zbMATH — the first resource for mathematics

Semi-classical analysis for nonlinear Schrödinger equations. WKB analysis, focal points, coherent states. 2nd edition. (English) Zbl 1448.35461
Hackensack, NJ: World Scientific (ISBN 978-981-12-2790-5/hbk; 978-981-12-2792-9/ebook). xiv, 352 p. (2021).
Publisher’s description: The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent.
Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrödinger equations in negative order Sobolev spaces.
The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler’s formula, generalized lens transform.
See the review of the first edition in [Zbl 1153.35070].
35Q55 NLS equations (nonlinear Schrödinger equations)
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
Full Text: DOI