Nonlinear evolution of Benjamin-Bona-Mahony wave packet due to an instability of a pair of modulations
type: Item
instance of: scholarly article
title: Nonlinear evolution of Benjamin-Bona-Mahony wave packet due to an instability of a pair of modulations
zbMATH Open document ID: 1487.76069
DOI: 10.1155/2018/1716571
author: Vera Halfiani
author: Dwi Fadhiliani
author: Harish Abdillah Mardi
author: Marwan Ramli
published in: International Journal of Differential Equations
publication date: 2018-08-10T00:00:00Z
review text: Summary: This article discusses the evolution of Benjamin-Bona-Mahony (BBM) wave packet's envelope. The envelope equation is derived by applying the asymptotic series up to the third order and choosing appropriate fast-to-slow variable transformations which eliminate the resonance terms that occur. It is obtained that the envelope evolves satisfying the nonlinear Schrödinger (NLS) equation. The evolution of NLS envelope is investigated through its exact solution, soliton on finite background, which undergoes modulational instability during its propagation. The resulting wave may experience phase singularity indicated by wave splitting and merging and causing amplification on its amplitude. Some parameter values take part in triggering this phenomenon. The amplitude amplification can be analyzed by employing maximal temporal amplitude (MTA) which is a quantity measuring the maximum wave elevation at each spatial position during the observation time. Wavenumber value affects the extreme position of the wave but not the amplitude amplification. Meanwhile, modulational frequency value affects both terms. Comparison of the evolution of the BBM wave packet to the previous results obtained from KdV equation gives interesting outputs regarding the extreme position and the maximum wave peaking.
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 6915947
zbMATH Keywords: envelope equation
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: modulational instability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2018/1716571
OpenAlex ID: W2806131591
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Wikidata QID: Q129711735
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description: scientific article; zbMATH DE number 6915947