Численные подходы к решению нелинейной системы уравнений Шрёдингера для распространения волн в оптическом волокне

2020 ◽  
Author(s):  
A.J. Sakhabutdinov ◽  
V.I. Anfinogentov ◽  
O.G. Morozov ◽  
R.R. Gubaidullin
Keyword(s):  
Finite Difference ◽  
Nonlinear Term ◽  
Initial Boundary ◽  
Equation System ◽  
Integration Scheme ◽  
Integration Step ◽  
Disturbance Propagation ◽  
Calculation Results ◽  
Definition Of ◽  
Selection Of

The paper discusses approaches to the numerical integration of the second-kind Manakov equation system. Emphasis is placed on the transition from writing equations in dimensional quantities to equations in dimensionless units. A combined explicit/implicit finite-difference integration scheme based on the implicit CrankNicolson finite-difference scheme is proposed and substantiated, which allows integrating a nonlinear system of equations with a choice of nonlinear term at the previous integration step. An algorithm for leveling the disadvantage associated with the definition of the nonlinear term from the previous integration step is proposed. The approach of automatic selection of the integration step, which reduces the total number of integration steps while maintaining the required accuracy of the approximate solution, is substantiated. Examples of the calculation results for some values of the disturbance propagation are given. The limitations imposed by the computing scheme on the length of the integrable fiber section are described, and approaches, that eliminate these limitations without the need to increase arrays dimensions, are proposed. Requirements for initial boundary conditions are discussed. Предложена разработка метода приближенного решения системы уравнения Манакова как одного из частных случаев системы уравнений Шрёдингера, связанного с моделированием оптических линий связи на основе многомодовых волокон. Решение ищется методами численного интегрирования. Показано, что численное интегрирование может быть осуществлено с использованием комбинированной явно-неявной схемы численного интегрирования на основе схемы КранкаНиколсон с записью нелинейного слагаемого в конечно-разностной форме, взятого с предыдущего шага интегрирования. Использован алгоритм автоматического выбора шага интегрирования, реализован итерационный алгоритм уточнения решения на каждом шаге, предложен алгоритм, позволяющий производить расчет параметров на протяженных участках. Нахождение приближенного решения системы уравнения Манакова может быть осуществлено с использованием комбинированной явно-неявной схемы КранкаНиколсон, а запись нелинейного слагаемого в конечно-разностной форме, взятого с предыдущего шага интегрирования, дает неплохой результат. Алгоритм автоматического выбора шага интегрирования обеспечивает лучшую сходимость результатов интегрирования на большом расстоянии и снижение необходимого количества шагов интегрирования. Алгоритм уточнения решения на каждом шаге позволяет нивелировать недостаток метода явной записи неявного слагаемого и интегрировать с большим шагом. Алгоритм расчета параметров распространения возмущения со сдвигом фрейма позволяет сделать вывод о целесообразности развития этого алгоритма.

scholarly journals Numerical Method for Coupled Nonlinear Schrödinger Equations in Few Mode Fiber

2020 ◽  
Author(s):  
Airat Sakhabutdinov ◽  
Vladimir Anfinogentov ◽  
Oleg Morozov ◽  
Vladimir Burdin ◽  
Anton Bourdine ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Equation System ◽  
Integration Scheme ◽  
Integration Step ◽  
Non Linear ◽  
Computing Scheme

This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system in case of few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes interactions. In this case, the non-linear coupled Schrödinger equation system solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails, due to the non-stability of the computing scheme. Due to this fact, the combined explicit/implicit finite-difference integration scheme, based on the implicit Crank–Nicolson finite-difference scheme, is used. It allows ensuring the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has performance gains (or resolutions) up to three or more orders of magnitude in comparison with the split-step Fourier method due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step. The main advantage of the proposed method is the ability to calculate the propagation of an arbitrary number of modes in the fiber.

scholarly journals Numerical Method for Coupled Nonlinear Schrödinger Equations in Few-Mode Fiber

Fibers ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 1
Author(s):  
Airat Zh. Sakhabutdinov ◽  
Vladimir I. Anfinogentov ◽  
Oleg G. Morozov ◽  
Vladimir A. Burdin ◽  
Anton V. Bourdine ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Equation System ◽  
Integration Scheme ◽  
Integration Step ◽  
Nonlinear Schrödinger ◽  
Computing Scheme

