Description
Book SynopsisThe 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.
Table of ContentsWKB Analysis: Preliminary Analysis ; Weak Nonlinearity; Modulated Energy Functionals; Pointwise Description; Some Instability Phenomena; Caustic Crossing: The Case of Focal Points: Caustic Crossing: Formal Analysis; Focal Point withtout External Potential; Focal Point in the Presence of an External Potential; Some Ideas for Supercritical Cases; Coherent States: The Linear Case; Tools for the Nonlinear Case; Power-like Nonlinearity; Hartree-type Nonlinearity;
