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Introduction to Nonlinear Dispersive Equations

2014-12-15 Mathematics
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Release : 2014-12-15
Genre : Mathematics
Kind : eBook
Book Rating : 819/5 ( reviews)

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Book Synopsis Introduction to Nonlinear Dispersive Equations by : Felipe Linares

Download or read book Introduction to Nonlinear Dispersive Equations written by Felipe Linares. This book was released on 2014-12-15. Available in PDF, EPUB and Kindle. Book excerpt: This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introduction to Nonlinear Dispersive Equations builds upon the success of the first edition by the addition of updated material on the main topics, an expanded bibliography, and new exercises. Assuming only basic knowledge of complex analysis and integration theory, this book will enable graduate students and researchers to enter this actively developing field.

Nonlinear Dispersive Equations

2006 Mathematics
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Release : 2006
Genre : Mathematics
Kind : eBook
Book Rating : 432/5 ( reviews)

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Book Synopsis Nonlinear Dispersive Equations by : Terence Tao

Download or read book Nonlinear Dispersive Equations written by Terence Tao. This book was released on 2006. Available in PDF, EPUB and Kindle. Book excerpt: "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.".

Nonlinear Dispersive Equations

2009 Mathematics
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Release : 2009
Genre : Mathematics
Kind : eBook
Book Rating : 976/5 ( reviews)

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Book Synopsis Nonlinear Dispersive Equations by : Jaime Angulo Pava

Download or read book Nonlinear Dispersive Equations written by Jaime Angulo Pava. This book was released on 2009. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied include Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.

Dispersive Equations and Nonlinear Waves

2014-07-14 Mathematics
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Release : 2014-07-14
Genre : Mathematics
Kind : eBook
Book Rating : 368/5 ( reviews)

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Book Synopsis Dispersive Equations and Nonlinear Waves by : Herbert Koch

Download or read book Dispersive Equations and Nonlinear Waves written by Herbert Koch. This book was released on 2014-07-14. Available in PDF, EPUB and Kindle. Book excerpt: 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.​

Mathematical Aspects of Nonlinear Dispersive Equations (AM-163)

2009-01-10 Mathematics
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Release : 2009-01-10
Genre : Mathematics
Kind : eBook
Book Rating : 795/5 ( reviews)

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Book Synopsis Mathematical Aspects of Nonlinear Dispersive Equations (AM-163) by : Jean Bourgain

Download or read book Mathematical Aspects of Nonlinear Dispersive Equations (AM-163) written by Jean Bourgain. This book was released on 2009-01-10. Available in PDF, EPUB and Kindle. Book excerpt: This collection of new and original papers on mathematical aspects of nonlinear dispersive equations includes both expository and technical papers that reflect a number of recent advances in the field. The expository papers describe the state of the art and research directions. The technical papers concentrate on a specific problem and the related analysis and are addressed to active researchers. The book deals with many topics that have been the focus of intensive research and, in several cases, significant progress in recent years, including hyperbolic conservation laws, Schrödinger operators, nonlinear Schrödinger and wave equations, and the Euler and Navier-Stokes equations.

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