The ionosphere of the Earth has been actively studied since the 1920's, following the discovery of ground radio-sounding. By means of this method results were ohtained by an international network of ionospheric stations, in particular, by the successful implementation of a number of rigorously planned international scientific research programs, '" enabling the collection of extensive experimental material on some of the most important parameters of the ionosphere - the critical freLjuencies of E-, Fl and F2-layers. Comprehensive analyses of these observation data give a fairly complete picture of the various changes taking place in the principal ionospheric layers at different points on our globe. Another important aspect of the study of the ionosphere, which has been in progress for the past three decades, is an extensive program of in situ determinations of the various physical parameters - first using rockets, and subsequently artificial satellites. The data thus obtained on the principal ionizing agent - short-wave solar radiation - and on the physical conditions prevailing in the upper atmosphere and in the ionosphere at different altitudes, allow the proposal of a self-consistent mechanism of ionosphere formation. A general outline of the theory of ionosphere formation at different altitudes is now complete. Its application to specific cases, dependent on a more accurate determination of input parameters to give solutions valid for a definite set of conditions etc., is yet to be accomplished. The use of artificial satellites in cosmic research yielded abundant scientific data."

The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields."