Nonlinear Evolution of Benjamin-Bona-Mahony Wave Packet due to an Instability of a Pair of Modulations

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 occurred. It is...

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Bibliographic Details
Published in:International journal of differential equations 2018-01, Vol.2018 (2018), p.1-10
Main Authors: Ramli, Marwan, Mardi, Harish Abdillah, Fadhiliani, Dwi, Halfiani, Vera
Format: Article
Language:English
Subjects:
Amplification
Amplitudes
Astrophysics
Asymptotic series
Coasts
Differential equations, Partial
Evolution
Extreme values
Laboratories
Mathematical research
Mathematics
Position measurement
Propagation
Schrodinger equation
Science
Stability
Studies
Transformations (Mathematics)
Wave propagation
Wavelengths
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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 occurred. It is obtained that the envelope evolves satisfying the Nonlinear Schrodinger (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.
ISSN:1687-9643
1687-9651
DOI:10.1155/2018/1716571