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Semilinear Schrödinger Equations
Thierry Cazenave, Université de Paris VI, Pierre et Marie Curie, France
A co-publication of the AMS and Courant Institute of Mathematical Sciences at New York University.
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Courant Lecture Notes
2003; 323 pp; softcover
Volume: 10
ISBN-10: 0-8218-3399-5
ISBN-13: 978-0-8218-3399-5
List Price: US$46
Member Price: US$36.80
Order Code: CLN/10
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The nonlinear Schrödinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or the Korteweg-de Vries equation. From the mathematical point of view, Schrödinger's equation is a delicate problem, possessing a mixture of the properties of parabolic and elliptic equations. Useful tools in studying the nonlinear Schrödinger equation are energy and Strichartz's estimates.

This book presents various mathematical aspects of the nonlinear Schrödinger equation. It studies both problems of local nature (local existence of solutions, uniqueness, regularity, smoothing effect) and problems of global nature (finite-time blowup, global existence, asymptotic behavior of solutions). In principle, the methods presented apply to a large class of dispersive semilinear equations. The first chapter recalls basic notions of functional analysis (Fourier transform, Sobolev spaces, etc.). Otherwise, the book is mostly self-contained.

It is suitable for graduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Readership

Graduate students and research mathematicians interested in nonlinear partial differential equations and applications to mathematical physics.

Reviews

"The book, written by one of the leading expert on the subject, is also an up-to- date source of references for recent results and open problems, well represented in its extensive bibliography. It can certainly be used as a guideline for a course to an audience with a sufficient background on functional analysis and partial differential equations."

-- Zentralblatt MATH

"In summary, the author gives a well balanced treatment of the many types of mathematical results... This book would be an excellent place to start for readers interested in an introduction to these topics. There is an extensive bibliography which nicely complements the author's discussions."

-- Woodford W. Zachary for Mathematical Reviews

Table of Contents

  • Preliminaries
  • The linear Schrödinger equation
  • The Cauchy problem in a general domain
  • The local Cauchy problem
  • Regularity and the smoothing effect
  • Global existence and finite-time blowup
  • Asymptotic behavior in the repulsive case
  • Stability of bound states in the attractive case
  • Further results
  • Bibliography
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