Explicit Approximations for Strictly Nonlinear Oscillators with Slowly Varying Parameters with Applications to Free‐Electron Lasers

The first part of this paper summarizes the mathematical modeling of free‐electron lasers (FEL), and the remainder concerns general perturbation methods for solving FEL and other strictly nonlinear oscillatory problems with slowly varying parameters and small perturbations. We review and compare the Kuzmak‐Luke method and that of near‐identity averaging transformations. In order to implement the calculation of explicit solutions we develop two approximation schemes. The first involves use of finite Fourier series to represent either the leading approximation of the solution or the transformation of the governing equations to a standard form appropriate for the method of averaging. In the second scheme we fit a cubic polynomial to the potential such that the leading approximation is expressible in terms of elliptic functions. The ideas are illustrated with a number of examples, which are also solved numerically to assess the accuracy of the various approximations.