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| Author | : Benjamin Dodson |
| Publisher | : Cambridge University Press |
| Total Pages | : 256 |
| Release | : 2019-03-28 |
| Genre | : Mathematics |
| ISBN | : 1108681670 |
Download Defocusing Nonlinear Schrödinger Equations Book in PDF, ePub and Kindle
This study of Schrödinger equations with power-type nonlinearity provides a great deal of insight into other dispersive partial differential equations and geometric partial differential equations. It presents important proofs, using tools from harmonic analysis, microlocal analysis, functional analysis, and topology. This includes a new proof of Keel–Tao endpoint Strichartz estimates, and a new proof of Bourgain's result for radial, energy-critical NLS. It also provides a detailed presentation of scattering results for energy-critical and mass-critical equations. This book is suitable as the basis for a one-semester course, and serves as a useful introduction to nonlinear Schrödinger equations for those with a background in harmonic analysis, functional analysis, and partial differential equations.
| Author | : Panayotis G. Kevrekidis |
| Publisher | : SIAM |
| Total Pages | : 437 |
| Release | : 2015-08-04 |
| Genre | : Mathematics |
| ISBN | : 1611973945 |
Download The Defocusing Nonlinear Schr?dinger Equation Book in PDF, ePub and Kindle
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 Schr?dinger-type models that arise therein.?The Defocusing Nonlinear Schr?dinger Equation?is a broad study of nonlinear?excitations in self-defocusing nonlinear media. It summarizes state-of-the-art knowledge on the defocusing nonlinear Schr?dinger-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.
| Author | : Panayotis G. Kevrekidis |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 417 |
| Release | : 2009-07-07 |
| Genre | : Science |
| ISBN | : 3540891994 |
Download The Discrete Nonlinear Schrödinger Equation Book in PDF, ePub and Kindle
This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrödinger equation and the physical settings that it describes.
| Author | : Jean Bourgain |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 193 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 0821819194 |
Download Global Solutions of Nonlinear Schrodinger Equations Book in PDF, ePub and Kindle
This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrodinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with Large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented and several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research related to dispersive equations and Hamiltonian PDEs.
| Author | : |
| Publisher | : |
| Total Pages | : |
| Release | : 2014 |
| Genre | : |
| ISBN | : 9783037196311 |
Download The Defocusing NLS Equation and Its Normal Form Book in PDF, ePub and Kindle
| Author | : Herbert Koch |
| Publisher | : Springer |
| Total Pages | : 310 |
| Release | : 2014-07-14 |
| Genre | : Mathematics |
| ISBN | : 3034807368 |
Download Dispersive Equations and Nonlinear Waves Book in PDF, ePub and Kindle
The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.
| Author | : Benoit Grébert |
| Publisher | : Erich Schmidt Verlag GmbH & Co. KG |
| Total Pages | : 184 |
| Release | : 2014 |
| Genre | : Schrödinger equation |
| ISBN | : 9783037191316 |
Download The Defocusing NLS Equation and Its Normal Form Book in PDF, ePub and Kindle
The theme of this monograph is the nonlinear Schrodinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in various physical systems such as water waves, plasma physics, solid state physics and nonlinear optics. More specifically, this book treats the defocusing nonlinear Schrodinger (dNLS) equation on the circle with a dynamical systems viewpoint. By developing the normal form theory, it is shown that this equation is an integrable partial differential equation in the strongest possible sense. In particular, all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic in time and Hamiltonian perturbations of this equation can be studied near solutions far away from the equilibrium. The book is intended not only for specialists working at the intersection of integrable PDEs and dynamical systems but also for researchers farther away from these fields as well as for graduate students. It is written in a modular fashion; each of its chapters and appendices can be read independently of each other.
| Author | : Samuel Fromm |
| Publisher | : |
| Total Pages | : |
| Release | : 2021 |
| Genre | : |
| ISBN | : 9789178738632 |
Download The Defocusing Nonlinear Schrödinger Equation with Step-like Oscillatory Data Book in PDF, ePub and Kindle
| Author | : Terence Tao |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 394 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 0821841432 |
Download Nonlinear Dispersive Equations Book in PDF, ePub and Kindle
"Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.".
| Author | : Remi Carles |
| Publisher | : World Scientific |
| Total Pages | : 256 |
| Release | : 2008-03-04 |
| Genre | : Mathematics |
| ISBN | : 9814471747 |
Download Semi-classical Analysis For Nonlinear Schrodinger Equations Book in PDF, ePub and Kindle
These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.
