Complex Envelope Variable Approximation in Nonlinear Dynamics

Abstract

We present the complex envelope variable approximation (CEVA) as a useful and compact method for analysis of essentially nonlinear dynamical systems. The basic idea is that the introduction of complex variables, which are analogues of the creation and annihilation operators in quantum mechanics, considerably simplifies the analysis of a number of nonlinear dynamical systems. The first stage of the procedure, in fact, does not require any additional assumptions, except for the proposition of the existence of a single-frequency stationary solution. This allows us to study both the stationary and nonstationary dynamics even in the cases when there are no small parameters in the initial problem. In particular, the CEVA method provides an analysis of nonlinear normal modes and their resonant interactions in discrete systems for a wide range of oscillation amplitudes. The dispersion relations depending on the oscillation amplitudes can be obtained in analytical form for both the conservative and the dissipative nonlinear lattices in the framework of the main-order approximation. In order to analyze the nonstationary dynamical processes, we suggest a new notion-the "slow" Hamiltonian, which allows us to generate the nonstationary equations in the slow time scale. The limiting phase trajectory, the bifurcations of which determine such processes as the energy localization in the nonlinear chains or the escape from the potential well under the action of external forces, can be also analyzed in the CEVA. A number of complex problems were studied earlier in the framework of various modifications of the method, but the accurate formulation of the CEVA with the step-by-step illustration is described here for the first time. In this paper we formulate the CEVA's formalism and give some nontrivial examples of its application.