Nonlinear dispersive waves. Asymptotic analysis and solitons. (English) Zbl 1232.35002

Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press (ISBN 978-1-107-66410-4/pbk; 978-1-107-01254-7/hbk). xiv, 348 p. (2011).
This book is developed from the lecture notes of a specialized graduate course on Nonlinear Waves at the University of Colorado at Boulder. It is now a self-contained text suitable for advanced graduate courses or for self-directed learning in the areas of nonlinear dispersive waves. The level of presentation fits the standards of Cambridge texts on applied mathematics, with informal discussions, clear examples, and transparent analytical computations. The book is written by a master in soliton theory, whose contributions pioneered the development of the entire area of research.
The book consists of three parts. The first part describes asymptotic methods of derivations of the nonlinear dispersive wave equations such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations. In particular, linear dispersive wave equations are derived by using the stationary phase method of Fourier analysis. Nonlinear terms are included by using methods of multiple scales. These and other methods are used to derive the KdV-type equations from water waves, NLS-type equations from model equations, water wave equations, and the Maxwell equations of nonlinear optics.
The second part covers a basic introduction to solitons and inverse scattering. In particular, Lax pairs, Miura transformations, the Gelfand-Levitan-Marchenko integral equation, and soliton solutions are reviewed for the KdV equation. This presentation is brief and self-contained, but the interested reader should refer to other books by the same author, in particular, to [M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society Lecture Note Series. 149. Cambridge (UK) etc.: Cambridge University Press (1991; Zbl 0762.35001)].
The third part includes two applications of the NLS equation to fiber optics communications and mode-locked lasers. These chapters cover more recent original contributions of the author and his research group to the relevant subjects.


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions


Zbl 0762.35001