Valeri Smirnov

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.

We present an analysis of torsion oscillations in quasi-one-dimensional lattices with periodic potentials of the nearest neighbor interaction. A one-dimensional chain of point dipoles (spins) under an external field and without the latter is the simplest realization of such a system. We obtained dispersion relations for the nonlinear normal modes for a wide range of oscillation amplitudes and wave numbers. The features of the short wavelength part of the spectrum at large-amplitude oscillations are discussed. The problem of localized excitations near the edges of the spectrum is studied by the asymptotic method. We show that the localized oscillations (breathers) appear near the long wavelength edge, while the short wavelength edge of the spectrum contains only dark solitons. The continuum limit of the dynamic equations leads to a generalization of the nonlinear Schrödinger equation and can be considered as a complex representation of the sine-Gordon equation.

An analysis is presented of the nonlinear dynamics of harmonically coupled pendulums without restrictions to oscillation amplitudes. This is a basic model in many areas of mechanics and physics (paraffin crystals, DNA molecules etc.). Stationary solutions of equations of motion corresponding to nonlinear normal modes (NNMs) are obtained. The inversion of the NNM frequencies with increasing oscillation amplitude is found. An essentially nonstationary process of the resonant energy exchange is described in terms of limiting phase trajectories (LPTs), for which an effective analytic representation is obtained in slow time-scale. Explicit expressions of threshold values of dimensionless parameters are found which correspond to the instability of NNMs and to the transition (in parametric space) from the full energy exchange between the pendulums to the localization of energy. The analytic results obtained are verified by analysis of the Poincar´e sections describing evolution of the initial system.