 1990, there has been a marked shift towards a greater proportion of burden due to YLDs from non-communicable diseases and injuries. In 2019, there were 11 countries where non-communicable disease and injury YLDs constituted more than half of all disease burden. Decreases in age-standardised DALY rates have accelerated over the past decade in countries at the lower end of the SDI range, while improvements have started to stagnate or even reverse in countries with higher SDI. Interpretation: As disability becomes an increasingly large component of disease burden and a larger component of health expenditure, greater research and developm nt investment is needed to identify new, more effective intervention strategies. With a rapidly ageing global population, the demands on health services to deal with disabling outcomes, which increase with age, will require policy makers to anticipate these changes. The mix of universal and more geographically specific influences on health reinforces the need for regular reporting on population health in detail and by underlying cause to help decision makers to identify success stories of disease control to emulate, as well as opportunities to improve. Funding: Bill & Melinda Gates Foundation. © 2020 The Author(s). Published by Elsevier Ltd. This is an Open Access article under the CC BY 4.0 licens

Global age-sex-specific fertility, mortality, healthy life expectancy (HALE), and population estimates in 204 countries and territories, 1950-2019 : a comprehensive demographic analysis for the Global Burden of Disease Study 2019

Background: Accurate and up-to-date assessment of demographic metrics is crucial for understanding a wide range of social, economic, and public health issues that affect populations worldwide. The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019 produced updated and comprehensive demographic assessments of the key indicators of fertility, mortality, migration, and population for 204 countries and territories and selected subnational locations from 1950 to 2019. Methods: 8078 country-years of vital registration and sample registration data, 938 surveys, 349 censuses, and 238 other sources were identified and used to estimate age-specific fertility. Spatiotemporal Gaussian process regression (ST-GPR) was used to generate age-specific fertility rates for 5-year age groups between ages 15 and 49 years. With extensions to age groups 10–14 and 50–54 years, the total fertility rate (TFR) was then aggregated using the estimated age-specific fertility between ages 10 and 54 years. 7417 sources were used for under-5 mortality estimation and 7355 for adult mortality. ST-GPR was used to synthesise data sources after correction for known biases. Adult mortality was measured as the probability of death between ages 15 and 60 years based on vital registration, sample registration, and sibling histories, and was also estimated using ST-GPR. HIV-free life tables were then estimated using estimates of under-5 and adult mortality rates using a relational model life table system created for GBD, which closely tracks observed age-specific mortality rates from complete vital registration when available. Independent estimates of HIV-specific mortality generated by an epidemiological analysis of HIV prevalence surveys and antenatal clinic serosurveillance and other sources were incorporated into the estimates in countries with large epidemics. Annual and single-year age estimates of net migration and population for each country and territory were generated using a Bayesian hierarchical cohort component model that analysed estimated age-specific fertility and mortality rates along with 1250 censuses and 747 population registry years. We classified location-years into seven categories on the basis of the natural rate of increase in population (calculated by subtracting the crude death rate from the crude birth rate) and the net migration rate. We computed healthy life expectancy (HALE) using years lived with disability (YLDs) per capita, life tables, and standard demographic methods. Uncertainty was propagated throughout the demographic estimation process, including fertility, mortality, and population, with 1000 draw-level estimates produced for each metric. Findings: The global TFR decreased from 2·72 (95% uncertainty interval [UI] 2·66–2·79) in 2000 to 2·31 (2·17–2·46) in 2019. Global annual livebirths increased from 134·5 million (131·5–137·8) in 2000 to a peak of 139·6 million (133·0–146·9) in 2016. Global livebirths then declined to 135·3 million (127·2–144·1) in 2019. Of the 204 countries and territories included in this study, in 2019, 102 had a TFR lower than 2·1, which is considered a good approximation of replacement-level fertility. All countries in sub-Saharan Africa had TFRs above replacement level in 2019 and accounted for 27·1% (95% UI 26·4–27·8) of global livebirths. Global life expectancy at birth increased from 67·2 years (95% UI 66·8–67·6) in 2000 to 73·5 years (72·8–74·3) in 2019. The total number of deaths increased from 50·7 million (49·5–51·9) in 2000 to 56·5 million (53·7–59·2) in 2019. Under-5 deaths declined from 9·6 million (9·1–10·3) in 2000 to 5·0 million (4·3–6·0) in 2019. Global population increased by 25·7%, from 6·2 billion (6·0–6·3) in 2000 to 7·7 billion (7·5–8·0) in 2019. In 2019, 34 countries had negative natural rates of increase; in 17 of these, the population declined because immigration was not sufficient to counteract the negative rate of decline. Globally, HALE increased from 58·6 years (56·1–60·8) in 2000 to 63·5 years (60·8–66·1) in 2019. HALE increased in 202 of 204 countries and territories between 2000 and 2019

