Recent Trends in Partial Differential Equations

Author: Xavier Cabré
Publisher: American Mathematical Soc.
ISBN: 9780821838914
Release Date: 2006
Genre: Mathematics

"The research and expository articles for courses and talks given at the 'UIMP-RSME Lluis A. Santaló Summer School' with title 'Recent Trends in Partial Differential Equations'."--Pref.

The Defocusing Nonlinear Schr dinger Equation

Author: Panayotis G. Kevrekidis
Publisher: SIAM
ISBN: 9781611973945
Release Date: 2015-08-04
Genre: Mathematics

Bose?Einstein condensation is a phase transition in which a fraction of particles of a boson gas condenses into the same quantum state known as the Bose?Einstein condensate (BEC). The aim of this book is to present a wide array of findings in the realm of BECs and on the nonlinear SchrÓdinger-type models that arise therein.÷The Defocusing Nonlinear SchrÓdinger Equation÷is a broad study of nonlinear÷excitations in self-defocusing nonlinear media. It summarizes state-of-the-art knowledge on the defocusing nonlinear SchrÓdinger-type models in a single volume and contains a wealth of resources, including over 800 references to relevant articles and monographs and a meticulous index for ease of navigation.

Vortices in Bose Einstein Condensates

Author: Amandine Aftalion
Publisher: Springer Science & Business Media
ISBN: 9780817644925
Release Date: 2007-10-14
Genre: Mathematics

This book provides an up-to-date approach to the diagnosis and management of endocarditis based on a critical analysis of the recent studies. It is the only up-to-date clinically oriented textbook available on this subject. The book is structured in a format that is easy to follow, clinically relevant and evidence based. The author has a special interest in the application of ultrasound in the study of cardiac structure and function.

Perspectives in Nonlinear Partial Differential Equations

Author: Henri Berestycki
Publisher: American Mathematical Soc.
ISBN: 9780821841907
Release Date: 2007
Genre: Mathematics

In celebration of Haim Brezis's 60th birthday, a conference was held at the Ecole Polytechnique in Paris, with a program testifying to Brezis's wide-ranging influence on nonlinear analysis and partial differential equations. The articles in this volume are primarily from that conference. They present a rare view of the state of the art of many aspects of nonlinear PDEs, as well as describe new directions that are being opened up in this field. The articles, written by mathematicians at the center of current developments, provide somewhat more personal views of the important developments and challenges.

Spectral Methods in Surface Superconductivity

Author: Søren Fournais
Publisher: Springer Science & Business Media
ISBN: 9780817647971
Release Date: 2010-05-19
Genre: Mathematics

This book examines in detail the nonlinear Ginzburg–Landau functional, the model most commonly used in the study of superconductivity. Specifically covered are cases in the presence of a strong magnetic field and with a sufficiently large Ginzburg–Landau parameter kappa. Spectral Methods in Surface Superconductivity is intended for students and researchers with a graduate-level understanding of functional analysis, spectral theory, and the analysis of partial differential equations. The book also includes an overview of all nonstandard material as well as important semi-classical techniques in spectral theory that are involved in the nonlinear study of superconductivity.

Vortices in Bose u2014 Einstein Condensates

Author: Amandine Aftalion
ISBN: OCLC:1066455948
Release Date: 2006
Genre: Applied mathematics

Since the first experimental achievement of Bose–Einstein condensates (BEC) in 1995 and the award of the Nobel Prize for Physics in 2001, the properties of these gaseous quantum fluids have been the focus of international interest in physics. This monograph is dedicated to the mathematical modelling of some specific experiments which display vortices and to a rigorous analysis of features emerging experimentally. In contrast to a classical fluid, a quantum fluid such as a Bose–Einstein condensate can rotate only through the nucleation of quantized vortices beyond some critical velocity. There are two interesting regimes: one close to the critical velocity, where there is only one vortex that has a very special shape; and another one at high rotation values, for which a dense lattice is observed. One of the key features related to superfluidity is the existence of these vortices. We address this issue mathematically and derive information on their shape, number, and location. In the dilute limit of these experiments, the condensate is well described by a mean field theory and a macroscopic wave function solving the so-called Gross–Pitaevskii equation. The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. We prove existence of solutions that have properties consistent with the experimental observations. Open problems related to recent experiments are presented. The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum fluids, and can also complement a graduate seminar in elliptic PDEs or modelling of physical experiments.

Ginzburg Landau Vortices

Author: Haim Br‚zis
Publisher: World Scientific
ISBN: 9789812562036
Release Date: 2005
Genre: Mathematics

The Ginzburg-Landau equation us a mathematical model of superconductors has become an extremely useful tool in many areas of physics where vortices carrying a topological charge appear. The remarkable progress in the mathematical understanding of this equation involves a combined use of mathematical tools from many branches of mathematics. The Ginzburg-Landau model has been an amazing source of new problems and new ideas in analysis, geometry and topology. This collection will meet the urgent needs of the specialists, scholars and graduate students working in this area or related areas.