Dark-Bright Solitons in Inhomogeneous Bose-Einstein Condensates

We investigate dark-bright vector solitary wave solutions to the coupled non-linear Schr\"odinger equations which describe an inhomogeneous two-species Bose-Einstein condensate. While these structures are well known in non-linear fiber optics, we show that spatial inhomogeneity strongly affects their motion, stability, and interaction, and that current technology suffices for their creation and control in ultracold trapped gases. The effects of controllably different interparticle scattering lengths, and stability against three-dimensional deformations, are also examined.Comment: 5 pages, 5 figure

CALCULATION OF THE DIGITAL TWIN OF THE SALES FUNNEL

Commercial activity has always been influenced by the competitive environment and its spread to the online space is the next stage of development and a defining trend for the nearest time horizon. The changes in the business landscape influenced by COVID19 pose new challenges for marketers and entrepreneurs. It is necessary to use the forced sharp increase in online interaction with consumers. The course towards the digital economy determines the use of scientific, mathematical methods to optimize the target indicators of economic activity. These global shifts in business interactions are generating innovative tools for measuring business results and transforming old practices to meet new market realities. This is the basic condition for the sustainability of doing business in any industry. This study is devoted to the development of a theoretical description of the process of multi-stage interaction with a consumer pool. To solve this problem, a mathematical model has been developed, the basis of which is digital information interaction, starting from the stage of determining the target audience and ending with the complete completion of a commercial transaction. This article presents the results of modeling sales funnel, as the basis for the software of a modern market analyst, using a cross-system approach. In contrast to the classical sales funnel, the presented algorithms allow using the multidimensional conversion funnel not only for assessing business results for the reporting period. Thanks to the flow of model arguments in real time, it becomes possible to optimize the business process by moving to the concept of leading economic indicators.In practice, this means the ability to implement effective business planning on digital platforms. The arguments of the mathematical model are Internet statistics, the dynamics of consumer preferences, the history of the business process accumulated in the big data system. At the same time, the means of queuing theory, differential calculus, economic and mathematical modeling are involved, based on indicators such as KPI (Key Performance Indicators), CTR (click-through rate), CR (Conversion rate). This made it possible to formulate the concept of a digital twin of a commercial process and its transformation, convenient for practical applications, into a conversion funnel for embedding into algorithms implemented on a computer

Effect of optically-induced potential on the energy of trapped exciton-polaritons below the condensation threshold

Exciton-polaritons (polaritons herein) offer a unique nonlinear platform for studies of collective macroscopic quantum phenomena in a solid state system. Shaping of polariton flow and polariton confinement via potential landscapes created by nonresonant optical pumping has gained considerable attention due to the degree of flexibility and control offered by optically-induced potentials. Recently, large density-dependent energy shifts (blueshifts) exhibited by optically trapped polaritons at low densities, below the bosonic condensation threshold, were interpreted as an evidence of strong polariton-polariton interactions [Nat. Phys. 13, 870 (2017)]. In this work, we further investigate the origins of these blueshifts in optically-induced circular traps and present evidence of significant blueshift of the polariton energy due to reshaping of the optically-induced potential with laser pump power. Our work demonstrates strong influence of the effective potential formed by an optically-injected excitonic reservoir on the energy blueshifts observed below and up to the polariton condensation threshold and suggests that the observed blueshifts arise due to interaction of polaritons with the excitonic reservoir, rather than due to polariton-polariton interaction.Comment: 10 pages, 8 figure