Effects of hospital facilities on patient outcomes after cancer surgery: an international, prospective, observational study

Background Early death after cancer surgery is higher in low-income and middle-income countries (LMICs) compared with in high-income countries, yet the impact of facility characteristics on early postoperative outcomes is unknown. The aim of this study was to examine the association between hospital infrastructure, resource availability, and processes on early outcomes after cancer surgery worldwide.Methods A multimethods analysis was performed as part of the GlobalSurg 3 study-a multicentre, international, prospective cohort study of patients who had surgery for breast, colorectal, or gastric cancer. The primary outcomes were 30-day mortality and 30-day major complication rates. Potentially beneficial hospital facilities were identified by variable selection to select those associated with 30-day mortality. Adjusted outcomes were determined using generalised estimating equations to account for patient characteristics and country-income group, with population stratification by hospital.Findings Between April 1, 2018, and April 23, 2019, facility-level data were collected for 9685 patients across 238 hospitals in 66 countries (91 hospitals in 20 high-income countries; 57 hospitals in 19 upper-middle-income countries; and 90 hospitals in 27 low-income to lower-middle-income countries). The availability of five hospital facilities was inversely associated with mortality: ultrasound, CT scanner, critical care unit, opioid analgesia, and oncologist. After adjustment for case-mix and country income group, hospitals with three or fewer of these facilities (62 hospitals, 1294 patients) had higher mortality compared with those with four or five (adjusted odds ratio [OR] 3.85 [95% CI 2.58-5.75]; p<0.0001), with excess mortality predominantly explained by a limited capacity to rescue following the development of major complications (63.0% vs 82.7%; OR 0.35 [0.23-0.53]; p<0.0001). Across LMICs, improvements in hospital facilities would prevent one to three deaths for every 100 patients undergoing surgery for cancer.Interpretation Hospitals with higher levels of infrastructure and resources have better outcomes after cancer surgery, independent of country income. Without urgent strengthening of hospital infrastructure and resources, the reductions in cancer-associated mortality associated with improved access will not be realised

Numerical simulation of Bose-Einstein condensates

Dans cette thèse, nous considérons une fonctionnelle d'énergie Gross--Pitaevskii (GP) comme modèle pour la rotation d'un espèce et de deux espèces de condensats de Bose--Einstein (BEC) en deux dimensions. Ce modèle peut être adimensionné pour mettre en évidence un régime de confinement fort avec une forte interaction entre les deux composants. Nous introduisons une nouvelle discrétisation de cette énergie, comportant à la fois des schémas de différence finie et de Fourier, dans un domaine borné dans R² en utilisant des conditions aux bords de Dirichlet. Nous développons un algorithme de méthode de gradient explicite avec pas adaptatif et projection (EPG) sur la variété de contraintes pour la minimisation de l'énergie. Cette méthode permet de dériver un critère d'arrêt. Nous proposons également deux algorithmes de post-traitement pour les minimiseurs numériques. L'un est destiné aux vortex simples tandis que l'autre est destiné aux nappes de vortex. Les deux algorithmes détectent ces structures et calculent leurs indices.Dans un article récent intitulé "Vortex patterns and sheets in segregated two component Bose-Einstein condensates", les auteurs étudient le comportement d'un BEC ségrégué à deux espèces mis en rotation. Ils ont pu prouver que pour une grande rotation, l'interface entre les deux composants s'allonge, conduisant éventuellement à des nappes de vortex. Ils ont également étudié les structures des vortex du BEC dans un régime ségrégué. Dans cette thèse, nous avons pu produire des simulations numériques à l'aide d'EPG, validant ces résultats théoriques récents, supportant des conjectures et couvrant différents cas physiques (les cas d'un espèce et de deux espèces en régime de coexistence qui existent déjà dans la littérature mathématique). Nous illustrons également l'efficacité d'EPG par rapport à la méthode bien connue de GPELab qui consiste à résoudre un système linéairement implicite à chaque pas de temps.Enfin, nous avons pu adapter quelques théorèmes trouvés dans la littérature à notre problème discret. Nous prouvons l'existence d'un minimiseur global de la fonctionnelle d'énergie Gross--Pitaevskii pour le schéma aux différences finies et étudions certaines de ses propriétés. Nous travaillons également sur des problèmes symétriques rencontrés dans certaines simulations numériques.In this thesis, we consider a Gross--Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose--Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier schemes, in a bounded domain in R² using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the energy. This method allows for the derivation of a stopping criterion. We propose as well two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices.In a recent article titled "Vortex patterns and sheets in segregated two component Bose-Einstein condensates", the authors study the behaviour of a segregated two-component BEC set into rotation. They were able to prove that for large rotation, the interface between the components gets long, leading possibly towards vortex sheets. They also studied the vortex structures of BEC in a segregated regime. In this thesis, we were able to produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures and covering different physical cases (the cases of one component and two components in coexistence regime that alredy exists in the mathematical litterature). We also illustrate the efficiency of EPG compared to that well-known GPELab's method which consists on solving linearly implicit system at each time-step.Finally, we were able to adapt few theorems found in the literature to our discrete problem. We prove the existence of a global minimizer of the Gross--Pitaevskii energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations

