Global Solutions of Nonlinear Schrodinger Equations
Global Solutions of Nonlinear Schrodinger Equations

Author: Jean Bourgain

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 193

ISBN-13: 0821819194

DOWNLOAD EBOOK

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.

The Nonlinear Schrödinger Equation
The Nonlinear Schrödinger Equation

Author: Gadi Fibich

Publisher: Springer

Published: 2015-03-06

Total Pages: 870

ISBN-13: 3319127489

DOWNLOAD EBOOK

This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of the subject. It combines rigorous analysis, asymptotic analysis, informal arguments, numerical simulations, physical modelling, and physical experiments. It repeatedly emphasizes the relations between these approaches, and the intuition behind the results. The Nonlinear Schrödinger Equation will be useful to graduate students and researchers in applied mathematics who are interested in singular solutions of partial differential equations, nonlinear optics and nonlinear waves, and to graduate students and researchers in physics and engineering who are interested in nonlinear optics and Bose-Einstein condensates. It can be used for courses on partial differential equations, nonlinear waves, and nonlinear optics. Gadi Fibich is a Professor of Applied Mathematics at Tel Aviv University. “This book provides a clear presentation of the nonlinear Schrodinger equation and its applications from various perspectives (rigorous analysis, informal analysis, and physics). It will be extremely useful for students and researchers who enter this field.” Frank Merle, Université de Cergy-Pontoise and Institut des Hautes Études Scientifiques, France

Semilinear Schrodinger Equations
Semilinear Schrodinger Equations

Author: Thierry Cazenave

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 346

ISBN-13: 0821833995

DOWNLOAD EBOOK

The nonlinear Schrodinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. This book presents various mathematical aspects of the nonlinear Schrodinger equation. It studies both problems of local nature and problems of global nature.

Nonlinear Wave Equations
Nonlinear Wave Equations

Author: Walter A. Strauss

Publisher: American Mathematical Soc.

Published: 1990-01-12

Total Pages: 106

ISBN-13: 0821807250

DOWNLOAD EBOOK

The theory of nonlinear wave equations in the absence of shocks began in the 1960s. Despite a great deal of recent activity in this area, some major issues remain unsolved, such as sharp conditions for the global existence of solutions with arbitrary initial data, and the global phase portrait in the presence of periodic solutions and traveling waves. This book, based on lectures presented by the author at George Mason University in January 1989, seeks to present the sharpest results to date in this area. The author surveys the fundamental qualitative properties of the solutions of nonlinear wave equations in the absence of boundaries and shocks. These properties include the existence and regularity of global solutions, strong and weak singularities, asymptotic properties, scattering theory and stability of solitary waves. Wave equations of hyperbolic, Schrodinger, and KdV type are discussed, as well as the Yang-Mills and the Vlasov-Maxwell equations. The book offers readers a broad overview of the field and an understanding of the most recent developments, as well as the status of some important unsolved problems. Intended for mathematicians and physicists interested in nonlinear waves, this book would be suitable as the basis for an advanced graduate-level course.

The Nonlinear Schrödinger Equation
The Nonlinear Schrödinger Equation

Author: Catherine Sulem

Publisher: Springer Science & Business Media

Published: 2007-06-30

Total Pages: 363

ISBN-13: 0387227687

DOWNLOAD EBOOK

Filling the gap between the mathematical literature and applications to domains, the authors have chosen to address the problem of wave collapse by several methods ranging from rigorous mathematical analysis to formal aymptotic expansions and numerical simulations.

Nonlinear Dispersive Equations
Nonlinear Dispersive Equations

Author: Terence Tao

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 394

ISBN-13: 0821841432

DOWNLOAD EBOOK

"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.".

Dispersive Equations and Nonlinear Waves
Dispersive Equations and Nonlinear Waves

Author: Herbert Koch

Publisher: Springer

Published: 2014-07-14

Total Pages: 310

ISBN-13: 3034807368

DOWNLOAD EBOOK

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.​