This paper discusses novel approaches to the numerical integration of the coupled nonlinear Schrödinger equations system for few-mode wave propagation. The wave propagation assumes the propagation of up to nine modes of light in an optical fiber. In this case, the light propagation is described by the non-linear coupled Schrödinger equation system, where propagation of each mode is described by own Schrödinger equation with other modes’ interactions. In this case, the coupled nonlinear Schrödinger equation system (CNSES) solving becomes increasingly complex, because each mode affects the propagation of other modes. The suggested solution is based on the direct numerical integration approach, which is based on a finite-difference integration scheme. The well-known explicit finite-difference integration scheme approach fails due to the non-stability of the computing scheme. Owing to this, here we use the combined explicit/implicit finite-difference integration scheme, which is based on the implicit Crank–Nicolson finite-difference scheme. It ensures the stability of the computing scheme. Moreover, this approach allows separating the whole equation system on the independent equation system for each wave mode at each integration step. Additionally, the algorithm of numerical solution refining at each step and the integration method with automatic integration step selection are used. The suggested approach has a higher performance (resolution)—up to three times or more in comparison with the split-step Fourier method—since there is no need to produce direct and inverse Fourier transforms at each integration step. The key advantage of the developed approach is the calculation of any number of modes propagated in the fiber.

scholarly journals Original Solution of Coupled Nonlinear Schrödinger Equations for Simulation of Ultrashort Optical Pulse Propagation in a Birefringent Fiber

Fibers ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 34 ◽  
Author(s):  
Airat Zhavdatovich Sakhabutdinov ◽  
Vladimir Ivanovich Anfinogentov ◽  
Oleg Gennadievich Morozov ◽  
Vladimir Alexandrovich Burdin ◽  
Anton Vladimirovich Bourdine ◽  
...  
Keyword(s):  
Finite Difference ◽  
Nonlinear Schrödinger Equations ◽  
Integration Scheme ◽  
Integration Step ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Physical Processes ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Implicit Finite Difference

This paper discusses approaches to the numerical integration of the coupled nonlinear Schrödinger equations system, different from the generally accepted approach based on the method of splitting according to physical processes. A combined explicit/implicit finite-difference integration scheme based on the implicit Crank–Nicolson finite-difference scheme is proposed and substantiated. It allows the integration of a nonlinear system of equations with a choice of nonlinear terms from the previous integration step. The main advantages of the proposed method are: its absolute stability through the use of an implicit finite-difference integration scheme and an integrated mechanism for refining the numerical solution at each step; integration with automatic step selection; performance gains (or resolutions) up to three or more orders of magnitude due to the fact that there is no need to produce direct and inverse Fourier transforms at each integration step, as is required in the method of splitting according to physical processes. An additional advantage of the proposed method is the ability to calculate the interaction with an arbitrary number of propagation modes in the fiber.

Research on the Interaction of Tunnel Surrounding Rock Masses and Supporting Structure

2011 ◽  
Vol 261-263 ◽  
pp. 1515-1522
Author(s):  
Jin Fan ◽  
Qin Qin Liu ◽  
Yang Song Zhang
Keyword(s):  
Plastic Region ◽  
Surrounding Rock ◽  
Finite Element Models ◽  
Analysis And Design ◽  
Supporting Structure ◽  
Calculation Results ◽  
Definition Of ◽  
Design Calculations ◽  
Selection Of ◽  
The Ideal

To reveal the mechanical property on the interface between the surrounding rock and the supporting structure, inducting the definition of the interface, based on ABAQUS software, two finite element models were established to compare and analysis the interaction through the aspects of stress, deformation and plastic region of surrounding rocks and the resistance of supporting structure. At last, compare the simulative and the metrical results. The result shows that the ideal full-constraint between the surrounding rock and the supporting structure isn’t possibility. It becomes more reasonable to define an interface during the numerical analysis and design calculations, the selection of the constitutive relation of the interface is directly related to the accuracy of the numerical calculation results.