Simulation numérique de condensats de Bose-Einstein

In this thesis, we consider a Gross–Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose–Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier transform approaches, in a bounded domain of R 2 using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the discrete energy. This method allows for the derivation of a stopping criterion. Moreover, we propose two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices. In a recent article [6], the authors study the behaviour of a segregated two components BEC set into rotation. They are able to prove that for large rotation, the interface between the components gets long, conjecturing the possibility towards vortex sheets. They also study the vortex structures of BEC in a segregated regime. In this thesis, we produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures, and covering different physical cases (the cases of one component and two components in coexistence regime [45, 39]). We also illustrate the efficiency of EPG compared to the well-known GPELab method [11] which requires to solve a linearly implicit system at each step. Finally, we adapt a few theorems found in the literature to our discrete setting. We prove the existence of a global minimizer of the GP energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations.Dans cette thèse, nous considérons une fonctionnelle d'énergie Gross–Pitaevskii (GP) comme modèle d’une espèce et de deux espèces de condensats de Bose–Einstein (BEC) en deux dimensions mise en rotation. Ce modèle peut être non dimensionné pour mettre en évidence un régime de fort confinement avec une forte interaction entre les deux espèces. Nous introduisons une nouvelle discrétisation de cette énergie, comportant à la fois des approches par différences finies et par transformée de Fourier, dans un domaine borné de $\R^2$ en utilisant des conditions de Dirichlet nulles aux bords. Nous développons ainsi un algorithme de méthode de gradient explicite avec pas adaptatif et projection (EPG) sur la variété de contraintes pour la minimisation de l'énergie discrète. Cette méthode permet de dériver un critère d’arrêt. De plus, nous proposons deux algorithmes de post-traitement pour les minimiseurs numériques. L’un est destiné aux vortex simples tandis que l’autre est destiné aux nappes de vortex. Les deux algorithmes détectent ces structures et calculent leurs indices. Dans un article récent [6], les auteurs étudient le comportement d’un BEC à deux espèces en ségrégation et mis en rotation. Ils sont capables de prouver que pour une forte rotation, l’interface entre les espèces devient longue, conjecturant la possibilité vers des nappes de vortex. Ils étudient également les structures des vortex du BEC en régime de ségrégation. Dans cette thèse, nous produisons des simulations numériques à l’aide d’EPG, validant ces résultats théoriques récents, supportant des conjectures, et couvrant différents cas physiques (les cas d’une espèce et de deux espèces en régime de co-existence [45, 39]). Nous illustrons également l’efficacité d’EPG par rapport à la méthode bien connue GPELab [11] qui nécessite de résoudre un système linéaire implicite à chaque étape. Enfin, nous adaptons quelques théorèmes trouvés dans la littérature à notre cadre discret. Nous prouvons l’existence d’un minimiseur global de la fonctionnelle d’énergie GP pour le schéma aux différences finies et étudions certaines de ses propriétés. Nous travaillons également sur des problèmes symétriques rencontrés dans certaines simulations numériques

