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Introduction to Nonlinear Dispersive Equations

Felipe Linares, Gustavo Ponce (Autoren)

Buch | Softcover

301 Seiten

2014 | 2nd ed. 2015
Springer-Verlag New York Inc.
978-1-4939-2180-5 (ISBN)

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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.

Thesecond 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.

Felipe Linares is a Researcher at the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil. Gustavo Ponce is a Professor of Mathematics at the University of California in Santa Barbara.

1. The Fourier Transform.- 2. Interpolation of Operators.- 3. Sobolev Spaces and Pseudo-Differential Operators.- 4. The Linear Schrodinger Equation.- 5. The Non-Linear Schrodinger Equation.- 6. Asymptotic Behavior for NLS Equation.- 7. Korteweg-de Vries Equation.- 8. Asymptotic Behavior for k-gKdV Equations.- 9. Other Nonlinear Dispersive Models.- 10. General Quasilinear Schrodinger Equation.- Proof of Theorem 2.8.- Proof of Lemma 4.2.- References.- Index.

Reihe/Serie	Universitext
Zusatzinfo	1 Illustrations, black and white; XIV, 301 p. 1 illus.
Verlagsort	New York
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Mechanik
Schlagworte	Korteweg-de Vries Equation • Marcinkiewicz interpolation theorem • pseudo-differential operators • Riesz–Thorin convexity theorem • Stein interpolation theorem
ISBN-10	1-4939-2180-0 / 1493921800
ISBN-13	978-1-4939-2180-5 / 9781493921805
Zustand	Neuware