90 research outputs found
Small noise asymptotic of the timing jitter in soliton transmission
We consider the problem of the error in soliton transmission in long-haul
optical fibers caused by the spontaneous emission of noise inherent to
amplification. We study two types of noises driving the stochastic focusing
cubic one dimensional nonlinear Schr\"{o}dinger equation which appears in
physics in that context. We focus on the fluctuations of the mass and arrival
time or timing jitter. We give the small noise asymptotic of the tails of these
two quantities for the two types of noises. We are then able to prove several
results from physics among which the Gordon--Haus effect which states that the
fluctuation of the arrival time is a much more limiting factor than the
fluctuation of the mass. The physical results had been obtained with arguments
difficult to fully justify mathematically.Comment: Published in at http://dx.doi.org/10.1214/07-AAP449 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Mini-Workshop: Dynamics of Stochastic Systems and their Approximation
The aim of this workshop was to bring together specialists in the area of stochastic dynamical systems and stochastic numerical analysis to exchange their ideas about the state of the art of approximations of stochastic dynamics. Here approximations are considered in the analytical sense in terms of deriving reduced dynamical systems, which are less complex, as well as in the numerical sense via appropriate simulation methods. The main theme is concerned with the efficient treatment of stochastic dynamical systems via both approaches assuming that ideas and methods from one ansatz may prove beneficial for the other. A particular goal was to systematically identify open problems and challenges in this area
Integrability, rational solitons and symmetries for nonlinear systems in Biology and Materials Physics
[ES] Los sistemas no lineales constituyen un tema de investigación de creciente interés en las últimas décadas dada su versatilidad en la descripción de fenómenos fÃsicos en diversos campos de estudio. Generalmente, dichos fenómenos vienen modelizados por ecuaciones diferenciales no lineales, cuya estructura matemática ha demostrado ser sumamente rica, aunque de gran complejidad respecto a su análisis. Dentro del conjunto de los sistemas no lineales, cabe destacar un reducido grupo, pero a la vez selecto, que se distingue por las propiedades extraordinarias que presenta: los denominados sistemas integrables.
La presente tesis doctoral se centra en el estudio de algunas de las propiedades más relevantes observadas para los sistemas integrables. En esta tesis se pretende proporcionar un marco teórico unificado que permita abordar ecuaciones diferenciales no lineales que potencialmente puedan ser consideradas como integrables. En particular, el análisis de integralidad de dichas ecuaciones se realiza a través de técnicas basadas en la Propiedad de Painlevé, en combinación con la subsiguiente búsqueda de los problemas espectrales asociados y la identificación de soluciones analÃticas de naturaleza solitónica. El método de la variedad singular junto con las transformaciones de auto-Bäcklund y de Darboux jugarán un papel fundamental en este estudio. Además, también se lleva a cabo un análisis complementario basado en las simetrÃas de Lie y reducciones de similaridad, que nos permitirán estudiar desde esta nueva perspectiva los problemas espectrales asociados.
Partiendo de la archiconocida ecuación de Schrödinger no lineal, se han investigado diferentes generalizaciones integrables de dicha ecuación con numerosas aplicaciones en diversos campos cientÃficos, como la FÃsica Matemática, FÃsica de Materiales o BiologÃa.[EN] Nonlinear systems emerge as an active research topic of growing interest during the last decades due to their versatility when it comes to describing physical phenomena. Such scenarios are typically modelled by nonlinear differential equations, whose mathematical structure has proved to be incredibly rich, but highly nontrivial to treat. In particular, a narrow but surprisingly special group of this kind stands out: the so-called integrable systems.
The present doctoral thesis focuses on the study of some of the extraordinary properties observed for integrable systems. The ultimate purpose of this dissertation lies in providing a unified theoretical framework that allows us to approach nonlinear differential equations that may potentially be considered as integrable. In particular, their integrability characterization is addressed by means of Painlevé analysis, in conjunction with the subsequent quest of the associated spectral problems and the identification of analytical solutions of solitonic nature. The singular manifold method together with auto-Bäckund and Darboux transformations play a critical role in this setting. In addition, a complementary methodology based on Lie symmetries and similarity reductions is proposed so as to analyze integrable systems by studying the symmetry properties of their associated spectral problems.
Taking the ubiquitous nonlinear Schrödinger equation as the starting point, we have investigated several integrable generalizations of this equation that possess copious applications in distinct scientific fields, such as Mathematical Physics, Material Sciences and Biology
Random Attractors for Stochastic Partly Dissipative Systems
We prove the existence of a global random attractor for a certain class of
stochastic partly dissipative systems. These systems consist of a partial (PDE)
and an ordinary differential equation (ODE), where both equations are coupled
and perturbed by additive white noise. The deterministic counterpart of such
systems and their long-time behaviour have already been considered but there is
no theory that deals with the stochastic version of partly dissipative systems
in their full generality. We also provide several examples for the application
of the theory.Comment: 29 page
[Book of abstracts]
USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos
Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation
Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo
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