Defocusing Nonlinear Schrödinger Equations

by Benjamin Dodson
Defocusing Nonlinear Schrödinger Equations
Author: Benjamin Dodson
Publisher: Cambridge University Press
Total Pages: 256
Release: 2019-03-28
Genre: Mathematics
ISBN: 1108681670
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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.

Defocusing Nonlinear Schrödinger Equations

by Benjamin Dodson
Defocusing Nonlinear Schrödinger Equations
Author: Benjamin Dodson
Publisher: Cambridge University Press
Total Pages: 255
Release: 2019-03-28
Genre: Mathematics
ISBN: 1108472087
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Explores Schrödinger equations with power-type nonlinearity, with scattering results for mass- and energy-critical Schrödinger equations.

The Discrete Nonlinear Schrödinger Equation

by Panayotis G. Kevrekidis
The Discrete Nonlinear Schrödinger Equation
Author: Panayotis G. Kevrekidis
Publisher: Springer Science & Business Media
Total Pages: 417
Release: 2009-07-07
Genre: Science
ISBN: 3540891994
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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.

Nonlinear Dispersive Equations

by Terence Tao
Nonlinear Dispersive Equations
Author: Terence Tao
Publisher: American Mathematical Soc.
Total Pages: 394
Release: 2006
Genre: Mathematics
ISBN: 0821841432
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"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.".

The Defocusing Nonlinear Schr?dinger Equation

by Panayotis G. Kevrekidis
The Defocusing Nonlinear Schr?dinger Equation
Author: Panayotis G. Kevrekidis
Publisher: SIAM
Total Pages: 437
Release: 2015-08-04
Genre: Mathematics
ISBN: 1611973945
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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.

The Defocusing NLS Equation and Its Normal Form

by Benoit Grébert
The Defocusing NLS Equation and Its Normal Form
Author: Benoit Grébert
Publisher: Erich Schmidt Verlag GmbH & Co. KG
Total Pages: 184
Release: 2014
Genre: Schrödinger equation
ISBN: 9783037191316
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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.

Fully Nonlinear Elliptic Equations

by Luis A. Caffarelli
Fully Nonlinear Elliptic Equations
Author: Luis A. Caffarelli
Publisher: American Mathematical Soc.
Total Pages: 114
Release: 1995
Genre: Mathematics
ISBN: 0821804375
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The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Dispersive Equations and Nonlinear Waves

by Herbert Koch
Dispersive Equations and Nonlinear Waves
Author: Herbert Koch
Publisher: Springer
Total Pages: 310
Release: 2014-07-14
Genre: Mathematics
ISBN: 3034807368
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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.​