Chapter Two - Exact Solutions to the Beam Equations

Abstract

We speak about an exact solution when the original system of partial differential equations describing the beam can be reduced to a system of ordinary differential equations. The set of exact solutions also includes the solutions that can be expressed in terms of elementary and special functions. The most important function of the exact solutions is testing of the approximate and numerical models. In doing so, not just one but a specially chosen set of exact solutions should be used because no single solution contains all the peculiarities inherent to practical problems. These peculiarities may include the emission in ρ- and T - modes from a curvilinear surface with a non-homogeneous current density in the presence of an external magnetic field and compensating background, as well as relativistic and high-frequency effects, extended trajectories, high-compression and deep-deceleration modes, and so forth. A complete set of exact solutions can be constructed by investigating the group properties of the beam equations. The main goal of those investigations is to reveal all the transformations of independent and dependent variables that preserve the form of the beam equations. A general approach based on the theory of groups and the notion of invariant solution was developed by L. V. Ovsyannikov. This section also familiarizes readers with the theory of continuous groups as applied to the problems of intense beam optics. The group properties of 3D nonstationary relativistic flows are studied, and the tables of the H-solutions that can be described by the ordinary differential equations are constructed. Aside from the transformations of translation, scaling, and rotations, the transformations with arbitrary functions of time are revealed for the non-relativistic case. Examples of the solutions in elementary functions for the axisymmetric and 3D flows are given. Most of the known exact solutions turn out to be invariant; however, there exist the solutions, the relation of which to the group properties of the beam equation remains still unknown.