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Introduction to Nonlinear Dispersive Equations

  • Felipe Linares
  • Gustavo Ponce

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Felipe Linares, Gustavo Ponce
    Pages 1-23
  3. Felipe Linares, Gustavo Ponce
    Pages 25-43
  4. Felipe Linares, Gustavo Ponce
    Pages 63-92
  5. Felipe Linares, Gustavo Ponce
    Pages 93-124
  6. Felipe Linares, Gustavo Ponce
    Pages 125-150
  7. Felipe Linares, Gustavo Ponce
    Pages 151-189
  8. Felipe Linares, Gustavo Ponce
    Pages 191-214
  9. Felipe Linares, Gustavo Ponce
    Pages 215-248
  10. Felipe Linares, Gustavo Ponce
    Pages 249-270
  11. Back Matter
    Pages 271-301

About this book

Introduction

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.

Keywords

Korteweg-de Vries Equation Marcinkiewicz interpolation theorem Riesz–Thorin convexity theorem Stein interpolation theorem pseudo-differential operators

Authors and affiliations

  • Felipe Linares
    • 1
  • Gustavo Ponce
    • 2
  1. 1.Instituto Nacional de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  2. 2.Dept. MathematicsUniversity of California, Santa Barbara College of Letters & ScienceSanta BarbaraUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4939-2181-2
  • Copyright Information Springer-Verlag New York 2015
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-2180-5
  • Online ISBN 978-1-4939-2181-2
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site
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