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Nonlinear dispersive equations. Local and global analysis. (English) Zbl 1106.35001

CBMS Regional Conference Series in Mathematics 106. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4143-2/pbk). xv, 373 p. (2006).
This welcome monograph is an expanded version of a set of lectures given by the author in New Mexico University in the spring of 2005. It provides an excellent introduction to the methods and results used in the modern analysis of the local and global study of the Cauchy problem, for equations like the nonlinear Schrödinger equation, the nonlinear wave equation and the Korteweg-de Vries equation, and it also introduces the concept of wave maps.
This account is written at an introductory graduate level, and uses Fourier analysis, the bootstrap method, and perturbation methods in the simpler case of ODE before generalizing the approach to PDE. The main purpose of this work is to study four model nonlinear equations that describe wave propagation, in some parts in an informal way, to the point where a rigorous theoretical treatment can be given of real world nonlinear wave phenomena.
The work is well suited for a graduate level course on nonlinear PDE, and it is to be thoroughly recommended.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35L15 Initial value problems for second-order hyperbolic equations
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