Nonlinear oscillations in a thin ring. II: Four-wave resonant interactions. Self-modulation
type: Item
instance of: scholarly article
title: Nonlinear oscillations in a thin ring. II: Four-wave resonant interactions. Self-modulation
zbMATH Open document ID: 0909.73047
DOI: 10.1007/BF01172808
published in: Acta Mechanica
publication date: 1999-04-07T00:00:00Z
review text: The first part of the work [the author and \textit{A. I. Potapov}, ibid. 189-200 (1998)] has concluded that bending waves in ring, being the low-frequency modes of resonant triplet, are stable against small perturbations. Consequently, on the one hand, a bending wave should behave as a linear quasi-harmonic wave train. The first-order approximation analysis predicts that the triple mode interactions cannot play a primary role in the evaluation of bending waves. On the other hand, one may anticipate that the intense bending wavetrain can be subjected to the self-modulation during the long-time evolution. This means that the wave cannot be stable for a long time. The paper confirms such expectations by extending the first-order nonlinear analysis to include in the model the higher-order effects. Within the second-order approximation analysis, the amplitude modulation is examined for intense bending waves. As a result, the soliton-like envelopes can be formed from unstable bending wavetrains.
Mathematics Subject Classification ID: 74H45
zbMATH DE Number: 1143341
zbMATH Keywords: stability
zbMATH Keywords: resonant triplet
zbMATH Keywords: first-order approximation analysis
zbMATH Keywords: intense bending wavetrain
zbMATH Keywords: long-time evolution
zbMATH Keywords: second-order approximation analysis
zbMATH Keywords: soliton-like envelopes
author: Dmitry Anatolyevich Kovriguine
MaRDI profile type: Publication
cites work: Dynamic Stability of Transverse Axisymmetric Waves in Circular Cylindrical Shells
cites work: Q4154174
cites work: Q4151274
cites work: Q4144941
cites work: Q3925229
cites work: Nonlinear Dynamics of Deep-Water Gravity Waves
description: scientific article; zbMATH DE number 1143341