Use of spectral method for solving nonlinear problems in oscillation theory

Abstract

To solve nonlinear partial differential equations which describe nearly periodic processes with dispersion and delay in the nonlinear part of the system one proceeds by slowly varying in time and in the coordinate current space-time spectral densities (“spectra”) of amplitudes and of phases of quasiharmonic components of the function appearing in the equation. One replaces all the derivatives of the latter on the left-hand side of the equation by their expressions in terms of the spectrum and on the right-hand side in the nonlinear perturbation one separates the terms which provide a contribution to every form of dynamic equilibrium of the system motion in accordance with the resonance conditions; this results in a set of differential equations of the first order for the amplitude and phase spectra of the quasiharmonic component which reflects their space-time dynamics.