The goal of this memoir is to justify the use of methods of "geometric optics" type for a large class of semilinear systems that remain invariant under gauge and Lorentz transformations. The author explicitly constructs families of approximate solutions of a model system coupling a gauge field (Yang-Mills equation) with a scalar field (wave equation) and a spinor field (Dirac equation) in the form of single phase expansions oscillating at high frequency and with maximal amplitude. Further, the author proves the existence of exact solutions that remain asymptotic to the expansions already obtained. There is a stability result in the case where oscillatory solutions are obtained as high-frequency perturbations of a given smooth solution of the system. It shows that the system can keep a stable nonlinear behavior, even fields with very weak regularity, where nonlinear terms usually lead to destructive interactions.

The book is of interest to graduate students and research mathematicians in partial differential equations and their applications.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in partial differential equations and their applications.

Table of Contents

• Introduction
• Construction de familles de solutions approchées de $(YM)$ sous forme de développements oscillant à haute fréquence, monophasés, et d'amplitude maximale
• Prolongement asymptotique des familles de solutions approchées par des familles de solutions exactes
• Appendice A
• Appendice B
• Bibliographie