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.
This study concerns on the solution of exact Korteweg de Vries (KdV) equation in its application in generating extreme waves. The method of asymptotic expansion is employed up to the fifth order. In the previous research, the same method was applied up to the third order and fifth order but it only considered the side band solutions. Here, solutions at each order will be analyzed. The existence of resonance terms at the odd orders and side band terms are interesting to observe considering the importance of these quantities in analyzing the wave deformation which link to the phenomenon of wave’s amplitude amplification. Bichromatic signal is used as the initial wave signal as it experiences instability during its propagation which results the amplitude amplification. The amplitude amplification is presented as Maximal Temporal Amplitude (MTA) which is a quantity measuring the highest elevation at every spatial position during the observation time.
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