Abstract:In a simple model of propagation of asymmetric Gaussian beams in nonlinear waveguides, described by a reduction to ordinary differential eqautions of generalized nonlinear Schrödinger equations (GNLSEs) with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile, the beam widths exhibit two different types of beating behavior, with transitions between them. We present an analytic model to explain these phenomena, which originate in a 1 : 1 resonance in a 2 degreeof-freedom Hamiltonian… Show more
“…(4), and then using the approximations in Eqs. (26) and (29). Clearly, the least-mutual-error point should neither be too close to nor too far from unity.…”
Section: Matched Asymptoticsmentioning
confidence: 97%
“…In [25], the aforementioned approach was used for the exploration of beating in a spring pendulum under conditions of 1:2 resonance. A recent application [26] addresses the propagation of asymmetric Gaussian beams in nonlinear waveguides.…”
We consider a system of two linear and linearly coupled oscillators with ideal impact constraints. Primary resonant energy exchange is investigated by analysis of the slow flow using the action-angle (AA) formalism. Exact inversion of the action-energy dependence for the linear oscillator with impact constraints is not possible. This difficulty, typical for many models of nonlinear oscillators, is circumvented by matching the asymptotic expansions for the linear and impact limits. The obtained energy-action relation enables the complete analysis of the slow flow and the accurate description of the critical delocalization transition. The transition from the localization regime to the energy-exchange regime is captured by prediction of the critical coupling value. Accurate prediction of the delocalization transition requires a detailed account of the coupling energy with appropriate redefinition and optimization of the limiting phase trajectory on the resonant manifold.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.
“…(4), and then using the approximations in Eqs. (26) and (29). Clearly, the least-mutual-error point should neither be too close to nor too far from unity.…”
Section: Matched Asymptoticsmentioning
confidence: 97%
“…In [25], the aforementioned approach was used for the exploration of beating in a spring pendulum under conditions of 1:2 resonance. A recent application [26] addresses the propagation of asymmetric Gaussian beams in nonlinear waveguides.…”
We consider a system of two linear and linearly coupled oscillators with ideal impact constraints. Primary resonant energy exchange is investigated by analysis of the slow flow using the action-angle (AA) formalism. Exact inversion of the action-energy dependence for the linear oscillator with impact constraints is not possible. This difficulty, typical for many models of nonlinear oscillators, is circumvented by matching the asymptotic expansions for the linear and impact limits. The obtained energy-action relation enables the complete analysis of the slow flow and the accurate description of the critical delocalization transition. The transition from the localization regime to the energy-exchange regime is captured by prediction of the critical coupling value. Accurate prediction of the delocalization transition requires a detailed account of the coupling energy with appropriate redefinition and optimization of the limiting phase trajectory on the resonant manifold.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.
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