Simulation numérique de condensats de Bose-Einstein

In this thesis, we consider a Gross--Pitaevskii (GP) energy functional as a model for rotating one component and two components Bose--Einstein condensates (BEC) in two dimensions. This model can be non-dimensionalized to highlight a strong confinement regime with strong interaction between the two components. We introduce a new discretization of this energy, featuring both finite difference and fast Fourier schemes, in a bounded domain in R² using Dirichlet boundary conditions. We develop an explicit gradient method algorithm with adaptive step and projection (EPG) over the constraints manifold for the minimization of the energy. This method allows for the derivation of a stopping criterion. We propose as well two post processing algorithms for the numerical minimizers. One is aimed for single vortices while the other is aimed for vortex sheets. Both algorithms detect these structures and compute their indices.In a recent article titled "Vortex patterns and sheets in segregated two component Bose-Einstein condensates", the authors study the behaviour of a segregated two-component BEC set into rotation. They were able to prove that for large rotation, the interface between the components gets long, leading possibly towards vortex sheets. They also studied the vortex structures of BEC in a segregated regime. In this thesis, we were able to produce numerical simulations using EPG, validating these recent theoretical results, supporting conjectures and covering different physical cases (the cases of one component and two components in coexistence regime that alredy exists in the mathematical litterature). We also illustrate the efficiency of EPG compared to that well-known GPELab's method which consists on solving linearly implicit system at each time-step.Finally, we were able to adapt few theorems found in the literature to our discrete problem. We prove the existence of a global minimizer of the Gross--Pitaevskii energy functional for the finite difference scheme and study some of its properties. We also work on symmetrical problems we encountered in some of the numerical simulations.Dans cette thèse, nous considérons une fonctionnelle d'énergie Gross--Pitaevskii (GP) comme modèle pour la rotation d'un espèce et de deux espèces de condensats de Bose--Einstein (BEC) en deux dimensions. Ce modèle peut être adimensionné pour mettre en évidence un régime de confinement fort avec une forte interaction entre les deux composants. Nous introduisons une nouvelle discrétisation de cette énergie, comportant à la fois des schémas de différence finie et de Fourier, dans un domaine borné dans R² en utilisant des conditions aux bords de Dirichlet. Nous développons un algorithme de méthode de gradient explicite avec pas adaptatif et projection (EPG) sur la variété de contraintes pour la minimisation de l'énergie. Cette méthode permet de dériver un critère d'arrêt. Nous proposons également deux algorithmes de post-traitement pour les minimiseurs numériques. L'un est destiné aux vortex simples tandis que l'autre est destiné aux nappes de vortex. Les deux algorithmes détectent ces structures et calculent leurs indices.Dans un article récent intitulé "Vortex patterns and sheets in segregated two component Bose-Einstein condensates", les auteurs étudient le comportement d'un BEC ségrégué à deux espèces mis en rotation. Ils ont pu prouver que pour une grande rotation, l'interface entre les deux composants s'allonge, conduisant éventuellement à des nappes de vortex. Ils ont également étudié les structures des vortex du BEC dans un régime ségrégué. Dans cette thèse, nous avons pu produire des simulations numériques à l'aide d'EPG, validant ces résultats théoriques récents, supportant des conjectures et couvrant différents cas physiques (les cas d'un espèce et de deux espèces en régime de coexistence qui existent déjà dans la littérature mathématique). Nous illustrons également l'efficacité d'EPG par rapport à la méthode bien connue de GPELab qui consiste à résoudre un système linéairement implicite à chaque pas de temps.Enfin, nous avons pu adapter quelques théorèmes trouvés dans la littérature à notre problème discret. Nous prouvons l'existence d'un minimiseur global de la fonctionnelle d'énergie Gross--Pitaevskii pour le schéma aux différences finies et étudions certaines de ses propriétés. Nous travaillons également sur des problèmes symétriques rencontrés dans certaines simulations numériques