Selection and Preparation of Specific Tissue Regions For Tem Using Large Epoxy-Embedded Blocks

1973 ◽  
Vol 31 ◽  
pp. 324-325 ◽  
Author(s):  
P. M. Lowrie ◽  
W. S. Tyler
Keyword(s):  
Electron Microscopy ◽  
Anatomical Structures ◽  
Ultrastructural Studies ◽  
Transmission Electron ◽  
Microscopic Evaluation ◽  
Embedded Blocks ◽  
Specific Tissue ◽  
Definition Of ◽  
Areas Of Interest ◽  
Selection Of

The importance of examining stained 1 to 2μ plastic sections by light microscopy has long been recognized, both for increased definition of many histologic features and for selection of specimen samples to be used in ultrastructural studies. Selection of specimens with specific orien ation relative to anatomical structures becomes of critical importance in ultrastructural investigations of organs such as the lung. The uantity of blocks necessary to locate special areas of interest by random sampling is large, however, and the method is lacking in precision. Several methods have been described for selection of specific areas for electron microscopy using light microscopic evaluation of paraffin, epoxy-infiltrated, or epoxy-embedded large blocks from which thick sections were cut. Selected areas from these thick sections were subsequently removed and re-embedded or attached to blank precasted blocks and resectioned for transmission electron microscopy (TEM).

scholarly journals Promotion of Resilience in Migrants: A Systematic Review of Study and Psychosocial Intervention

2021 ◽  
Author(s):  
Maria Ciaramella ◽  
Nadia Monacelli ◽  
Livia Concetta Eugenia Cocimano
Keyword(s):  
Systematic Review ◽  
Psychosocial Intervention ◽  
Life Experiences ◽  
Psychosocial Interventions ◽  
Migrant Population ◽  
Methodological Choices ◽  
Research Questions ◽  
Definition Of ◽  
Selection Of

AbstractThis systematic review aimed to contribute to a better and more focused understanding of the link between the concept of resilience and psychosocial interventions in the migrant population. The research questions concerned the type of population involved, definition of resilience, methodological choices and which intervention programmes were targeted at migrants. In the 90 articles included, an heterogeneity in defining resilience or not well specified definition resulted. Different migratory experiences were not adequately considered in the selection of participants. Few resilience interventions on migrants were resulted. A lack of procedure’s descriptions that keep in account specific migrants’ life-experiences and efficacy’s measures were highlighted.

scholarly journals Getting to Hope: Perspectives from Patients and Caregivers Living with Chronic Childhood Illness

Children ◽  
2021 ◽  
Vol 8 (6) ◽  
pp. 525
Author(s):  
Emily von Scheven ◽  
Bhupinder K. Nahal ◽  
Rosa Kelekian ◽  
Christina Frenzel ◽  
Victoria Vanderpoel ◽  
...  
Keyword(s):  
Conceptual Model ◽  
Research Question ◽  
Young Patients ◽  
Future Research ◽  
Childhood Onset ◽  
Virtual Focus Groups ◽  
Definition Of ◽  
Selection Of ◽  
Chronic Childhood ◽  
Research Studies

Promoting hope was identified in our prior work as the top priority research question among patients and caregivers with diverse childhood-onset chronic conditions. Here, we aimed to construct a conceptual model to guide future research studies of interventions to improve hope. We conducted eight monthly virtual focus groups and one virtual workshop with patients, caregivers, and researchers to explore key constructs to inform the model. Discussions were facilitated by Patient Co-Investigators. Participants developed a definition of hope and identified promotors and inhibitors that influence the experience of hope. We utilized qualitative methods to analyze findings and organize the promotors and inhibitors of hope within three strata of the socio-ecologic framework: structural, interpersonal, and intrapersonal. Participants identified three types of interventions to promote hope: resources, navigation, and activities to promote social connection. The hope conceptual model can be used to inform the selection of interventions to assess in future research studies aimed at improving hope and the specification of outcome measures to include in hope research studies. Inclusion of the health care system in the model provides direction for identifying strategies for improving the system and places responsibility on the system to do better to promote hope among young patients with chronic illness and their caregivers.

scholarly journals Examining and Fine-tuning the Selection of Glycan Compositions with GlyConnect Compozitor

2020 ◽  
Vol 19 (10) ◽  
pp. 1602-1618 ◽  
Author(s):  
Thibault Robin ◽  
Julien Mariethoz ◽  
Frédérique Lisacek
Keyword(s):  
Search Engine ◽  
Posttranslational Modifications ◽  
Search Engines ◽  
Web Application ◽  
Contextual Information ◽  
Fine Tuning ◽  
Data Sources ◽  
Web Interface ◽  
Definition Of ◽  
Selection Of

