Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene trated deep into various applied branches (theoretical physics, mechanics, ap plied astronomy, dynamics of space flights, and others) and laid the founda tion for lrumerous generalizations and for the creation of various modifications of thesem. E f, hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay...

The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have...

Many dynamical systems are described by differential equations that can be separated into one part, containing linear terms with constant coefficients, and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic oscillators in the absence of perturbations. This book is devoted...

This unique monograph deals with the development of asymptotic methods of perturbation theory, making wide use of group- theoretical techniques. Various assumptions about specific group properties are investigated, and are shown to lead to modifications of existing methods, such as the Bogoliubov averaging method and the Poincare--Birkhoff normal form, as well as to the formulation of new ones. The development of normalization techniques of Lie groups is also treated. The wealth of examples demonstrates how these new group theoretical techniques can be applied to analyze specific problems....

Many problems in celestial mechanics, physics and engineering involve the study of oscillating systems governed by nonlinear ordinary differential equations or partial differential equations. This volume represents an important contribution to the available methods of solution for such systems.

Many dynamical systems are described by differential equations that can be separated into one part, containing linear terms with constant coefficients, and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic oscillators in the absence of perturbations. This book is devoted...