РАСЧЕТ ЦИФРОВОГО ДВОЙНИКА ВОРОНКИ ПРОДАЖ

Commercial activity has always been influenced by the competitive environment and its spread to the online space is the next stage of development and a defining trend for the nearest time horizon. The changes in the business landscape influenced by COVID19 pose new challenges for marketers and entrepreneurs. It is necessary to use the forced sharp increase in online interaction with consumers. The course towards the digital economy determines the use of scientific, mathematical methods to optimize the target indicators of economic activity. These global shifts in business interactions are generating innovative tools for measuring business results and transforming old practices to meet new market realities. This is the basic condition for the sustainability of doing business in any industry. This study is devoted to the development of a theoretical description of the process of multi-stage interaction with a consumer pool. To solve this problem, a mathematical model has been developed, the basis of which is digital information interaction, starting from the stage of determining the target audience and ending with the complete completion of a commercial transaction. This article presents the results of modeling sales funnel, as the basis for the software of a modern market analyst, using a cross-system approach. In contrast to the classical sales funnel, the presented algorithms allow using the multidimensional conversion funnel not only for assessing business results for the reporting period. Thanks to the flow of model arguments in real time, it becomes possible to optimize the business process by moving to the concept of leading economic indicators.In practice, this means the ability to implement effective business planning on digital platforms. The arguments of the mathematical model are Internet statistics, the dynamics of consumer preferences, the history of the business process accumulated in the big data system. At the same time, the means of queuing theory, differential calculus, economic and mathematical modeling are involved, based on indicators such as KPI (Key Performance Indicators), CTR (click-through rate), CR (Conversion rate). This made it possible to formulate the concept of a digital twin of a commercial process and its transformation, convenient for practical applications, into a conversion funnel for embedding into algorithms implemented on a computer.Коммерческая деятельность всегда испытывала воздействие конкурентной среды, и онлайн-торговля является следующей ступенью развития и определяющим трендом на ближайшее время. Произошедшие под влиянием COVID-19 изменения в бизнес-ландшафте ставят перед маркетологами и предпринимателями новые задачи. Необходимо использовать вынужденный резкий рост онлайн-взаимодействия с потребителями. Курс на цифровую экономику обуславливает применение научных, математических методов для оптимизации целевых показателей экономической деятельности. Такие глобальные перемены в бизнес-взаимодействии порождают инновационные инструменты для оценки результатов коммерции и трансформируют прежние методики для соответствия новым реалиям рынка. Это является базовым условием устойчивости ведения бизнеса в любой отрасли. Настоящее исследование посвящено разработке теоретического описания процесса многоступенчатого взаимодействия с потребительским пулом. Для решения задачи формализации данного процесса разработана математическая модель, основу которой составляет информационное цифровое взаимодействие от этапа определения целевой аудитории до полного завершения коммерческой сделки.В предлагаемой статье изложены результаты работы по моделированию sales funnel как основы программного обеспечения современного аналитика рынка с применением кросс-системного подхода. В отличие от классической sales funnel, представленные алгоритмы позволяют использовать многомерную conversion funnel не только для оценки результатов работы бизнеса за отчетный период: благодаря потоку аргументов модели в режиме реального времени становится возможным оптимизировать коммерческий процесс за счет перехода к концепции опережающих экономических показателей.На практике это означает возможность реализации на цифровых платформах эффективного планирования коммерческой деятельности. Аргументами математической модели служат интернет-статистика, динамика потребительских предпочтений, история бизнес-процесса, аккумулированная в системе больших данных. При этом задействованы средства queuing theory, дифференциального исчисления, экономико-математического моделирования с опорой на такие показатели, как KPI (Key Performance Indicators), CTR (click-through rate), CR (Conversion rate). Это позволило сформулировать концепцию цифрового двойника коммерческого процесса. Нами разработаны математические формализмы, удобные для практических приложений. Это позволяет обеспечить приемлемую для программирования на ЭВМ реализацию алгоритмов, описывающих conversion funnel

Interaction of matter-wave gap solitons in optical lattices

We study mobility and interaction of gap solitons in a Bose-Einstein condensate (BEC) confined by an optical lattice potential. Such localized wavepackets can exist only in the gaps of the matter-wave band-gap spectrum and their interaction properties are shown to serve as a measure of discreteness imposed onto a BEC by the lattice potential. We show that inelastic collisions of two weakly localized near-the-band-edge gap solitons provide simple and effective means for generating strongly localized in-gap solitons through soliton fusion.Comment: 12 pages, 7 figure