A key point in achieving accurate intact glycopeptide identification is the definition of the glycan composition file that is used to match experimental with theoretical masses by a glycoproteomics search engine. At present, these files are mainly built from searching the literature and/or querying data sources focused on posttranslational modifications. Most glycoproteomics search engines include a default composition file that is readily used when processing MS data. We introduce here a glycan composition visualizing and comparative tool associated with the GlyConnect database and called GlyConnect Compozitor. It offers a web interface through which the database can be queried to bring out contextual information relative to a set of glycan compositions. The tool takes advantage of compositions being related to one another through shared monosaccharide counts and outputs interactive graphs summarizing information searched in the database. These results provide a guide for selecting or deselecting compositions in a file in order to reflect the context of a study as closely as possible. They also confirm the consistency of a set of compositions based on the content of the GlyConnect database. As part of the tool collection of the Glycomics@ExPASy initiative, Compozitor is hosted at https://glyconnect.expasy.org/compozitor/ where it can be run as a web application. It is also directly accessible from the GlyConnect database.

General Schema Theory for Genetic Programming with Subtree-Swapping Crossover: Part II

2003 ◽  
Vol 11 (2) ◽  
pp. 169-206 ◽  
Author(s):  
Riccardo Poli ◽  
Nicholas Freitag McPhee
Keyword(s):  
Genetic Programming ◽  
Schema Theory ◽  
Exact Formulation ◽  
Expected Number ◽  
Size Evolution ◽  
General Schema ◽  
Definition Of ◽  
Effective Fitness ◽  
Selection Of ◽  
Exact Definition

This paper is the second part of a two-part paper which introduces a general schema theory for genetic programming (GP) with subtree-swapping crossover (Part I (Poli and McPhee, 2003)). Like other recent GP schema theory results, the theory gives an exact formulation (rather than a lower bound) for the expected number of instances of a schema at the next generation. The theory is based on a Cartesian node reference system, introduced in Part I, and on the notion of a variable-arity hyperschema, introduced here, which generalises previous definitions of a schema. The theory includes two main theorems describing the propagation of GP schemata: a microscopic and a macroscopic schema theorem. The microscopic version is applicable to crossover operators which replace a subtree in one parent with a subtree from the other parent to produce the offspring. Therefore, this theorem is applicable to Koza's GP crossover with and without uniform selection of the crossover points, as well as one-point crossover, size-fair crossover, strongly-typed GP crossover, context-preserving crossover and many others. The macroscopic version is applicable to crossover operators in which the probability of selecting any two crossover points in the parents depends only on the parents' size and shape. In the paper we provide examples, we show how the theory can be specialised to specific crossover operators and we illustrate how it can be used to derive other general results. These include an exact definition of effective fitness and a size-evolution equation for GP with subtree-swapping crossover.

Influence of Clearances and Thermal Effects on the Dynamic Behavior of Gear-Hydrodynamic Journal Bearing Systems

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
R. Fargère ◽  
P. Velex
Keyword(s):  
Degrees Of Freedom ◽  
Integration Scheme ◽  
Time Step ◽  
Specific Element ◽  
Normal Contact ◽  
Contact Algorithm ◽  
Gear Teeth ◽  
Proposed Model ◽  
Elastic Couplings ◽  
Definition Of

A global model of mechanical transmissions is introduced which deals with most of the possible interactions between gears, shafts, and hydrodynamic journal bearings. A specific element for wide-faced gears with nonlinear time-varying mesh stiffness and tooth shape deviations is combined with shaft finite elements, whereas the bearing contributions are introduced based on the direct solution of Reynolds' equation. Because of the large bearing clearances, particular attention has been paid to the definition of the degrees-of-freedom and their datum. Solutions are derived by combining a time step integration scheme, a Newton–Raphson method, and a normal contact algorithm in such a way that the contact conditions in the bearings and on the gear teeth are simultaneously dealt with. A series of comparisons with the experimental results obtained on a test rig are given which prove that the proposed model is sound. Finally, a number of results are presented which show that parameters often discarded in global models such as the location of the oil inlet area, the oil temperature in the bearings, the clearance/elastic couplings interactions, etc. can be influential on static and dynamic tooth loading.

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