Non-standard characteristics in asymptotic problems

CONTENTS Introduction. Examples of non-standard characteristics. Statement of the general problem Chapter I. Characteristics of linear equations and equations with a non-local non-linearity ??1. Pseudodifferential operators with symbols and characteristics on arbitrary symplectic manifolds. Coordinate-momentum quantification conditions ??2. Electron terms ??3. Pseudodifferential operators with complex characteristics. Global asymptotic behaviour ??4. Problems in which there is a logarithmic asymptotic solution. The class of equations of tunnel type. The instanton as a logarithmic limit. Fourth generalization of the concept of characteristic ??5. Characteristics and global asymptotic behaviour as for equations with a non-local non-linearity. Equations of Vlasov type for wave front oscillation propagation. Bicharacteristics defining canonical transformations Chapter II. Characteristics of non-linear equations of general type ??1. Linear equations with rapidly oscillating coefficients. Non-linear wave equations having the equations of gas dynamics as characteristics. Equations of characteristics of composition of non-linear waves - equations of gas mixture dynamics ??2. The case of a weak quadratic and cubic non-linearity. Main resonances. Three- and four-train processes ??3. Propagation of singularities in non-linear equations. Conditions of Hugoniot type - the analogue of equations of characteristics for this problem. Singularities of branching type. Solutions with finite ejections at isolated points References