Моделирование и численное решение задачи с односторонними связями и трением при динамическом действии нагрузки

Problems with unilateral constraints are not uncommon in the practice of calculating  building construction and structures. Certain difficulties in solving them arise during contact friction, as well as the dynamic action of the load. It is known that such problems from a mathematical point of view s are not correct enough, their solution becomes more complicated and depends on the history of loading and deformation of the structure. At the same time, the ability to take into account the complex working conditions of the structure makes its calculation more complete and accurate. The paper considers the solution of  a dynamic contact problem on the basis of the finite element method and the step-by-step loading method. Unilateral constraints with Coulomb friction are modeled using contact finite elements of a frame-rod type. The method of compensating loads is applied in order to comply with the limitations under ultimate friction-sliding conditions. Based on the considered discrete contact model and the step-by-step analysis method, a numerical algorithm has been developed, which allows in one step-by-step process to integrate simultaneously the equations of motion and implement contact conditions with Coulomb friction. With the help of the proposed approach, numerical solutions of the problem pertaining to a structure contact with the base have been obtained and analyzed at various parameters of dynamic load. Comparison of the results with the solution obtained by the well-known iteration method on the ultimate friction forces allows to conclude about the efficiency and reliability of the developed algorithm under complex contact conditions and dynamic loading.Задачи с односторонними связями нередки в практике расчетов строительных конструкций и сооружений. Определенные трудности в их решении возникают при трении на контакте, а также при динамическом действии нагрузки. Известно, что такие задачи с математической точки зрения являются недостаточно корректными, решение их усложняется и зависит от истории нагружения и деформирования конструкции. В то же время возможность учета сложных условий работы конструкции делает ее расчет более полным и точным. В статье рассмотрено решение динамической контактной задачи на основе метода конечных элементов и метода шагового нагружения. Односторонние связи с кулоновским трением моделируются с помощью контактных конечных элементов рамно-стержневого типа. Для соблюдения ограничений в условиях по предельному трению-скольжению применяется способ компенсирующих нагрузок. На основе рассмотренной дискретной модели контакта и метода пошагового анализа разработан численный алгоритм, позволяющий в одном пошаговом процессе выполнять одновременно интегрирование уравнений движения и реализацию контактных условий с трением Кулона. С помощью предложенного подхода получены и проанализированы численные решения задачи контакта сооружения с основанием при различных параметрах динамической нагрузки. Сравнение результатов с решением, выполненным известным методом итераций по предельным силам трения, позволяет сделать вывод об эффективности и надежности разработанного алгоритма при сложных условиях контакта и динамическом нагружени

Topological phase transition in an all-optical exciton-polariton lattice

Topological insulators are a class of electronic materials exhibiting robust edge states immune to perturbations and disorder. This concept has been successfully adapted in photonics, where topologically nontrivial waveguides and topological lasers were developed. However, the exploration of topological properties in a given photonic system is limited to a fabricated sample, without the flexibility to reconfigure the structure in-situ. Here, we demonstrate an all-optical realization of the orbital Su-Schrieffer-Heeger (SSH) model in a microcavity exciton-polariton system, whereby a cavity photon is hybridized with an exciton in a GaAs quantum well. We induce a zigzag potential for exciton polaritons all-optically, by shaping the nonresonant laser excitation, and measure directly the eigenspectrum and topological edge states of a polariton lattice in a nonlinear regime of bosonic condensation. Furthermore, taking advantage of the tunability of the optically induced lattice we modify the intersite tunneling to realize a topological phase transition to a trivial state. Our results open the way to study topological phase transitions on-demand in fully reconfigurable hybrid photonic systems that do not require sophisticated sample engineering.Comment: 7 pages, 4 figure