Asymptotic expansions of the solutions of nonlinear evolution equations

Abstract

A lot of methods have been developed for finding approximate periodic solutions of nonlinear equations on the basis of classic perturbation theory. All of them are developed under assuming the nonlinearity to be weak that allows us to separate all of the motions onto fast and slow ones. However the majority of those methods are limited by the first level solutions because of either principal difficulties (the method of averaging) or technical causes connected with the awkward calculations and absence of regular algorithm. Such algorithm is developed in the present work as well as its program realization is carried out. Besides that the weak nonlinearity was shown not to be the necessary condition for separating motions onto fast and slow. It is quite enough to redetermine the parameter of expansion into series within the frame of used spectral method. Such redetermining is shown to lead to expansion that is available in the case of strong nonlinearity for oscillations in conservative systems as well as for stationary processes in self-excited oscillation systems. The developed method is applicable for nonlinear equations describing single-frequency conservative and self-excited oscillation systems with power nonlinearities. In the present work the results of analysis are presented for a single-frequency conservative system are described.