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

Abstract

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.

The dynamics of surface wave propagation based on the Benjamin Bona Mahony equation

“…7(b), the real part curve at position 𝜉 = 0 passes 𝑅𝑒(𝐹 ) = 0 as twice (pair of singular points) for one period thus causing an amplitude of 0 at that position. It is appropriate given that the phase singularity phenomenon for the BBM group envelope occurs when the modulation frequency is at interval [0, √ (3∕2)] [72]. The same phenomenon does not occur at ṽ = 1.3 (Fig.…”

“…At the position 𝜉 = 0 for ṽ = 0.7, ṽ = 0.7, ṽ = 0.7, and ṽ = 0.7, the line goes through 𝑅𝑒(𝐹 ) = 0 as twice (a pair of singular points) for one period. This is not appropriate given that the phase singularity phenomenon for the BBM group enveloped exists in the interval [0, √ 3∕2] for modulation frequency [72]. In Fig.…”

“…This NLS equation has different nonlinear and dispersion coefficients with the NLS envelope of the KdV waveform. The BBM wave propagates at a relatively slower speed compared to the KdV wave [72]. BBM wave groups also experience the phase singularity during its propagation.…”

“…BBM wave groups also experience the phase singularity during its propagation. In [72], the phase singularity is explained through a sighting from the profile and contour of the wave groups. This research intends to investigate further about the phenomenon.…”